Asheim-The Consistent Preferences Approach to Deductive Reasoning in Games (2)

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To Kristin





viii CONSISTENT PREFERENCES

5.3 Admissible consistency of preferences 62

6. RELAXING COMPLETENESS 69

6.1 Epistemic modeling of strategic games (cont.) 69

6.2 Consistency of preferences (cont.) 73

6.3 Admissible consistency of preferences (cont.) 75

7. BACKWARD INDUCTION 79

7.1 Epistemic modeling of extensive games 82

7.2 Initial belief of opponent rationality 87

7.3 Belief in each subgame of opponent rationality 89

7.4 Discussion 94

8. SEQUENTIALITY 99

8.1 Epistemic modeling of extensive games (cont.) 101

8.2 Sequential consistency 104

8.3 Weak sequential consistency 107

8.4 Relation to backward induction 113

9. QUASI-PERFECTNESS 115

9.1 Quasi-perfect consistency 116

9.2 Relating rationalizability concepts 118

10. PROPERNESS 121

10.1 An illustration 123

10.2 Proper consistency 124

10.3 Relating rationalizability concepts (cont.) 127

10.4 Induction in a betting game 128

11. CAPTURING FORWARD INDUCTIONTHROUGH FULL PERMISSIBILITY 133

11.1 Illustrating the key features 135

11.2 IECFA and fully permissible sets 138

11.3 Full admissible consistency 14211.4 Investigating examples 149

11.5 Related literature 152







List of Figures

2.1 G1 (“battle-of-the-sexes”). 12

2.2 G2, illustrating deductive reasoning. 13

2.3 G3, illustrating weak dominance. 13

2.4 G3 and a corresponding extensive form Γ

3 (a “cen-tipede” game). 14


2. 16


1 (“battle-of-the-sexes with an outside option”). 17

3.1 Γ4 and its strategic form. 25

4.1 The basic structure of the analysis in Chapter 4. 39

7.1 Γ5 (a four-legged “centipede” game). 93

8.1 Γ6 and its strategic form. 111

8.2 Γ6 and its pure strategy reduced strategic form. 112

10.1 G7, illustrating common certain belief of properconsistency. 123

10.2 A betting game. 129

10.3 The strategic form of the betting game. 130

11.1 G8, illustrating that IEWDS may be problematic. 134

11.2 G9, illustrating the key features of full admissibleconsistency. 136

11.3 G10, illustrating the relation between IECFA andIEWDS. 142




xii CONSISTENT PREFERENCES

12.2 Reduced form of Γ12 (a 3-period “prisoners’ dilemma”game). 166

12.3 G13 (the pure strategy reduced strategic form of “burning money”). 169





List of Tables

0.1 The main interactions between the chapters. xvi

2.1 Relationships between different equilibrium concepts. 18

2.2 Relationships between different rationalizability concepts. 19

3.1 Relationships between different sets of axioms andtheir representations. 29

7.1 An epistemic model for G3 with corresponding ex-

tensive form Γ3. 89

7.2 An epistemic model for Γ5. 93

10.1 An epistemic model for the betting game. 13112.1 Applying IECFA to “burning money”. 170





Preface

During the last decade I have explored the consequences of what Ihave chosen to call the ‘consistent preferences’ approach to deductivereasoning in games. To a great extent this work has been done in coop-eration with my co-authors Martin Dufwenberg, Andres Perea, and YlvaSøvik, and it has lead to a series of journal articles. This book presentsthe results of this research program.

Since the present format permits a more extensive motivation for andpresentation of the analysis, it is my hope that the content will be of interest to a wider audience than the corresponding journal articles can

reach. In addition to active researcher in the field, it is intended forgraduate students and others that wish to study epistemic conditionsfor equilibrium and rationalizability concepts in game theory.

Structure of the book

This book consists of twelve chapters. The main interactions betweenthe chapters are illustrated in Table 0.1.

As Table 0.1 indicates, the chapters can be organized into four dif-ferent parts. Chapters 1 and 2 motivate the subsequent analysis byintroducing the ‘consistent preferences’ approach, and by presenting ex-amples and concepts that are revisited throughout the book. Chapters 3

and 4 present the decision-theoretic framework and the belief operatorsthat are used in later chapters. Chapters 5, 6, 10, and 11 analyze gamesin the strategic form, while the remaining chapters—Chapters 7, 8, 9,and 12—are concerned with games in the extensive form.

The material can, however, also be organized along the vertical axisin Table 0.1. Chapters 5, 8, 9, and 10 are concerned with players thatare endowed with complete preferences over their own strategies. In con-



xvi CONSISTENT PREFERENCES

Table 0.1. The main interactions between the chapters.

Chapter 11 ⇒ Chapter 12

⇑ ↑

Chapter 1 Chapter 4 ⇒ Chapter 6 ⇒ Chapter 7

⇓ ⇑ ↑ ↓

Chapter 2 ⇒ Chapter 3 ⇒ Chapter 5 ⇒ Chapter 8

⇓ ⇓

Chapter 10 ← Chapter 9

Prelimi- Strategic Extensive Motivation naries games games

strast, Chapters 4, 6, 7, 11, and 12 present analyses that allow playersto have incomplete preferences, corresponding to an inability to assignsubjective probabilities to the strategies of their opponents. The gener-alization to possibly incomplete preferences is motivated in Section 3.1,and is an essential feature of the analysis in Chapter 11. Note also thatthe concepts of Chapters 7, 8, 9, and 10 imply backward induction but

not forward induction, while the concept of Chapters 11 and 12 promotesforward induction but not necessarily backward induction.

Notes on the history of the research program

While the arrows in Table 0.1 seek to guide the reader through thematerial presented here, they are not indicative of the chronologicaldevelopment of this work.

I started my work on non-equilibrium concepts in games in 1993 byconsidering the games that are illustrated in Figures 12.1–12.4. After joining forces with Martin Dufwenberg—who had independently devel-oped the same basic intuition about what deductive reasoning could

lead to in these examples—we started in 1994 work on our joint papers“Admissibility and common belief” and “Deductive reasoning in exten-sive games”, published in Games and Economic Behavior and Economic Journal in 2003, and incorporated as Chapters 11 and 12 in this book. 1

1“Deductive reasoning in extensive games” was awarded the Royal Economic Society Prizefor the best paper published in the Economic Journal in 2003.









Chapter 1

INTRODUCTION

This book presents, applies, and synthesizes what my co-authors and Ihave called the ‘consistent preferences’ approach to deductive reasoningin games. Briefly described, this means that the object of the analysis is the ranking by each player of his own strategies , rather than his choice.The ranking can be required to be consistent (in different senses) withhis beliefs about the opponent’s ranking of her strategies. This can becontrasted to the usual ‘rational choice’ approach where a player’s strat-

egy choice is (in different senses) rational given his beliefs about theopponent’s strategy choice. Our approach has turned out to be fruitfulfor providing epistemic conditions for backward and forward induction,and for defining or characterizing concepts like proper, quasi-perfect andsequential rationalizability. It also facilitates the integration of game the-ory and epistemic analysis with the underlying decision-theoretic foun-dation.

The present text considers a setting where the players have preferencesover their own strategies in a game, and investigates the following mainquestion: What preferences may be viewed as “reasonable”, providedthat each player takes into account the rationality of the opponent, he

takes into account that the opponent takes into account the player’s ownrationality, and so forth? And in the extension of this: Can we developformal, intuitive criteria that eventually lead to a selection of preferencesfor the players that may be viewed as “reasonable”?

The ‘consistent preferences’ approach as such is not new. It is firmlyrooted in a thirty year old game-theoretic tradition where a strategy of aplayer is interpreted as an expression of the belief (or the “conjecture”)



2 CONSISTENT PREFERENCES

of his opponent; cf., e.g., Harsanyi (1973), Aumann (1987a), and Blumeet al. (1991b). What is new in this book (and the papers on whichit builds) is that such a ‘consistent preferences’ approach is used tocharacterize a wider set of equilibrium concepts and, in particular, toserve as a basis for various types of interactive epistemic analysis where equilibrium assumptions are not made .

Throughout this book, games are analyzed from the subjective per-spective of each player. Hence, we can only make subjective statementsabout what a player “will do”, by considering “reasonable” preferences(and the corresponding representation in terms of subjective probabil-ities) of his opponent. This subjective perspective is echoed by recent

contributions like Feinberg (2004a) and Kaneko and Kline (2004), whichhowever differ from the present approach in many respects.1

To illustrate the differences between the two approaches—the ‘ratio-nal choice’ approach on the one hand and the ‘consistent preferences’approach on the other—in a setting that will be familiar to most read-ers, Section 1.1 will be used to consider how epistemic conditions forNash equilibrium in a strategic game can be formulated within each of these approaches.

The remaining Sections 1.2 and 1.3 will provide motivation for the‘consistent preferences’ approach through the following two points:

1 It facilitates the analysis of backward and forward induction.

2 It facilitates the integration of game theory and epistemic analysiswith the underlying decision-theoretic foundation.

1.1 Conditions for Nash equilibrium

To fix ideas, consider a simple coordination game, where two driversmust choose what side to drive on in order to avoid colliding. In an

1In the present text, reasoning about hypothetical events will be captured by each player hav-ing an initial (interim – after having become aware of his own “type”) system of conditionalpreferences; cf. Chapters 3 and 4. This system encodes how the player will update his beliefs

as actual play develops. In contrast, the subjective framework of Feinberg (2004a) does notrepresent the reasoning from such an interim viewpoint, and beliefs are not constrained tobe evolving or revised. Instead, beliefs are represented whenever there is a decision to bemade based on the presumption that beliefs should only matter when a decision is made. InFeinberg’s framework, only the ex-post beliefs are present and all ex-post subjective views

are equally modeled. Even though also Kaneko and Kline (2004) consider a player havinga subjective view on the objective situation, their main point is the inductive derivation of this individual subjective view from individual experiences.



Introduction 3

equilibrium in the ‘rational choice’ approach, a driver chooses to driveon the right side of the road if he believes that his opponent choosesto drive on the right side of the road. This can be contrasted withan equilibrium in the ‘consistent preferences’ approach, where a driverprefers to drive on the right side of the road if he believes that hisopponent prefers to drive on the right side of the road. As mentioned,this follows a tradition in equilibrium analysis from Harsanyi (1973) toBlume et al. (1991b). This section presents, as a preliminary analysis,how these two interpretations of Nash equilibrium can be formalized.

First, introduce the concept of a strategic game. A strategic two-player game G = (S 1, S 2, u1, u2) consists of, for each player i, a set of

pure strategies , S i, and a payoff function, ui : S 1 × S 2 → R.Then, turn to the epistemic modeling. An epistemic model for a

strategic game within the ‘rational choice’ approach will typically specify,for each player i,

a finite set of types, T i,

a function that assigns a strategy choice to each type, si : T i → S i,and,

for each type ti in T i, a probability distribution on the set of opponenttypes, µti ∈ ∆(T j),

where ∆(T j ) denotes the set of probability distributions on T j.When combined with i’s payoff function, the function s j and the prob-

ability distribution µti determine player i’s preferences at ti over his ownstrategies; these preferences will be denoted ti :

si ti si iff

tj

µti(t j )ui(si, s j(t j)) ≥

tj

µti(t j )ui(si, s j(t j )) .

This in turn determines i’s set of best responses at ti, which will through-out be referred to as i’s choice set at ti:

S

ti

i := {si ∈ S i| ∀s

i ∈ S i, si ti

s

i} .

Finally, in the context of the ‘rational choice’ approach, we can definethe set of type profiles for which player i chooses rat ionally:

[rati] := {(t1, t2) ∈ T 1 × T 2| si(ti) ∈ S tii } .

Write [rat] := [rat1] ∩ [rat2].




It is now straightforward to give sufficient epistemic conditions for apure strategy Nash equilibrium:

(s1, s2) ∈ S 1 × S 2 is a pure strategy Nash equilibrium

if there exists an epistemic model with (t1, t2) ∈ [rat]

such that (s1, s2) = (s1(t1), s2(t2)) and, for each i, µti(t j) = 1 .

In words, (s1, s2) is a pure strategy Nash equilibrium if there is mutualbelief of a profile of types that rationally choose s1 and s2. In fact,we need not require mutual belief of the type profile: in line with theinsights of Aumann and Brandenburger (1995) (cf. their PreliminaryObservation) it is sufficient that there is mutual belief of the strategyprofile, as we need not be concerned with what one player believes thatthe other player believes (or any higher order beliefs).

Consider next how to formulate epistemic conditions for a mixed strat-egy Nash equilibrium. Following, e.g., Harsanyi (1973), Armbruster andBoge (1979), Aumann (1987a), Brandenburger and Dekel (1989), Blumeet al. (1991b), and Aumann and Brandenburger (1995), a mixed strat-egy Nash equilibrium is often interpreted as an equilibrium in beliefs.According to this rather prominent view, a player need not random-ize in a mixed strategy Nash equilibrium, but may choose some purestrategy. However, the other player does not know which one, and themixed strategy of the one player is an expression of the belief (or the

“conjecture”) of the other.The ‘consistent preferences’ approach is well-suited for formulating

epistemic conditions for a mixed strategy Nash equilibrium according tothis interpretation. An epistemic model for a strategic game within the‘consistent preferences’ approach will typically specify, for each player i,

a finite set of types, T i, and

for each type ti in T i, a probability distribution on the set of opponentstrategy-type pairs, µti ∈ ∆(S j × T j).

Hence, instead of specifying a function that assigns strategy choices totypes, each type’s probability distribution is extended to the Cartesian

product of the opponent’s strategy set and type set.We can still determine type i’s preferences at ti over his own strategies,

si ti si iff

sj

tj

µti(s j, t j)ui(si, s j) ≥

sj

tj

µti(s j , t j )ui(si, s j ) ,

and i’s choice set at ti:

S tii := {si ∈ S i| ∀si ∈ S i, si ti s

i} .



Introduction 5

However, we are now concerned with what i at ti believes that oppo-nent types do, rather than with what i at ti does himself. Naturally,such beliefs will only be well-defined for opponent types that ti deemssubjectively possible, i.e., for player j types in the set

T jti :=

t j ∈ T j

µti(S j , t j ) > 0

,

where µti(S j, t j) :=

sj∈S jµti(s j , t j ). Say that the mixed strategy p j

ti|tj

is induced for t j by ti if t j ∈ T jti , and for each s j ∈ S j ,

p jti|tj (s j ) =

µti(s j, t j)

µti(S j

, t j

) .

Finally, in the context of the ‘consistent preferences’ approach, we candefine the set of type profiles for which ti i nduces a r ational mixedstrategy for any subjectively possible opponent type:

[iri] :=

(t1, t2) ∈ T 1 × T 2

∀t j ∈ T j

ti , p jti|tj ∈ ∆

S j

tj

.

If the true type profile is in [iri], then player i’s preferences over hisstrategies are consistent with the preferences of his opponent. Ratherthan player j actually being rational, it entails that player i believes that j is rational.

Write [ir] := [ir1] ∩ [ir2]. Through the event [ir] one can formulatesufficient epistemic conditions for a mixed strategy Nash equilibrium,interpreted as an equilibrium in beliefs:

( p1, p2) ∈ ∆(S 1) × ∆(S 2) is a mixed strategy Nash equilibrium

if there exists an epistemic model with (t1, t2) ∈ [ir]

such that ( p1, p2) = p1

t2|t1 , p2t1|t2

and, for each i, µti(S j, t j) = 1 .

In words, ( p1, p2) is a mixed strategy Nash equilibrium if there is mutualbelief of a profile of types, where each type induces the opponent’s mixedstrategy for the other, and where any pure strategy in the induced mixedstrategy is rational for the opponent type. Since any pure strategy Nash

equilibrium can be viewed as a degenerate mixed strategy Nash equi-librium, these epistemic conditions are sufficient for pure strategy Nashequilibrium as well. Again, we need not require mutual belief of the typeprofile; it is sufficient that there is mutual belief of each player’s belief about the strategy choice of his opponent.

It is by no means infeasible to provide epistemic conditions for mixedstrategy Nash equilibrium, interpreted as an equilibrium in beliefs, with-




in the ‘rational choice’ approach. Indeed, this is what Aumann andBrandenburger (1995) do through their Theorem A in the case of two-player games. One can still argue for the epistemic conditions arisingwithin the ‘consistent preferences’ approach. If a mixed strategy Nashequilibrium is interpreted as an expression of what each player believeshis opponent will do, then one can argue—based on Occam’s razor—thatthe epistemic conditions should specify these beliefs only, and not alsowhat each player actually does. In particular, we need not require, asAumann and Brandenburger (1995) do, that the players are rational.

1.2 Modeling backward and forward induction

This book is mainly concerned with the analysis of deductive rea-soning in games—leading to rationalizability concepts—rather than thestudy of steady states where coordination problems have been solved—corresponding to equilibrium concepts. Deductive reasoning within the‘consistent preferences’ approach means that events like [ir] will be madesubject to interactive epistemology, without assuming that there is mu-tual belief of the type profile.

Backward induction is a prime example of deductive reasoning ingames. To capture the backward induction procedure, one must be-lieve that each player chooses rationally at every information set of anextensive game, also at information sets that the player’s own strategy

precludes from being reached. As will be indicated through the analysisof Chapters 7–10—based partly on joint work with Andres Perea—thismight be easier to capture by analyzing events where each player be-lieves that the opponent chooses rationally, rather than events whereeach player actually chooses rationally. The backward induction pro-cedure can be captured by conditions on how each player revises hisbeliefs after “surprising” choices by the opponent. Therefore, it mightbe fruitful to characterize this procedure through restrictions on the be-lief revision policies of the players, rather than through restrictions ontheir behavior at all information sets (also at information sets that canonly be reached if the behavioral restrictions at earlier information sets

were not adhered to). As will be apparent in Chapters 7–10, the ‘consis-tent preferences’ approach captures the backward induction procedurethrough conditions imposed directly on the players’ belief revision poli-cies.

In certain games—like the “battle-of-the-sexes with outside option”game (cf. Figure 2.6)—forward induction has considerable bite. Tomodel forward induction, one must essentially assume that each player



Introduction 7

believes that any rational choice by the opponent is infinitely more likelythan any choice that is not rational. Again, this might be easier to cap-ture by analyzing events relating to the beliefs of the player, rather thanevents relating to the behavior of the opponent. Chapters 11 and 12will report on joint work with Martin Dufwenberg that shows how the‘consistent preferences’ approach can be used to promote the forwardinduction outcome.

For ease of presentation only two-player games will be considered inthis book. This is in part a matter of convenience, as much of the sub-sequent analysis can essentially be generalized to n-player games (withn > 2). In particular, this applies to the analysis of backward induction

in Chapter 7, and to some extent, the analysis of forward induction inChapters 11 and 12. On the other hand, in the equilibrium analysis of Chapters 5, 8, 9, and 10, a strategy of one player is interpreted as anexpression of the belief of his opponent. This interpretation is straight-forward in two-player games, but requires that the beliefs of differentopponents coincide in games with more than two players—e.g., compareTheorems A and B of Aumann and Brandenburger (1995). Moreover,by only considering two-player games we can avoid the issue of whether(and if so, how) each player’s beliefs about the strategy choices of hisopponents are stochastically independent.

Throughout, player 1 will be referred to in the male gender (e.g.,

“he chooses among his strategies”), while player 2 will be referred toin the female gender (e.g., “she believes that player 1 . . . ”). Also, inthe examples the strategies of player 1 will be denoted by upper casesymbols (e.g., L and R), while the strategies of player 2 will be denotedby lower case symbols (e.g., and r).

1.3 Integrating decision theory and game theory

When a player in a two-player strategic game considers what decisionto make (i.e., what strategy to choose), only his belief about the strategychoice of his opponent matters for his decision. However, in order toform a well-judged belief regarding the choice of his opponent, he should

take her rationality into account. This makes it necessary for the playerto consider his belief about her belief about his strategy choice. Andso forth. Hence, the uncertainty faced by a player i concerns (a) thestrategy choice of his opponent j , (b) j ’s belief about i’s strategy choice,and so on; cf. Tan and Werlang (1988). A type of a player i correspondsto (a) a belief about j’s strategy choice, (b) a belief about j’s belief about i’s strategy choice, and so on. Models of such infinite hierarchies




of beliefs—see, e.g., Boge and Eisele (1979), Mertens and Zamir (1985),Brandenburger and Dekel (1993), and Epstein and Wang (1996)—yieldS 1 × T 1 × S 2 × T 2 as the ‘belief-complete’ state space, where T i is theset of all feasible types of player i. Furthermore, for each i, there is ahomeomorphism between T i and the set of beliefs on S i × S j × T j .

In the decision problem of any player i, i’s decision is to choose oneof his own strategies. For the modeling of this problem, i’s belief abouthis own strategy choice is not relevant and can be ignored. This doesnot mean that player i is not aware of his own choice. It signifies thatsuch awareness plays no role in the analysis, and is thus redundant.2

Hence, in the setting of a strategic game the belief of each type of player

i can be restricted to the set of opponent strategy-type pairs, S j × T j.Combined with the payoff function specified by the strategic game, abelief on S j × T j yields preferences over player i’s strategies.

As discussed in Section 5.1, the above results on ‘belief-complete’state spaces are not needed (since only finite games are treated without‘belief-completeness’ being imposed) and not always applicable in thesetting of the present text (since some of the analysis—e.g. in Chapters6, 7, 11, and 12—allows for incomplete preferences). Indeed, infinite hi-erarchies of beliefs can be modeled by an implicit but ‘belief-incomplete’model—with a finite type set T i for each player i—where the belief of a player corresponds to the player’s type, and where the belief of the

player concerns the opponent’s strategy-type pair.If we let each player be aware of his own type (as we will assume

throughout), this leads to an epistemic model where the state space of player i is T i × S j × T j. For each player, this is a standard decision-theoretic formulation in the tradition of Savage (1954), Anscombe andAumann (1963), and Blume et al. (1991a):

Player i as a decision maker is uncertain about what strategy-typepair in S j × T j will be realized.

Player i’s type ti determines his belief on S j × T j

Player i’s decision is to choose a (possibly mixed) strategy pi ∈ ∆(S i);each such strategy determines the (randomized) outcome of the gameas a function of the opponent strategy s j ∈ S j.3

2Tan and Werlang (1988) in their Sections 2 and 3 characterize rationalizable strategieswithout specifying beliefs about one’s own choice.3Hence, a strategy for a player corresponds to an Anscombe-Aumann act , assigning a (possiblyrandomized) outcome to any uncertain state; cf. Chapter 3.



Introduction 9

The model leads, however, to a different state space for each player,which may perhaps be considered problematic.

In the framework for epistemic modeling of games proposed by Au-mann (1987a)—applied by Aumann and Brandenburger (1995) and il-lustrated in Section 1.1—it is also explicitly modeled that each player isaware of his own decision (i.e., his strategy choice). This entails that,for each player i, there is function si from T i to S i that assigns si(ti) toti. Furthermore, it means that the relevant state space is T 1 × T 2, whichis identical for both players. In spite of its prevalence, Aumann’s modelleads to the following potential problem: If player i is of type ti and inspite of this were to choose some strategy si different from si(ti), then

the player would no longer be of type ti (since only si(ti) is assigned toti). So what, starting with a state where player i is of type ti, wouldplayer i believe about his opponent’s strategy choice if he were to choosesi = si(ti)?

In line with the defense by Aumann and Brandenburger (1995) onpp. 1174-1175, one may argue that Aumann’s framework is purely de-scriptive and contains enough information to determine whether a playeris rational and that we need not be concerned about what the playerwould have believed if the state were different. An alternative is, how-ever, to follow Board (2003) in arguing that ti’s belief about his oppo-nent’s strategy choice should remain unchanged in the counterfactual

event that he were to choose si = si(ti).The above discussion can be interpreted as support for the epistemic

structure that will underlie this book, and where the state space of playeri is T i × S j × T j . This kind of epistemic model describes the factorsthat are relevant for each player as a decision maker (namely, what hisopponent does and who his opponent is), while being silent about theawareness of player i of his own decision. Also in this formulation, adifferent choice by player i changes the state, as an element of S 1 ×T 1 × S 2 × T 2, but it does not influence the type of player i, as a specificstrategy is not assigned to each type. Hence, a different choice by playeri does not change his belief about what the opponents do.

In this setting, the epistemic analysis concerns the type profile, andnot the strategy profile. As we have seen in Section 1.1, and which wewill return to in Chapter 5, this is, however, sufficient to state and prove,e.g., a result that corresponds to Aumann and Brandenburger’s (1995)Theorem A, provided that mutual belief of rationality is weakened tothe condition that each player believes that his opponent is rational. Aswe will see in Chapters 5 and 6 it also facilitates the introduction of




caution, which then corresponds to players having beliefs that take intoaccount that opponents may make irrational choices, rather than playerstrembling when they make their choice.

Chapters 3 and 4 are concerned with the decision-theoretic frameworkand epistemic operators derived from this framework.

Chapter 3 spells out how the Anscombe-Aumann framework will beused as a decision-theoretic foundation. Following Blume et al. (1991a),continuity will be relaxed. Moreover, two different kinds of generaliza-tions are presented. On the one hand, completeness will be relaxed, asthis is not an integral part of the backward induction procedure, andcannot be imposed in the epistemic characterization of forward induc-

tion presented in Chapters 11 and 12. On the other hand, flexibilityconcerning how to specify a system of conditional beliefs will be intro-duced, leading to a structure that encompasses both the concept of aconditional probability system and conditionals derived from a lexico-graphic probability system . This flexibility turns out to be essential forthe analysis of Chapters 8 and 9.

Chapter 4 reports on joint work with Ylva Søvik which derives belief-operators from the preferences of decision makers and develop their se-mantics. These belief operators will in later chapters be used in theepistemic characterizations.

First, however, motivating examples will be presented and discussed

in Chapter 2.



Chapter 2

MOTIVATING EXAMPLES

Through examples this chapter illuminates the features that distin-guish the ‘consistent preferences’ approach from the ‘rational choice’ ap-proach (cf. Chapter 1). The examples also illustrate issues of relevancewhen capturing backward and forward induction in models of interactiveepistemology. The same examples will be revisited in later chapters.

Section 2.1 presents six different games, and contains a discussion of how suggested outcomes in these games can be promoted by different

solution concepts. This discussion leads in Section 2.2 to an overview of the solution concepts that will be covered in subsequent chapters. WhileSection 2.1 will illustrate how various concepts work in the differentexamples, Section 2.2 will relate the different concepts to each other,and provide references to relevant literature.

2.1 Six examples

Consider the “battle-of-the-sexes” game, G1, illustrated in Figure 2.1.This game has two Nash equilibria in pure strategies: (L, ) and (R, r).In the ‘rational choice’ approach, the first of these Nash equilibrium isinterpreted as player 1 choosing L and player 2 choosing , and these

choices being mutual belief. It is a Nash equilibrium since there is mutualbelief of the strategy choices and each player’s choice is rational, givenhis belief about the choice of his opponent. In the ‘consistent preferences’approach, in contrast, this Nash equilibrium is interpreted as player 1believing that 2 chooses and player 2 believing that 1 chooses L, andthese conjectures being mutual belief. It is a Nash equilibrium sincethere is mutual belief of the conjectures about opponent choice and each




r

L

R

3, 1 0, 0

0, 0 1, 3

Figure 2.1. G1 (“battle-of-the-sexes”).

player believes that the opponent chooses rationally given the opponent’sconjecture. The preferences of player 1—that he ranks L about R—isconsistent with the preferences of player 2—that she ranks above r, and

vice versa. More precisely, that player 1 ranks L above R is consistentwith his beliefs about player 2, namely that he believes that she ranks above r and she chooses rationally (i.e., chooses a top ranked strategy).

The ‘consistent preferences’ interpretation of Nash equilibrium carriesover to the mixed strategy equilibrium when interpreted as an equilib-rium in beliefs—cf. the Harsanyi (1973) interpretation discussed in Sec-tion 1.1. If player 1 believes with probability 1/4 that 2 chooses andwith probability 3/4 that 2 chooses r and player 2 believes with prob-ability 3/4 that 1 chooses L and with probability 1/4 that 1 choosesR, and these conjectures are common belief, then the players’ beliefsconstitute a mixed-strategy Nash equilibrium. It is a Nash equilibriumsince there is mutual belief of the conjectures about opponent choiceand each player believes that the opponent chooses rationally given theopponent’s conjecture.

Rationalizability concepts have no bite in the “battle-of-the-sexes”game, G1: Interactive epistemology based on rationality alone cannotguide the players to one of the equilibria. Hence, to illustrate the forceof deductive reasoning in games—leading to rationalizability concepts—we must consider other examples.

In game G2 of Figure 2.2, there is a unique Nash equilibrium, ( L, ).Furthermore, deductive reasoning will readily lead player 1 to L andplayer 2 to . In the ‘rational choice’ approach this works as follows:If player 1 chooses rationally, then he chooses L. This is independent

of his conjecture about 2’s behavior since L strongly dominates R (as4 > 3 and 1 > 0). Therefore, if player 2 believes that 1 chooses ratio-nally, and 2 chooses rationally herself, then she chooses (since 1 > 0).This argument shows that L is the unique rationalizable strategy forplayer 1 and is the unique rationalizable strategy for player 2. In the‘consistent preferences’ approach, we get: Player 1 ranks L above R,independently of his conjecture about 2’s behavior. If player 2 believes



Motivating Examples 13

r

L

R

4, 1 1, 0

3, 0 0, 3

Figure 2.2. G2, illustrating deductive reasoning.

r

L

R

1, 3 4, 2

1, 3 3, 5

Figure 2.3. G3, illustrating weak dominance.

that 1 chooses rationally, then she believes that 1 chooses L and ranks above r. Therefore, if player 1 believes that 2 chooses rationally, and hebelieves that she believes that 1 chooses rationally, then he believes that2 chooses . As we will return to in Chapters 5 and 6, this is an alterna-tive way to establish L and as the players’ rationalizable strategies. Inany case, the deductive reasoning leading to rationalizability correspondsto iterated elimination of strongly dominated strategies (IESDS).

In game G3 of Figure 2.3, there is also a unique Nash equilibrium,(L, ). However, deductive reasoning is more problematic and inter-esting in the case of this game. For each player, both strategies arerationalizable, meaning that rationalizability has no bite in this game.In particular, if player 1 deems it subjectively impossible that 2 maychoose r, then R is a rational choice. Moreover, if player 2 believes that1 chooses R, then r is a rational choice. Still, we might argue that 1should not rule out the possibility that 2 might choose r, leading himto rank L above R (since L weakly dominates R) and player 2 to rank above r. Such deductive reasoning leads to permissible strategies inthe terminology of Brandenburger (1992). Permissibility correspondsto one round of elimination of all weakly dominated strategies followed

by iterated elimination of strongly dominated strategies—the so-calledDekel-Fudenberg procedure, after Dekel and Fudenberg (1990). It canbe formalized in two different ways.

On the one hand, within an analysis based on what players do, onecan postulate that players make ‘almost’ rational choices by, in the spiritof Selten (1975) and his “trembling hand”, assuming that ‘mistakes’ aremade with (infinitely) small probability. Borgers (1994) shows how such




r

Out

InL

InR

2, 0 2, 0

1, 3 4, 2

1, 3 3, 5

1 2 1 3In r R 5

Out L

2 1 40 3 2

Figure 2.4. G3 and a corresponding extensive form Γ3 (a “centipede” game).

an approach does indeed correspond to the Dekel-Fudenberg procedureand thus characterizes permissibility.

On the other hand, within an analysis based on what players believe,one can impose that players are ‘cautious’, in the sense of deeming noopponent strategy as subjectively impossible. This approach to permis-sibility—which is in the spirit of Blume et al. (1991b) and Brandenburger(1992)—combines such caution with an assumption that each player be-lieves that the opponent is rational. It is shown in Chapters 5 and 6how this yields an alternative characterization of permissibility, whereone need not consider whether players in fact are rational.

Let us then turn to an expanded version of G3, namely the game G 3

illustrated in Figure 2.4 with a corresponding extensive form Γ3. Fol-

lowing Rosenthal (1981) Γ3 is often called a “centipede” game. Here,

(Out, ) is normally suggested as a solution for this game. In the strategicform G3, this suggestion can be obtained by iterated (maximal) elimina-tion of weakly dominated strategies (IEWDS), and in the extensive formΓ

3, it is based on backward induction. While epistemic conditions forthe procedure of IEWDS have been given by Brandenburger and Keisler(2002)—see also the related work by Battigalli and Siniscalchi (2002)—IEWDS will fall outside the class of procedures that will be characterizedin this book. The procedure of backward induction, on the other hand,will play a central role in Chapters 7–10.

Permissibility, which corresponds to the Dekel-Fudenberg procedure,does not promote only (Out, ) in the games of Figure 2.4. While theDekel-Fudenberg procedure eliminates the weakly dominated strategy

InR, this procedure does not allow for further rounds of weak elimi-nation. Hence, since r is not strongly dominated by even after theelimination of InR, r will not be eliminated by the Dekel-Fudenbergprocedure. Hence, InL as well as Out are permissible for player 1, andr as well as are permissible for player 2.

In the extensive game, Γ3, one can give the following intuition for

how InL and r are compatible with the deductive reasoning underlying




permissibility: If player 1 believes that player 2 will choose , then heprefers Out to his two other strategies. Similarly, if player 2 assignsprobability one to player 1 choosing Out, and revises her beliefs byassigning probability one to InL conditional on being asked to play,then she prefers to r. However, if player 2 assigns probability one toplayer 1 choosing Out, but revises her beliefs so that InL and InR areequally likely conditional on being asked to play, then she prefers r to .So if player 1 assigns sufficient probability to player 2 being of the lattertype and believes—conditional on her being of this type—that she willbe rational by choosing her top-ranked strategy r, then he will preferInL to his two other strategies. Following Ben-Porath (1997), Chapter

7 demonstrates within a formal epistemic model how such interactivebeliefs are consistent with the assumptions underlying permissibility.

As shown by Ben-Porath (1997), when permissibility is applied toan extensive game like Γ

3, each player must believe that her opponentchooses rationally as long as the opponent’s behavior is consistent withthe player’s initial beliefs. However, conditional on finding herself at aninformation set that contradicts her previous belief about his behavior,she is allowed to believe that he will no longer choose rationally. E.g.,in Γ

3 it is OK for player 2 to assign positive probability to the irrationalstrategy InR conditional on being asked to play, provided that she hadoriginally assigned probability one to player 1 rationally choosing Out.

An alternative is that the player should still believe that her opponentwill choose rationally, even conditionally on being informed about “sur-prising” moves. Chapters 7–9 will consider the event that each playerbelieves that her opponent chooses rationally at all his information setswithin models of interactive epistemology. Building on joint work withAndres Perea, this provides

epistemic conditions for backward induction and

definitions for the concepts of sequential and quasi-perfect rational-izability.

Note that imposing that a player believes that her opponent chooses

rationally at all his information sets is a requirement imposed on herbelief revision policy, not on her actual behavior. It therefore fits wellwithin the ‘consistent preferences’ approach.

If we move to an expanded version of G2, namely the game G2 illus-

trated in Figure 2.5 with a corresponding extensive form Γ 2, not even

the event that each player believes that the opponent chooses rationallyat all his information sets, will be sufficient for reaching the solution that




r

Out

InL

InR

2, 2 2, 2

4, 1 1, 0

3, 0 0, 3

2 12

2

Out

InL InR

r r

4 1 3 01 0 0 3

Figure 2.5. G2 and a corresponding extensive form Γ2.

one would normally suggest, namely (InL, ). This outcome is supportedby the following deductive reasoning: Since InL strongly dominates InR,implying that player 1 prefers the former strategy to the latter, player 2should deem InL much more likely than InR conditional on being askedto play, and hence prefer to r. This in turn would lead player 1 toprefer InL to his two other strategies if he believes that player 2 will berational by choosing her top-ranked strategy .

However, the concepts of sequential and quasi-perfect rationalizabilityonly preclude that player 2 unconditionally assigns positive probabilityto player 1 choosing InR. If player 2 assigns probability one to player1 choosing Out, then she may—when revising her beliefs conditional onbeing asked to play—assign sufficient probability to InR so that r ispreferred to . If player 1 assigns sufficient probability to player 2 beingof such a type, then he will prefer Out to his two other strategies.

The outcome (InL, ) can be promoted by considering the event thatplayer 2 respects the preferences of her opponent by deeming one oppo-nent strategy infinitely more likely than another if the opponent prefersthe former to the latter. Respect of opponent preferences was first con-sidered by Blume et al. (1991b) in their characterization of proper equi-librium. Being a requirement on the beliefs of players, it fits nicely intothe ‘consistent preferences’ approach. Within a model of interactiveepistemology Chapter 10 characterizes the concept of proper rational-

izability by considering the event that each player respects opponentpreferences. Proper rationalizability implies backward induction. How-ever, even though it yields conclusions that coincide with IEWDS in allof the examples above, this conclusion does not hold in general, as willbe shown by the next example and further discussed in Chapter 10.

Lastly, turn to an expanded version of G1, namely the game G1 illus-

trated in Figure 2.6 with a corresponding extensive form Γ1. The exten-




r

Out

InL

InR

2, 2 2, 2

3, 1 0, 0

0, 0 1, 3

2 12

2

Out

InL InR

r r

3 0 0 11 0 0 3

Figure 2.6. G1 and a corresponding extensive form Γ1 (“battle-of-the-sexes with an

outside option”).

sive game Γ1 is referred to as the “battle-of-the-sexes with an outside

option” game. This game was introduced by Kreps and Wilson (1982)(who credit Elon Kohlberg) and is often used to illustrate forward in-duction, namely that player 2 through deductive reasoning should figureout that player 1 has chosen InL and aims for the payoff 3 if 2 is beingasked to play. Respect of preferences only requires player 2 to deem InRinfinitely less likely than Out since the latter strategy strongly domi-nates the former; it does not require 2 to deem InR infinitely less likelythan InL and thereby prefer to r .

In contrast, IEWDS eliminates all strategies except InL for player 1

and for player 2, thereby promoting the forward induction outcome.Chapter 11 contains a critical assessment of how iterated weak domi-nance promotes forward induction in this and other examples. Based on joint work with Martin Dufwenberg, it will be suggested how forward in-duction can be promoted by strengthening the concept of permissibilityto our notion of full permissibility .

Full permissibility is characterized by conditions levied on the beliefsof players, and therefore fits naturally into the ‘consistent preferences’approach. In the final Chapter 12 this notion will be further illustratedthrough a series of extensive games, illustrating how it yields forwardinduction, while not always supporting backward induction (indeed, Γ

3

is an example of an extensive game where full permissibility does notpromote the backward induction outcome).

2.2 Overview over concepts

To provide a structure for the concepts that will be defined and char-acterized in the subsequent chapters, it might be useful as a roadmap topresent an overview over these concepts and their relationships.




Table 2.1. Relationships between different equilibrium concepts.

Proper equilibrium

Myerson (1978)↓

Strategic form Quasi-perfect perfect equil. ← equilibrium Selten (1975) van Damme (1984)

↓ ↓Nash Weak sequential Sequential equi- ← equilibrium ← equilibrium

librium Reny (1992) Kreps & Wilson (1982)

First, consider the equilibrium concepts of Table 2.1. Here, weak se-quential equilibrium refers to the equilibrium concept—defined by Reny(1992)—that results when each player only optimizes at information setsthat the player’s own strategy does not preclude from being reached.Moreover, quasi-perfect equilibrium is the concept defined by van Damme(1984) and which differs from Selten’s (1975) extensive form perfect equi-librium by having each player ignore the possibility of his own future mis-takes. The arrows indicate that any proper equilibrium corresponds to aquasi-perfect equilibrium and so forth. Nash equilibrium and (strategicform) perfect equilibrium will be characterized in Chapter 5, while se-quential equilibrium, quasi-perfect equilibrium, and proper equilibriumwill be characterized in Chapters 8, 9, and 10, respectively.

The non-equilibrium analogs to these equilibrium concepts are illus-trated in Table 2.2. Again, the arrows indicate that proper rationaliz-ability implies quasi-perfect rationalizability and so forth. Of course, thenotion of rationalizability due to Bernheim (1984) and Pearce (1984) is anon-equilibrium analog to Nash equilibrium. Likewise, the notion of per-

missibility due to Borgers (1994) and Brandenburger (1992) correspondsto Selten’s (1975) strategic form perfect equilibrium, and the notion of weak sequential rationalizability due to Ben-Porath (1997)—coined ‘weakextensive form rationalizablity’ by Battigalli and Bonanno (1999)—is anon-equilibrium analog of weak sequential equilibrium. Furthermore, se-quential rationalizability due to Dekel et al. (1999, 2002), quasi-perfectrationalizability due to Asheim and Perea (2004), and proper rational-




Table 2.2. Relationships between different rationalizability concepts.

Common . . . b elieves the . . . b elieves the . . . b elieves the

cert. belief oppon. chooses oppon. chooses oppon. chooses

that each rationally only rationally at rationally at

player . . . initially, in the all reachable all info. sets

whole game info. sets

. . . is cautious Proper

and respects [n.a.] [n.a.] rationalizability

preferences Schuhm. (1999)

[Chapter 10]

Permissibility ↓Borgers (1994) Quasi-perfect

. . . is cautious [n.a.] Brandenb. (1992) ← rationalizability

Dek. & Fud. (1990) Ash. & Per. (2004)

[Chapters 5–6] [Chapter 9]

Rationaliz- ↓ ↓

...is not ability Weak sequential Sequential

necessarily Bernh. (1984) ← rationalizability ← rationalizability

cautious Pearce (1984) Ben-Porath (1997) Dekel et al.

[Chapters 5–6] [Chapter 8] (1999, 2002)

[Chapter 8]

Does not imply Does not imply Implies

backward ind. backward ind. backward ind.

izability due to Schuhmacher (1999) are non-equilibrium analogs to se-quential equilibrium, quasi-perfect equilibrium, and proper equilibrium,respectively.

As indicated by Table 2.2, these concepts will be treated in Chapters5, 6, 8, 9, and 10, and they are characterized by

on the one hand, whether each player is cautious and respects oppo-nent preferences, and

on the other hand, whether each player believes that his opponent

chooses rationally only initially (in the whole game), or at all reach-able information sets, or at all information sets.

This taxonomy defines events which are made subject to common certainbelief, where ‘certain belief’ is the epistemic operator that will be usedfor the interactive epistemology. This operator is defined in Chapter 4and will have the following meaning: An event is said to be ‘certainlybelieved’ if the complement is deemed subjectively impossible.




Throughout this book, we will analyze assumptions about players’preferences , leading to events that are subsets of type profiles. We canstill make subjective statements about what a player “will do”, by con-sidering the preferences (and the corresponding representation in termsof subjective probabilities) of the other player.

For the concepts in the left and center columns of Table 2.2, we can domore than this, if we so wish. E.g., when characterizing weak sequentialrationalizability, we can consider the event of rational pure choice atall reachable information sets, and assume that this event is commonlybelieved (where the term ‘belief’ is used in the sense of ‘belief withprobability one’). These assumptions yield subsets of strategy profiles,

leading to direct behavioral implications within the model.This does not carry over to the concepts in the right column. It is

problematic to define the event of rational pure choice at all informa-tion sets, since reaching a non-reachable information set may contradictrational choice at earlier information sets. Also, if we consider the eventof (any kind of) rational pure choice, then we cannot use common cer-tain belief, since this—combined with rational choice—would preventwell-defined conditional beliefs after irrational opponent choices. How-ever, common belief (with probability one) of the event that each playerbelieves his opponent chooses rationally at all information sets does not yield backward induction in generic perfect information games, as shown

in the counterexample illustrated in Figure 7.1. Common certain belief is essential for our analysis of the concepts in the right column of Table2; this complicates obtaining direct behavioral implications.

Before defining the various belief operators that will be used in thelater chapters, the decision-theoretic framework will be presented andanalyzed in Chapter 3.



Chapter 3

DECISION-THEORETIC FRAMEWORK

In the ‘consistent preferences’ approach to deductive reasoning ingames, the object of the analysis is each player’s preferences over hisown strategies, rather than his choice. The preferences can be requiredto be consistent (in different senses) with his beliefs about the oppo-nent’s preferences over her strategies. The player’s preferences dependon his belief about the strategy choice of his opponent. Furthermore,in order for the player to consider the preferences of his opponent, herbelief about his strategy profile matters, and so forth. What kind of decision-theoretic framework is suited for such analysis?

This chapter spells out how the framework proposed by Anscombeand Aumann (1963) will be used as a decision-theoretic foundation. Fol-lowing Blume et al. (1991a), the Archimedean property will be relaxed.Moreover, two different kinds of generalizations will be presented:

(i) Completeness will be relaxed, as this is not an integral part of thebackward induction procedure (cf. the analysis of Chapter 7), andcannot be imposed in the epistemic characterization of forward in-duction presented in Chapters 11 and 12.

(ii) Flexibility concerning how to specify conditional preferences, lead-ing to a structure that encompasses both the concept of a conditional probability system and conditionals derived from a lexicographic prob-ability system . This flexibility turns out to be essential for the analysisof Chapters 8 and 9.

Section 3.1 motivates these generalizations, as well as providing rea-sons for the choice of the Ascombe-Aumann framework. Section 3.2




introduces the different sets of axioms that will be considered, while thefinal Section 3.3 presents the corresponding representation results.

3.1 Motivation

Standard decision theory under uncertainty concerns two differentkinds of decisions.

1 In the first kind, the object of choice is lotteries . There is a givenset of outcomes, and a lottery is an objective probability distributionover outcomes. If the decision maker satisfies the von Neumann-Morgenstern axioms—cf. von Neumann and Morgenstern (1947)—

then one can assign utilities to outcomes, so that the decision makerprefers one lottery to another if the former has higher expected utility.

2 In the second kind, the object of choice is acts . There is a given set of outcomes and a given set of uncertain states, and an act is a functionfrom states to outcomes. If the decision maker satisfies the Savage(1954) axioms, then one can assign utilities to outcomes and subjec-tive probabilities to states, so that the decision maker prefers one actto another if the former has higher (subjective) expected utility.

An act in the sense of Anscombe and Aumann (1963) is a functionfrom states to objective randomizations over outcomes.1 By consideringacts in this sense they are able to extend the von Neumann-Morgenstern

theory so that the utilities assigned to outcomes are determined solelyfrom preferences over lotteries, while the subjective probabilities as-signed to states are determined when also acts are considered.

A strategy in a game is a function that, for each opponent strategychoice, determines an outcome. A pure strategy determines for eachopponent strategy a deterministic outcome, while a mixed strategy de-termines for each opponent strategy an objective randomization over theset of outcomes. Hence, a pure strategy is an example of an act in thesense of Savage (1954), while a mixed strategy is an example of an actin the generalized sense of Anscombe and Aumann (1963).

Allowing for objective randomizations and using Anscombe-Aumann

acts are convenient for two reasons in the present context:The Anscombe-Aumann framework allows a player’s payoff functionto be a von Neumann-Morgenstern (vNM) utility function deter-

1Anscombe and Aumann (1963) use the term ‘roulette lottery’ for what we here call ‘lotteries’,‘horse lotteries’ for acts from states to deterministic outcomes, i.e., acts in the Savage (1954)sense, and ‘compound horse lotteries’ for what we here refer to as Anscombe-Aumann acts.



Decision-theoretic Framework 23

mined from his preferences over randomized outcomes, independentlyof the likelihood that he assigns to the different strategies of his oppo-nent. This is consistent with the way games are normally presented,where payoff functions for each player are provided independently of the analysis of the strategic interaction.2

When relaxing completeness, it turns out to be important to allowmixed strategies as objects of choice when determining maximal el-ements of a player’s incomplete preferences, for similar reasons asdomination by mixed strategies is needed for dominated strategies tocorrespond to strategies that can never be best replies.

We will consider three kinds of generalizations of the Anscombe-Aumann framework.

First, as mentioned in the introduction to this chapter, throughoutthis book we will follow Blume et al. (1991a) by imposing the condi-tional Archimedean property (also called conditional continuity ) insteadof Archimedean property (also called continuity ). This is important formodeling caution, which requires a player to take into account the pos-sibility that the opponent makes an irrational choice, while assigningprobability one to the event that the opponent makes a rational choice.I.e., even though any irrational choice is infinitely less likely than some

rational choice, it is not ruled out. Such discontinuous preferences willalso be useful when modeling players’ preferences in extensive games.

Second, we will relax the axiom of completeness to conditional com-pleteness . While complete preferences will normally be represented bymeans of subjective probabilities (cf. Propositions 1, 2, 3, and 5 of thischapter), incomplete preferences are insufficient to determine the relativelikelihood of the uncertain states. One possibility is, following Aumann(1962) and Bewley (1986), to represent incomplete preferences by meansof a set of subjective probability distributions.

Subjective probabilities are not part of the most common deductiveprocedures in game theory—like IESDS, the Dekel-Fudenberg proce-

dure, and the backward induction procedure. One can argue that, sincethey make no use of subjective probabilities, one should seek to provide

2This argument is in line with the analysis of Aumann and Dreze (2004), who however de-part from the Anscombe-Aumann framework by considering preferences—not over all func-

tions from states to randomized outcomes—but only on the subset of mixed strategies. TheAscombe-Aumann framework requires that the decision maker has access to objective prob-abilities; however, Machina (2004) points to how this requirement can be weakened.




epistemic conditions for such procedures without reference to subjectiveprobabilities. Indeed, subjective probabilities play no role in epistemicanalysis of backward induction by Aumann (1995).

In Chapters 6 and 7 we follow Aumann in this respect and provideepistemic conditions for IESDS, the Dekel-Fudenberg procedure, and thebackward induction procedure through modeling players endowed with(possibly) incomplete preferences that are not represented by subjectiveprobabilities. Moreover, for the modeling of forward induction in Chap-ters 11 and 12, it is a necessary part of the analysis that preferences areincomplete.

Third, we will allow for flexibility concerning how to specify condi-tional preferences. Such flexibility can be motivated in the context of the modeling of sequentiality and quasi-perfectness in Chapters 8 and9. Sequential rationalizability will be defined and sequential equilibriumcharacterized by considering the event that each player believes that theopponent chooses rationally at all her information sets. Adding pref-erence for cautious behavior to this event yields the concepts of quasi-perfect rationalizability and equilibrium. For these definitions and char-acterizations, we must describe what a player believes both conditionalon reaching his own information sets (to evaluate his rationality) andconditional on his opponent reaching her information sets (to determinehis beliefs about her choices). In other words, we must specify a system

of conditional beliefs for each player.There are various ways to do so. One possibility is a conditional prob-

ability system (CPS) where each conditional belief is a subjective prob-ability distribution.3 This is sufficient to model sequentiality. Anotherpossibility, which is sufficient to model quasi-perfectness, is to apply asingle sequence of subjective probability distributions—a so-called lexi-cographic probability system (LPS) as defined by Blume et al. (1991a)—and derive the conditional beliefs as the conditionals of such an LPS.Since each conditional LPS is found by constructing a new sequence,which includes the well-defined conditional probability distributions of the original sequence, each conditional belief is itself an LPS.

However, quasi-perfectness cannot always be modeled by a CPS sincethe modeling of preference for cautious behavior may require lexico-graphic probabilities. To see this, consider Γ4 of Figure 3.1. In this

3This is the terminology introduced by Myerson (1986). In philosophical literature, relatedconcepts are called Popper measures. For an overview over relevant literature and analysis,see Hammond (1994) and Halpern (2003).




1 2

D d F f

00

11

11

d f

F

D

1, 1 0, 0

1, 1 1, 1

Figure 3.1. Γ4 and its strategic form.

game, if player 1 believes that player 2 chooses rationally, then player 1must assign probability one to player 2 choosing d. Hence, if each (condi-tional) belief is associated with a subjective probability distribution—as

is the case with the concept of a CPS—and player 1 believes that hisopponent chooses rationally, then player 1 is indifferent between his twostrategies. This is inconsistent with quasi-perfectness, which requiresplayers to have preference for cautious behavior, meaning that player 1in Γ4 prefers D to F .

Moreover, sequentiality cannot always be modeled by means of condi-tionals of a single LPS since preference for cautious behavior is induced.To see this, consider a modified version of Γ1 where an additional sub-game is substituted for the (0, 0)–payoff, with all payoffs in that subgamebeing smaller than 1. If player 1’s conditional beliefs over strategies forplayer 2 is derived from a single LPS, then a well-defined belief con-

ditional on reaching the added subgame entails that player 1 deemspossible the event that player 2 chooses f , and hence, player 1 prefers Dto F . This is inconsistent with sequentiality, under which F is a rationalchoice.

Therefore, this chapter will present a new way of describing a systemof conditional beliefs, called a system of conditional lexicographic prob-abilities (SCLP), and which is based on joint work with Andres Perea;cf. Asheim and Perea (2004). In contrast to a CPS, an SCLP may induceconditional beliefs that are represented by LPSs rather than subjectiveprobability distributions. In contrast to the system of conditionals de-rived from a single LPS, an SCLP need not include all levels in the se-

quence of the original LPS when determining conditional beliefs. Thus,an SCLP ensures well-defined conditional beliefs representing nontrivialconditional preferences, while allowing for flexibility w.r.t. whether toassume preference for cautious behavior.






Axiom 6 (Conditionality) pε ε (resp. ∼ε) qε iff pφ φ|ε (resp.∼φ|ε) qφ, whenever ∅ = ε ⊆ φ.

It is an immediate observation that Axioms 5 and 6 imply non-null state independence as stated in Axiom 5 of Blume et al. (1991a).

Lemma 1 Assume that the system of conditional preferences {φ |φ ∈Φ} satisfies Axioms 5 and 6. Then, ∀φ ∈ Φ, pφ φ|{e} qφ iff pφ φ|{f }

qφ whenever e, f ∈ κ ∩ φ, and pφ and qφ satisfy pφ(e) = pφ(f ) and qφ(e) = qφ(f ).

Turn now the relaxation of Axioms 1, 4, and 6, as motivated in the

previous section.

Axiom 1 (Conditional Order) φ is reflexive and transitive and,∀e ∈ φ, φ|{e} is complete.

Axiom 4 (Conditional Archimedean Property) ∀e ∈ φ, if pφ

φ|{e} qφ φ|{e} p φ, then ∃0 < γ < δ < 1 such that δ p

φ+(1−δ )pφ φ|{e}

qφ φ|{e} γ pφ + (1 − γ )p

φ.

Axiom 6 (Dynamic Consistency) pε ε qε whenever pφ φ|ε qφ

and ∅ = ε ⊆ φ.

Since completeness implies reflexivity, Axiom 1 constitutes a weaken-

ing of Axioms 1. This weakening is substantive since, in the terminologyof Anscombe and Aumann (1963), it means that the decision maker hascomplete preferences over ‘roulette lotteries’ where objective probabil-ities are exogenously given, but not necessarily complete preferencesover ‘horse lotteries’ where subjective probabilities, if determined, areendogenously derived from the preferences of the decision maker.

Say that e ∈ κ is deemed infinitely more likely than f ∈ F (and writee f ) if p{e,f } {e,f } q{e,f } whenever p{e} {e} q{e}. Consider thefollowing two auxiliary axioms.

Axiom 11 (Partitional priority) If e e, then ∀f ∈ F , e f

or f e

.Axiom 16 (Compatibility) There exists a binary relation ∗

F satisfy-ing Axioms 1, 2, and 4 such that p ∗

F |φ q whenever pφ φ qφ and

∅ = φ ⊆ F .

While it is straightforward that Axiom 1 implies Axiom 1 , Axiom 4implies Axiom 4, and Axiom 6 implies Axiom 6, it is less obvious that






Table 3.1. Relationships between different sets of axioms and their representations.

Complete and 1 2 3 4 5 6 → 1 2 3 4 5 6 16continuous Prob. distr. CPS

↓

Complete and 1 2 3 4 5 6 ↓partitionally continuous LCPS

↓

Complete and 1 2 3 4 5 6 → 1 2 3 4 5 6 16

discontinuous LPS SCLP ↓

Incomplete and 1 11 2 3 4 5 6discontinuous Dynamic

Conditionality consistency

Axiom 4 (Partitional Archimedean Property) There is a parti-tion {π

1, . . . , πL|φ

} of κ ∩ φ such that

∀ ∈ {1, . . . , L|φ}, if p φ φ|π

qφ φ|π

p

φ, then ∃0 < γ < δ < 1 such that δ p

φ + (1 − δ )pφ φ|π

qφ φ|π

γ p

φ + (1 − γ )pφ, and

∀ ∈ {1, . . . , L|φ − 1}, pφ φ|πqφ implies pφ φ|π∪π+1

qφ.

Table 3.1 illustrates the relationships between the sets of axioms thatwe will consider. The arrows indicate that one set of axioms impliesanother. The figure indicates what kind of representations the differentsets of axioms correspond to, as reported in the next section.

3.3 Representation results

In view of Lemma 1 and using the characterization result of Anscombeand Aumann (1963), we obtain the following result under Axioms 1, 2,3, 4, 5, and 6; cf. Theorem 2.1 of Blume et al. (1991a).

For the statement of this and later results, denote by υ : Z → Ra vNM utility function, and abuse notation slightly by writing υ( p) =

z∈Z p(z)υ(z) whenever p ∈ ∆(Z ) is an objective randomization. Inthis and later results, υ is unique up to positive affine transformations.

Proposition 1 (Anscombe and Aumann, 1963) The following twostatements are equivalent.




1 (a) φ satisfies Axioms 1, 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {φ |φ ∈ Φ} satisfies Axioms 5 and 6.

2 There exist a vNM utility function υ : ∆(Z ) → R and a unique subjective probability distribution µ on F with support κ that satisfies, for any φ ∈ Φ,

pφ φ qφ iff

e∈φµ|φ(e)υ(pφ(e)) ≥

e∈φ

µ|φ(e)υ(qφ(e)) ,

where µ|φ is the conditional of µ on φ.

In view of Lemma 1, and using Theorem 3.1 of Blume et al. (1991a),we obtain the following result under Axioms 1, 2, 3, 4 , 5, and 6.

For the statement of this and later results, we need to introduceformally the concept of a lexicographic probability system. A lexico-graphic probability system (LPS) consists of L levels of subjective prob-ability distributions: If L ≥ 1 and, ∀ ∈ {1, . . . , L}, µ ∈ ∆(F ), thenλ = (µ1, . . . , µL) is an LPS on F . Denote by L∆(F ) the set of LPSs onF . Write supp λ := ∪L

=1supp µ for the support of λ. If supp λ ∩ φ = ∅,denote by λ|φ = (µ

1, . . . µL|φ

) the conditional of λ on φ.4

Furthermore, for two utility vectors v and w, denote by v ≥L w that,whenever w > v, there exists k < such that vk > wk, and let >L and

=L denote the asymmetric and symmetric parts, respectively.

Proposition 2 (Blume et. al, 1991a) The following two statements are equivalent.


2 There exist a vNM utility function υ : ∆(Z ) → R and an LPS λ on F with support κ that satisfies, for any φ ∈ Φ,

pφ φ qφ iff e∈φ

µ(e)υ(pφ(e))

L|φ

=1≥L

e∈φ

µ(e)υ(qφ(e))

L|φ

=1,

4I.e., ∀ ∈ {1, . . . , L|φ}, µ = µk|φ, where the indices k are given by k0 = 0, k =

min{k|µk(φ) > 0 and k > k−1} for > 0, and {k|µk(φ) > 0 and k > kL|φ} = ∅, andwhere µk|φ is given by the usual definition of conditional probabilities; cf. Definition 4.2 of Blume et al. (1991a).




where λ|φ = (µ1, . . . µ

L|φ) is the conditional of λ on φ.

In view of Lemma 1 and using Theorem 5.3 of Blume et al. (1991a),we obtain the following rusult under Axioms 1, 2, 3, 4 , 5, and 6.

For the statement of this results, we need to introduce the concept thatis called a lexicographic conditional probability system in the terminol-ogy that Blume et al. (1991a) use in their Definition 5.2. A lexicographic conditional probability system (LCPS) consists of L levels of non-overlap-ping subjective probability distributions: If λ = (µ1, . . . , µL) is an LPSon F and the supports of the µ’s are disjoint, then λ is an LCPS on F .

Proposition 3 (Blume et. al, 1991a) The following two statements are equivalent.


2 There exist a vNM utility function υ : ∆(Z ) → R and a unique LCPS λ on F with support κ that satisfies, for any φ ∈ Φ,

pφ φ qφ iff

e∈φµ

(e)υ(pφ(e))

L|φ

=1≥L e∈φ

µ(e)υ(qφ(e))

L|φ

=1,

where λ|φ = (µ1, . . . µ

L|φ) is the conditional of λ on φ (with the LCPS

λ|φ satisfying, ∀ ∈ {1, . . . , L|φ}, suppµ = π

).

Say that φ is conditionally represented by a vNM utility functionυ if (a) φ is non-trivial and (b) pφ φ|{e} qφ iff υ(pφ(e)) ≥ υ(qφ(e))whenever e is deemed subjectively possible. Under Axioms 1, 2, 3, 4, 5,and 6 conditional representation follows directly from the vNM theoremof expected utility representation.

Proposition 4 Assume that (a) φ satisfies Axioms 1 , 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {

φ |φ ∈ Φ} satisfies Axioms 5 and 6. Then

there exists a vNM utility function υ : ∆(Z ) → R such that, ∀φ ∈ Φ,pφ φ|{e} qφ iff υ(pφ(e)) ≥ υ(qφ(e)) whenever e ∈ κ ∩ φ.

Under Axioms 1, 2, 3, 4, 5, 6, and 16 we obtain the characterizationresult of Asheim and Perea (2004).

For the statement of this result, we need to introduce the concept of a system of conditional lexicographic probabilities. For this definition,




if λ := (µ1, . . . , µL) is an LPS and ∈ {1, . . . , L}, then write λ :=(µ1, . . . , µ) for the LPS that includes only the top levels of the originalsequence of probability distributions.

Definition 1 A system of conditional lexicographic probabilities (SCLP)(λ, ) on F with support κ consists of

an LPS λ = (µ1, . . . , µL) on F with support κ, and

a function : Φ → {1, . . . , L} satisfying (i) supp λ(φ) ∩ φ = ∅, (ii)(ε) ≥ (φ) whenever ∅ = ε ⊆ φ, and (iii) ({e}) ≥ whenevere ∈ supp µ.

The interpretation is that the conditional belief on φ is given by theconditional on φ of the LPS λ(φ), λ(φ)|φ = (µ

1, . . . µ(φ)|φ

). To deter-mine preference between acts conditional on φ, first calculate expectedutilities by means of the top level probability distribution, µ

1, and then,if necessary, use the lower level probability distributions, µ

2, . . . , µ(φ)|φ

,lexicographically to resolve ties. The function thus determines, for ev-ery event φ, the number of levels of the original LPS λ that can be used,provided that their supports intersect with φ, to resolve ties betweenacts conditional on φ.

Condition (i) ensures well-defined conditional beliefs that representnontrivial conditional preferences. Condition (ii) means that the system

of conditional preferences is dynamically consistent, in the sense thatstrict preference between two acts would always be maintained if newinformation, ruling out states at which the two acts lead to the sameoutcomes, became available. To motivate condition (iii), note that if e ∈ supp µ and ({e}) < , then it follows from condition (ii) that µ

could as well ignore e without changing the conditional beliefs.

Proposition 5 (Asheim and Perea, 2004) The following two state-ments are equivalent.

1 (a) φ satisfies Axioms 1, 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences

{φ |φ ∈ Φ} satisfies Axioms 5, 6

, and 16.2 There exist a vNM utility function υ : ∆(Z ) → R and an SCLP (λ, )

on F with support κ that satisfies, for any φ ∈ Φ,


µ(e)υ(pφ(e))

(φ)|φ

=1≥L

e∈φ

µ(e)υ(qφ(e))

(φ)|φ

=1,




where λ(φ)|φ = (µ1, . . . µ

(φ)|φ) is the conditional of λ(φ) on φ.

Proof. 1 implies 2. Since φ is trivial if φ /∈ Φ, we may w.l.o.g. as-sume that Axiom 16 is satisfied with ∗

F |φ being trivial for any φ /∈ Φ.

Consider any e ∈ κ. Since {e} satisfies Axioms 1, 2, 3, and 4

(implying Axiom 4 since {e} has only one state), it follows from thevNM theorem of expected utility representation that there exists a vNMutility function υ{e} : ∆(Z ) → R such that υ{e} represents {e}. ByAxiom 5, we may choose a common vNM utility function υ to represent{e} for all e ∈ κ. Since Axiom 16 implies, for any e ∈ κ, ∗

F |{e}

satisfies Axioms 1, 2, 3, and 4, and furthermore, p ∗

F |{e}

q wheneverp{e} {e} q{e}, we obtain that υ represents ∗

F |{e} for all e ∈ κ. It now

follows that ∗F satisfies Axiom 5 of Blume et al. (1991a).

By Theorem 3.1 of Blume et al. (1991a) ∗F is represented by υ and

an LPS λ = (µ1, . . . , µL) on F with support κ. Consider any φ ∈ Φ. If pφ φ qφ iff p ∗

F |φ q, then


µ(e)υ(pφ(e))

L|φ

=1≥L

e∈φ

µ(e)υ(qφ(e))

L|φ

=1,

where λ|φ = (µ1, . . . µ

L|φ) is the conditional of λ on φ, implying that we

can set (φ) = L. Otherwise, let (φ) ∈ {0, . . . , L − 1} be the maximum for which it holds that

pφ φ qφ if e∈φ

µk(e)υ(pφ(e))

|φ

k=1>L

e∈φ

µk(e)υ(qφ(e))

|φ

k=1,

where the r.h.s. is never satisfied if < min{k |supp λk ∩φ = ∅}, entailingthat the implication holds for any such . Define a set of pairs of actson φ, I , as follows:

(pφ, qφ) ∈ I iff e∈φ

µ(e)υ(pφ(e))

(φ)|φ

=1=L

e∈φ

µ(e)υ(qφ(e))

(φ)|φ

=1,

with (pφ, qφ) ∈ I for any acts pφ and qφ on φ if (φ) < min{ |supp λ

∩φ = ∅}. Note that I is a convex set. To show that υ and λ(φ)|φ

represent φ, we must establish that pφ ∼φ qφ whenever (pφ, qφ) ∈ I .Hence, suppose there exists (pφ, qφ) ∈ I such that pφ φ qφ. It follows




from the definition of (φ) and the completeness of φ (Axiom 1) thatthere exists (p

φ, qφ) ∈ I such that

pφ φ q

φ ande∈φ

µ(φ)+1(e)υ(pφ(e)) <

e∈φ

µ(φ)+1(e)υ(qφ(e)) .

Objective independence of φ (Axiom 2) now implies that, if 0 < γ < 1,then

γ pφ + (1 − γ )pφ φ γ qφ + (1 − γ )p

φ φ γ qφ + (1 − γ )qφ ;

hence, by transitivity of φ (Axiom 1),

γ pφ + (1 − γ )pφ φ γ qφ + (1 − γ )q

φ . (3.1)

However, by choosing γ sufficiently small, we have thate∈φ

µ(φ)+1(e)υ(γ pφ(e) + (1 − γ )pφ(e))

<

e∈φµ(φ)+1(e)υ(γ qφ(e) + (1 − γ )q

φ(e)) .

Since I is convex so that (γ pφ + (1 − γ )pφ, γ qφ + (1 − γ )q

φ) ∈ I , thisimplies that

γ p + (1 − γ )p

≺∗

F |φ γ q + (1 − γ )q

. (3.2)Since (3.1) and (3.2) contradict Axiom 16, this shows that pφ ∼φ qφ

whenever (pφ, qφ) ∈ I . This implies in turn that (φ) ≥ min{ |supp λ

∩φ = ∅} since φ is nontrivial. By Axiom 6, (ε) ≥ (φ) whenever∅ = ε ⊆ φ. Finally, since, υ represents {e} for all e ∈ κ, it follows thatp{e} {e} q{e} iff p ∗

F |{e} q. Hence, we can set ({e}) = L, implying

({e}) ≥ whenever e ∈ supp µ.2 implies 1. This follows from routine arguments.

By strengthening Axiom 4 to Axiom 4, we get the following corol-lary. For the statement of this result, we need to introduce formally the

concept of a conditional probability system. A conditional probability system (CPS) consists of a collection of subjective probability distribu-tions: If, for each φ ∈ Φ, µφ is a subjective probability distribution onφ, and {µφ| φ ∈ Φ} satisfies µ(δ ) · µφ() = µφ(δ ) whenever δ ⊆ ⊆ φand , φ ∈ Φ, then {µφ| φ ∈ Φ} is a CPS on F with support κ.




Corollary 1 The following three statements are equivalent.

1 (a) φ satisfies Axioms 1, 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {φ | φ ∈ Φ} satisfies Axioms 5, 6 , and 16.

2 There exist a vNM utility function υ : ∆(Z ) → R and a unique LCPS λ = (µ1, . . . , µL) on F with support κ that satisfies, for any φ ∈ Φ,

pφ φ qφ iff

e∈φµφ(e)υ(pφ(e)) ≥

e∈φ

µφ(e)υ(qφ(e)) ,

where µφ

is the conditional of µ(φ)

on φ and (φ) = min{| suppλ

∩φ = ∅}.

3 There exist a vNM utility function υ : ∆(Z ) → R and a unique CPS {µφ| φ ∈ Φ} on F with support κ that satisfies, for any φ ∈ Φ,

pφ φ qφ iff

e∈φµφ(e)υ(pφ(e)) ≥

e∈φ

µφ(e)υ(qφ(e)) .

Proof. 1 implies 2. By Proposition 5, the system of conditionalpreferences is represented by an SCLP (λ, ) on F with support κ. Bythe strengthening Axiom 4 to Axiom 4, it follows from the represen-tation result of Anscombe and Aumann (1963) that only the top levelprobability distribution is needed to represent each conditional prefer-ences; i.e., for any φ ∈ Φ, (φ) = min{| suppλ ∩ φ = ∅}. This impliesthat any overlapping supports in λ can be removed without changing,for any φ ∈ Φ, the conditional of λ(φ) on φ, turning λ into an LCPS.Furthermore, the LCPS thus determined is unique.

2 implies 1. This follows from routine arguments.2 implies 3. {µφ| φ ∈ Φ} is a CPS on F with support κ since µ(δ ) ·

µφ() = µφ(δ ) is satisfied whenever δ ⊆ ⊆ φ and , φ ∈ Φ. If analternative CPS {µφ| φ ∈ Φ} were to satisfy, for any φ ∈ Φ,

pφ φ qφ iff e∈φµφ(e)υ(pφ(e)) ≥ e∈φ

µφ(e)υ(qφ(e)) ,

then one could construct an alternative LCPS λ = (µ1, . . . , µL) suchthat, for any φ ∈ Φ, µφ is the conditional of µ(φ) on φ, where (φ) :=min{| suppµ ∩ φ = ∅}, contradicting the uniqueness of λ.

3 implies 2. Construct the LCPS λ = (µ1, . . . , µL) by the followingalgorithm: (i) µ1 = µF , (ii) ∀ ∈ {2, . . . , L}, µ = µφ, where φ =F \∪−1

k=1suppµk = F \κ, and (iii) ∪Lk=1suppµk = κ. Then, for any φ ∈ Φa,




µφ is the conditional of µ(φ) on φ, where (φ) := min{| suppµ ∩φ = ∅},and λ is the only LCPS having this property.

A full support SCLP (i.e., an SCLP where κ = F ) combines thestructural implication of a full support LPS—namely that conditionalpreferences are nontrivial—with flexibility w.r.t. whether to assume thebehavioral implication of any conditional of such an LPS—namely thatthe conditional LPS’s full support induces preference for cautious be-havior. A full support SCLP is a generalization of both

(1) conditional beliefs described by a single full support LPS λ = (µ1, . . . ,µL) (cf. Proposition 2): Let, for all φ ∈ Φ, (φ) = L. Then theconditional belief on φ is described by the conditional of λ on φ, λ|φ.

(2) conditional beliefs described by a CPS (cf. Corollary 1): Let, forall φ ∈ Φ, (φ) = min{| suppλ ∩ φ = ∅}. Then, it follows fromconditions (ii) and (iii) of Definition 1 that the full support LPSλ = (µ1, . . . , µL) has non-overlapping supports—i.e., λ is an LCPS—and the conditional belief on φ is described by the top level probabilitydistribution of the conditional of λ on φ. This corresponds to theisomorphism between CPS and LCPS noted by Blume et al. (1991a)on p. 72 and discussed by Hammond (1994) and Halpern (2003).

However, a full support SCLP may describe a system of conditionalbeliefs that is not covered by these special cases. The following is a simpleexample: Let κ = F = {d,e,f } and λ = (µ1, µ2), where µ1(d) = 1/2,µ1(e) = 1/2, and µ2(f ) = 1. If (F ) = 1 and (φ) = 2 for any othernon-empty subset φ, then the resulting SCLP falls outside cases (1) and(2).



Chapter 4

BELIEF OPERATORS

Belief operators play an important role in epistemic analyses of games.For any event, a belief operator determines the set of states where thisevent is (in some precise sense) believed. Belief operators may satisfydifferent kinds of properties, like

if one event implies another, then belief of the former implies belief of the latter (monotonicity ),

if two events are believed, then the conjunction is also believed,

an event that is always true is always believed,

an event that is never true is never believed,

if an event is believed, then the event that the event is believed isalso believed (positive introspection ), and

if an event is not believed, then the event that the event is not believedis believed (negative introspection ).

Belief operators satisfying this list are called KD45 operators.1

In epistemic analyses of games, it is common to derive belief operatorsfrom preferences, leading to what can be called subjective belief opera-

tors . Examples of subjective KD45 operators are ‘belief with probabil-ity one’, as used by, e.g., Tan and Werlang (1988), ‘belief with primaryprobability one’, as used by Brandenburger (1992), and ‘conditional be-lief with probability one’, as used by Ben-Porath (1997). More recently,

1A KD45 operator satisfies that belief of an event implies that the complement is not believed,but need not satisfy the truth axiom—i.e. that a believed event is always true.




Brandenburger and Keisler (2002), Battigalli and Siniscalchi (2002) andAsheim and Dufwenberg (2003a) have proposed non-monotonic subjec-tive belief operators called ‘assumption’, ‘strong belief’ and ‘full belief’,respectively. With the exception of Asheim and Dufwenberg’s (2003a)‘full belief’, these operators have in common that they are based onsubjective probabilities—arising from a probability distribution, a lexi-cographic probability system, or a conditional probability system—thatrepresent the preferences of the player as a decision maker.

An alternative approach to belief operators, applied by e.g. Stalnaker(1996, 1998), is to define belief operators by means of accessibility re-lations, as used in modal logic. Of particular interest is Stalnaker’s

non-monotonic ‘absolutely robust belief’ operator.Reproducing joint work with Ylva Søvik—Asheim and Søvik (2004)—

this chapter integrates these two approaches by showing how accessibilityrelations can be derived from preferences and in turn be used to defineand characterize belief operators; see Figure 4.1 for an illustration of thebasic structure of the analysis in this chapter. These belief operatorswill in later chapters be used in the epistemic analysis.

Morris (1997) observes that it is unnecessary to go via subjectiveprobabilities to derive subjective belief operators from the preferencesof a decision maker. This suggestion has been followed in Asheim (2002)and Asheim and Dufwenberg (2003a), the content of which will be re-

produced in Chapters 7 and 11. Epistemic conditions for backwardinduction are provided in Chapter 7 without the use of subjective prob-abilities (since one can argue that subjective probabilities play no rolein the backward induction argument), while Chapter 11 promotes for-ward induction within a structure based on incomplete preferences thatcannot be represented by subjective probabilities.

When deriving belief operators from preferences, it is essential thatthe preferences determine ‘subjective possibility’ (so that it can be de-termined whether an event is subjectively impossible) as well as ‘epis-temic priority’ (so that one allows for non-trivial belief revision). Aswe shall see, preferences need not satisfy completeness in order to deter-

mine ‘subjective possibility’ and ‘epistemic priority’. This chapter showshow belief operators corresponding to those used in the literature canbe derived from preferences that need not be complete.

We assume that preferences satisfy Axioms 1, 11, 2, 3, 4, 5, and6, entailing that preferences are (possibly) incomplete, but allow con-ditional representation (cf. Proposition 4 of Chapter 3). Following thestructure illustrated in Figure 4.1, Section 4.1 shows how a binary acces-



Belief operators 39

Preferences over acts (functions from states to

randomized outcomes)

‘Infinitely more likely’ ‘Admissibility’↓ ↓

Q Accessibility relation (R1, . . . , RL) Vector of nestedof epistemic priority accessibility relations

Belief operators certain belief conditional belief robust belief

defines characterizes

Figure 4.1. The basic structure of the analysis in Chapter 4.

sibility relation of epistemic priority Q can be derived from preferencessatisfying these axioms, by means of the ‘infinitely-more-likely’ relation.The properties of this priority relation are similar to but more generalthan those found, e.g., in Lamerre and Shoham (1994) and Stalnaker(1996, 1998) in that reflexivity of Q is not required.2 Furthermore, itis shown how preferences through “admissibility” give rise to a vector

of nested binary accessibility relations (R1, . . . , RL), where, for each ,R fulfills the usual properties of Kripke representations of beliefs; i.e.,they are serial, transitive and Euclidean. Finally, we establish that thetwo kinds of accessibility relations yield two equivalent representationsof ‘subjective possibility’ and ‘epistemic priority’.

In Section 4.2 we first use the accessibility relation of epistemic priorityQ to define the following belief operators:

Certain belief coincides with what Morris (1997) calls ‘Savage-belief’and means that the complement of the event is subjectively impossi-ble.

Conditional belief generalizes ‘conditional belief with probability one’.

2The term ‘epistemic priority’ will here be used to refer to what elsewhere is sometimesreferred to as ‘plausibility’ or ‘prejudice’; see, e.g., Friedman and Halpern (1995) and Lamerreand Shoham (1994). This is similar to ‘preference’ among states (or worlds) in nonmonotonic

logic—cf. Shoham (1988)—leading agents towards some states and away from others. Incontrast, we use the term ‘preferences’ in the decision-theoretic sense of a binary relation onthe set of functions (‘acts’) from states to outcomes.




Robust belief coincides with what Stalnaker (1998) calls ‘absolutelyrobust belief’.

We then show how these operators can be characterized by means of the vector of nested binary accessibility relations (R1, . . . , RL), therebyshowing that the concept of ‘full belief’ as used by Asheim and Dufwen-berg (2003a) coincides with robust belief.

Section 4.3 establishes properties of these belief operators. In partic-ular, the robust belief operator (while poorly behaved) is bounded bycertain and conditional belief, which are KD45 operators.

Section 4.4 shows how the characterization of robust belief corre-sponds to the concept of ‘assumption’ as used by Brandenburger andKeisler (2002), and observes how the definition of robust belief is relatedto the concept of ‘strong belief’ as used by Battigalli and Siniscalchi(2002). We thereby reconcile and compare these non-standard notionsof belief which have recently been used in epistemic analyses of games.

The proofs of the results in this chapter are included in Appendix A.

4.1 From preferences to accessibility relations

The purpose of this section is to show how two different kinds of acces-sibility relations—see, e.g., Lamerre and Shoham (1994) and Stalnaker(1996, 1998)—can be derived from preferences.

Consider the decision-theoretic framework of Chapter 3. However, asmotivated below, assume that the decision maker’s preferences may varybetween states. Hence, denote by d

φ the preferences over acts on φ atstate d, and use superscript d throughout in a similar manner.

Assume that, for each d ∈ F , (a) dφ satisfies Axioms 1, 2, and 4 if

φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φd (recalling from Chapter3 that Φd denotes {φ ∈ 2F \{∅}| κd ∩ φ = ∅}), and (b) the system of conditional preferences {d

φ |φ ∈ Φd} satisfies Axioms 5, 6, and 11. Inview of Axiom 6 we simplify notation and write

p d

φ

q instead of p d

F |φ

q ⇔ pφ d

φ

qφ ,

and simplify further by substituting d for dF . By Proposition 4, d

is conditionally represented: There exist a vNM utility function υd :∆(Z ) → R such that p d

{e} q iff υ d(p(e)) ≥ υd(q(e)) whenever e ∈ κd.

If E ⊆ F , say that pE weakly dominates qE at d if, ∀e ∈ E , υd(pE (e))≥ υd(qE (e)), with strict inequality for some f ∈ E . Say that d is



Belief operators 41

admissible on E if E is non-empty and p d q whenever pE weaklydominates qE at d. The following connection between admissibility onsubsets and the infinitely-more-likely relation is important for relatingthe two kinds of accessibility relations derived from preferences below;the one kind is based on the infinitely-more-likely relation, while theother is based on admissibility on subsets. Write ¬E for F \E .

Proposition 6 Let E = ∅ and ¬E = ∅. d is admissible on E iff e ∈ E and f ∈ ¬E imply e d f .

An epistemic model. In a semantic formulation of belief operatorsone can, following Aumann (1999), start with an information partitionof F , and then assume that the decision maker, for each element of thepartition, is endowed with a probability distribution that is concentratedon this element of the partition. Since all states within one element of the partition are indistinguishable, they are assigned the same probabil-ity distribution, which however differ from the probability distributionsassigned to states outside this element. In particular, probability dis-tributions assigned to two states in different elements of the partitionhave disjoint supports. Hence, in Aumann’s (1999) formulation, the de-cision maker’s probability distribution depends on in which element of the information partition the true state is.

This is consistent with the approach chosen here, where the proba-bility distribution—or more generally, the preferences—of the decisionmaker will be different for states in different elements of the informationpartition, and be the same for all states within the same element. How-ever, in line with our subjective perspective, we will construct the infor-mation partition from the preferences of the decision maker, so that eachelement of the partition is defined as a maximal set of states where thedecision maker’s preferences are the same, having the interpretation thatstates within this set are indistinguishable. Moreover, Aumann’s (1999)assumption that the probability distribution is concentrated within thecorresponding element of the partition will in our framework be captured

by the property that all states outside (and possibly some states inside)the element are deemed subjectively impossible.

Thus, for each d ∈ F , let τ d := {e ∈ F | p e q iff p d q} be the setof states that are subjectively distinguishable , and write d ≈ e if e ∈ τ d.Note that ≈ is a reflexive, transitive, and symmetric binary relation; i.e.,≈ is an equivalence relation that partitions F into equivalence classes(or “types”).




Moreover, κd denotes the set of states that are subjectively possible (i.e., not Savage-null) at d. In line with the above discussion, assumethat, for each d ∈ F , κd ⊆ τ d. This assumption will ensure that thepreference-based operators satisfy positive and negative introspection;it corresponds to “being aware of one’s own type”.

Refer to the collection {d | d ∈ F } as an epistemic model for thedecision maker.

In view of Axiom 6, it holds that p dφ q ⇔ p d q whenever κd ⊆

φ ⊂ F ; in particular, p dτ d

q ⇔ p d q. The interpretation is thedecision maker’s preferences at d are not changed by ruling out statesthat he can distinguish from the true state at d. Hence, we can adopt

an interim perspective where the decision maker has already becomeaware of his own preferences (his own “type”); in particular, the decisionmaker’s unconditional preferences are not obtained by conditioning “exante preferences” on his type.

Accessibility relation of epistemic priority. Consider the follow-ing definition of the accessibility relation Q.

Definition 2 dQe (“d does not have higher epistemic priority thane”) if (1) d ≈ e, (2) e is not Savage-null at d, and (3) d is not deemedinfinitely more likely than e at d.

Proposition 7 The relation Q is serial,3

transitive, and satisfies for-ward linearity 4 and quasi-backward linearity.5

A vector of nested accessibility relations. Consider the collec-tion of all sets E satisfying that d is admissible on E . Since d isadmissible on κd, it follows that the collection is non-empty as it is con-tains κd. Also, since no e ∈ E is Savage-null at d if d is admissible onE , it follows that any set in this collection is a subset of κd. Finally,since e d f implies that f d e does not hold, it follows from Proposi-tion 6 that E ⊆ E or E ⊆ E if d is admissible on both E and E ,implying that the sets in the collection are nested. Hence, there exists avector of nested sets, (ρd

1, . . . , ρdLd), on which d is admissible, satisfying:

∅ = ρd1 ⊂ · · · ⊂ ρd

⊂ · · · ⊂ ρdLd = κd ⊆ τ d

(where ⊂ denotes ⊆ and =).

3∀d, ∃e such that dQe.4dQe and dQf imply eQf or f Qe.5If ∃d ∈ F such that dQe, then dQf and eQf imply dQe or eQd.



Belief operators 43

If we assume that d satisfies not only Axiom 1 but also Axiom 1, sothat, as reported in Proposition 2, d is represented by υd and an LPS,λd = (µd

1, . . . , µdLd)—i.e., a sequence of Ld levels of subjective probability

distributions—then (ρd1, . . . , ρd

Ld) can in an obvious way be derived fromthe supports of these probability distributions:

∀ ∈ {1, . . . , Ld}, ρd =

k=1suppµd

k .

McLennan (1989a) develops an ordering of κd that is related to (ρd1,

. . . , ρdLd) in a context where a system of conditional probabilities is taken

as primitive. In a similar context, van Fraassen (1976) and Arlo-Costa

and Parith (2003) propose a concept of (belief/probability) cores thatcorrespond to the sets ρd

1, . . . , ρdLd . Grove (1988) spheres and Spohn’s

(1988) ordinal conditional functions are also related to these sets.For d ∈ F with Ld < L := maxe∈F L

e, let ρdLd = ρd

= κd for ∈ {Ld +1, . . . , L}. The collection of sets, {ρd

| d ∈ F }, defines an accessibilityrelation, R.

Definition 3 dRe (“at d, e is deemed possible at the epistemic level”) if e ∈ ρd

.

Proposition 8 The vector of relations, (R1, . . . , RL), has the follow-ing properties: For each ∈ {1, . . . , L}, R is serial, transitive, and Euclidean.6 For each ∈ {1, . . . , L − 1}, (i) dRe implies dR+1e and (ii) ( ∃f such that dR+1f and eR+1f ) implies ( ∃f such that dRf and eRf ).

The correspondence between Q and (R1, . . . , RL). That d isnot Savage-null at d can be interpreted as d being deemed subjectivelypossible (at some epistemic level) at any state in the same equivalenceclass. By part (i) of the following result, d being not Savage-null atd has two equivalent representations in terms of accessibility relations:dQd and dRLd. Likewise, e d d can be interpreted as e having higherepistemic priority than d. By part (ii) of the following result, e d

d have two equivalent representations: (dQe and not eQd) and (∃ ∈{1, . . . , L} such that dRe and not eRd). Thus, both Q and (R1, . . . , RL)capture ‘subjective possibility’ and ‘epistemic priority’ as implied by thepreferences of the preference system.

6dRe and dRf imply eRf .




Proposition 9 (i) dQd iff dRLd. (ii) ( dQe and not eQd) iff ( ∃ ∈{1, . . . , L} such that dRe and not eRd).

If Axiom 4 is substituted for Axiom 4—so that the conditional Archi-medean property is strengthened to the Archimedean property—then ebeing deemed infinitely more likely than f at d implies that f is Savage-null. Hence, L = 1, and by Definitions 2 and 3, Q = R1. Hence, we areleft with a unique serial, transitive, and Euclidean accessibility relationif preferences are continuous.

4.2 Defining and characterizing belief operators

In line with the basic structure illustrated in Figure 4.1, we now usethe accessibility relations of Section 4.1 to define and characterize belief operators.

Defining certain, conditional, and robust belief. Consider theaccessibility relation of epistemic priority, Q, having the properties of Proposition 7. In Asheim and Søvik (2003) we show how equivalenceclasses can be derived from Q with the properties of Proposition 7, im-plying that Q with such properties suffices for defining the belief oper-ators. In particular, we show that the set of states that are subjectiveindistinguishable at d is given by

τ d = {e ∈ F | ∃f ∈ F such that dQf and eQf } ,

and the set of states that are deemed subjectively possible at d equals

κd = {e ∈ τ d| ∃f ∈ F such that f Qe} = {e ∈ τ d| eQe} ,

where κd = ∅ since Q is serial, and where the last equality follows since,by quasi-backward linearity, eQe if f Qe.

Define ‘certain belief’ as follows.

Definition 4 At d the decision maker certainly believes E if d ∈ KE ,where KE := {e ∈ F |κe ⊆ E }.

Hence, at d an event E is certainly believed if the complement is deemedsubjectively impossible at d. This coincides with what Morris (1997)calls ‘Savage-belief’.

‘Conditional belief’ is defined conditionally on sets that are subjec-tively possible at any state; i.e., sets in the following collection:

Φ :=

d∈F Φd , where ∀d ∈ F, Φd = {φ ∈ 2F \{∅}| κd ∩ φ = ∅} .



Belief operators 45

Hence, a non-empty set φ is not in Φ if and only if there exists d ∈ F such that κd ∩ φ = ∅. Note that F ∈ Φ and, ∀φ ∈ Φ, ∅ = φ ⊆ F .

Since every φ ∈ Φ is subjectively possible at any state, it follows that,∀φ ∈ Φ,

β d(φ) := {e ∈ τ d ∩ φ|∀f ∈ τ d ∩ φ,fQe}

is non-empty, as demonstrated by the following lemma.

Lemma 3 If κd ∩ φ = ∅, then ∃e ∈ τ d ∩ φ such that ∀f ∈ τ d ∩ φ, fQe.

Define ‘conditional belief’ as follows.

Definition 5 At d the decision maker believes E conditional on φ if

d ∈ B(φ)E , where B(φ)E := {e ∈ F |β e(φ) ⊆ E }.

Hence, at d an event E is believed conditional on φ if E contains anystate in τ d ∩ φ with at least as high epistemic priority as any other statein τ d ∩ φ. This way of defining conditional belief is in the tradition of,e.g., Grove (1988), Boutilier (1994), and Lamerre and Shoham (1994).

Let ΦE be the collection of subjectively possible events φ having theproperty that E is subjectively possible conditional on φ whenever E issubjectively possible:

ΦE :=

d∈F Φd

E , where ∀d ∈ F,

Φ

d

E := {φ ∈ Φ

d

| E ∩ κ

d

∩ φ = ∅ if E ∩ κ

d

= ∅ } .Hence, a non-empty set φ is not in ΦE if and only if (1) there exists d ∈ F such that κd ∩ φ = ∅ or (2) there exists d ∈ F such that E ∩ κd = ∅ andE ∩ κd ∩ φ = ∅. Note that ΦE is a subset of Φ that satisfies F ∈ ΦE ;hence, ∅ = ΦE ⊆ Φ.

Define ‘robust belief’ as follows.

Definition 6 At d the decision maker robustly believes E if d ∈ B0E ,where B0E := ∩φ∈ΦE

B(φ)E .

Hence, at d an event E is robustly believed in the following sense: E isbelieved conditional on any event φ that does not make E subjectively

impossible. Indeed, B0 coincides with what Stalnaker (1998) calls ‘ab-solutely robust belief’ when we specialize to his setting where Q is alsoreflexive. The relation between this belief operator and the operators‘full belief’, ‘assumption’, and ‘strong belief’, introduced by Asheim andDufwenberg (2003a), Brandenburger and Keisler (2002), and Battigalliand Siniscalchi (2002), respectively, will be discussed at the end of thissection as well as in Section 4.4.




Characterizing certain, conditional, and robust belief. Con-sider the vector of nested accessibility relations (R1, . . . , RL) having theproperties of Proposition 8 and being related to Q as in Proposition 9. InAsheim and Søvik (2003) we first derive (R1, . . . , RL) from Q and thenshow how (R1, . . . , RL) characterizes the belief operators. In particular,it holds for any ∈ {1, . . . , L} that

τ d = {e ∈ F | ∃f ∈ F such that dRf and eRf } ,

and

ρd = {e ∈ F | dRe} .

Furthermore,

κd = {e ∈ τ d| eRLe} = {e ∈ F | dRLe} .

The latter observations yield a characterization of certain belief.

Proposition 10 KE = {d ∈ F | ρdL ⊆ E }.

Proposition 10 entails that certain belief as defined in Definition 4 cor-responds to what Arlo-Costa and Parith (2003) call ‘full belief’.

Furthermore, by the next result, (unconditional) belief, B(F ), corre-sponds to what van Fraassen (1995) calls ‘full belief’.

Proposition 11 ∀φ ∈ Φ, B(φ)E = {d ∈ F | ∃ ∈ {1, . . . , L} such that ∅ = ρd

∩ φ ⊆ E }.

Finally, by Proposition 9(ii) and the following result, E is robustlybelieved iff any subjectively possible state in E has higher epistemicpriority than any state in the same equivalence class outside E .

Proposition 12 B0E = {d ∈ F | ∃ ∈ {1, . . . , L} such that ρd = E ∩

κd}.

Asheim and Dufwenberg (2003a) say that an event A is ‘fully believed’at a if the preferences at a are admissible on the set of states in A that

are deemed subjectively possible at a. It follows from Proposition 12that this coincides with robust belief as defined in Definition 6.

4.3 Properties of belief operators

The present section presents some properties of certain, conditional,and robust belief operators. We do not seek to establish sound andcomplete axiomatic systems for these operators; this should, however, be






2 If β d ∩ φ = ∅, then β d(φ) = β d ∩ φ.

3 If φ ∈ Φ, then β d(φ) = ∅.

4 If β d(φ) ∩ φ = ∅, then β d(φ ∩ φ) = β d(φ) ∩ φ.

Properties of robust belief. It is easy to show that certain belief implies robust belief, which in turn implies (unconditional) belief.

Proposition 16 KE ⊆ B0E ⊆ B(F )E .

Even though robust belief is thus bounded by two KD45 operators,robust belief is not itself a KD45 operator.

Proposition 17 The following properties hold:

B0E ∩ B0E ⊆ B0(E ∩ E )

B0E ⊆ KB0E

¬B0E ⊆ K(¬B0E ).

Note that B0∅ = ∅, B0F = F , B0E ⊆ B0B0E and ¬B0E ⊆ B0(¬B0E )follow from Propositions 14 and 17 since KE ⊆ B0E ⊆ B(F )E . However,even though the operator B 0 satisfies B 0E ⊆ ¬B0¬E as well as positiveand negative introspection, it does not satisfy monotonicity since E ⊆ E

does not imply B0E ⊆ B0E . To see this let ρd1 = {d} and ρd

2 = κd ={d,e,f } for some d ∈ F . Now let E = {d} and E = {d, e}. Clearly,E ⊆ E , and since ρd

1 = E ∩ κd we have d ∈ B0E . However, since neitherρd

1 = E ∩ κd nor ρd2 = E ∩ κd, d /∈ B0E .

4.4 Relation to other non-monotonic operators

The purpose of this section is to show how robust belief correspondsto the ‘assumption’ operator of Brandenburger and Keisler (2002) and isrelated to the ‘strong belief’ operator of Battigalli and Siniscalchi (2002).

The ‘assumption’ operator. Brandenburger and Keisler (2002)consider an epistemic model which

is more general than the one that we consider in Section 4.1, sincethe set of states need not be finite, and

is more special than ours, since, for all d ∈ F , Axioms 1, 11, and4 are strengthened to Axioms 1 and 4, so that completeness andthe partitional Archimedean property are substituted for conditionalcompleteness, partitional priority, and the conditional Archimedeanproperty.



Belief operators 49

Within our setting with a finite set of states, F , it now follows, asreported in Proposition 3, that d is represented by υd and an LCPS,λd = (µd

1, . . . , µdLd)—i.e., a sequence of Ld levels of non-overlapping sub-

jective probability distributions. Hence, ∀ ∈ {1, . . . , Ld}, suppµd = π d

,where (πd

1, . . . , πdLd) is a partition of κd. In their Appendix B, Branden-

burger and Keisler (2002) employ an LCPS to represent preferences intheir setting with an infinite set of states.

Provided that completeness and the partitional Archimedean prop-erty are satisfied, Brandenburger and Keisler (2002) introduce the fol-lowing belief operator in their Definition B1; see also Brandenburger andFriedenberg (2003).

Definition 7 (Brandenburger and Keisler, 2002) At d the deci-sion maker assumes E if d

E is nontrivial and p dE q implies p d q.

Proposition 18 Assume that d satisfies Axioms 1 and 4 (in addi-tion to the assumptions made in Section 4.1). Then E is assumed at diff d ∈ B 0E .

Proposition 18 shows that the ‘assumption’ operator coincides withrobust belief (and thus with Stalnaker’s ‘absolutely robust belief’) undercompleteness and the partitional Archimedean property.

However, if the partitional Archimedean property is weakened to the

conditional Archimedean property, then this equivalence is not obtained.To see this, let κd = {d,e,f }, and let the preferences d, in additionto the properties listed in Section 4.1, also satisfy completeness. It thenfollows from Proposition 2 that a is represented by υa and a LPS—i.e.,a sequence of subjective probability distributions with possibly overlap-ping supports. Consider the example provided by Blume et al. (1991a) intheir Section 5 of a two-level LPS, where the primary probability distri-bution, µd

1, is given by µd1(d) = 1/2 and µd

1(e) = 1/2, and the secondaryprobability distribution, µd

2, used to resolve ties, is given by µd2(d) = 1/2

and µd2(f ) = 1/2. Consider the acts p and q, where υd(p(d)) = 2,

υd(p(e)) = 0, and υd(p(f )) = 0, and where υd(q(d)) = 1, υ d(q(e)) = 1,

and υd

(q(f )) = 2. Even though d

is admissible on {d, e}, and thus{d, e} is robustly believed at d, it follows that {d, e} is not ‘assumed’ atd since

pd{d,e}q while p≺dq .

Brandenburger and Keisler (2002) do not indicate that their definition—as stated in Definition 7—should be used outside the realm of pref-erences that satisfy the partitional Archimedean property. Hence, our




definition of robust belief—combined with the characterization result of Proposition 12 and its interpretation in term of admissibility—yields apreference-based generalization of Brandenburger and Keisler (2002) op-erator (in our setting with a finite set of states) to preferences that needonly satisfy the properties of Section 4.1.

The ‘strong belief ’ operator. In the setting of extensive formgames, Battigalli and Siniscalchi (2002) have suggested a non-monotonic‘strong belief’ operator. We now show how their ‘strong belief’ operatoris related to robust belief, and thereby, to ‘absolutely robust belief’ of Stalnaker (1998), ‘full belief’ of Asheim and Dufwenberg (2003a), and

‘assumption’ of Brandenburger and Keisler (2002).

Battigalli and Siniscalchi (2002) base their ‘strong belief’ operator ona conditional belief operator derived from an epistemic model where,at each state d ∈ F , the decision maker is endowed with a system of conditional preferences {d

φ| φ ∈ Φd} (with, as before, Φd denoting {φ ∈2F \{∅}| κd ∩ φ = ∅}). However, Battigalli and Siniscalchi (2002) assumethat, if the true state is d, then the decision maker’s system of conditionalpreferences is represented by υd and a CPS {µd

φ| φ ∈ Φd}. Since a CPSdoes not satisfy conditionality as specified by Axiom 6, we must embedtheir conditional belief operator in the framework of the present chapter.We can do so using Corollary 1 of Chapter 3.

One the one hand, Battigalli and Siniscalchi (2002) and Ben-Porath(1997) define ‘conditional belief with probability one’ in the followingway: At d the decision maker believes E conditional on φ ∈ Φ if suppµd

φ

⊆ E , where {µdφ| φ ∈ Φd} is a CPS on F with support κd.

On the other hand, according to Definition 5 of the present chapter,at d the decision maker believes E conditional on φ ∈ Φ if β d(φ) ⊆ E .If, however, Axioms 1, 4, and 11 are strengthened to Axioms 1 and4, so that by Proposition 3 d is represented by υd and an LCPS,λd = (µd

1, . . . , µdLd), on F with support κd, then Lemma 14 of Appendix

A implies that β d(φ) = suppµd ∩ φ, where := min{k| suppµd

k ∩ φ = ∅}.

Hence, by Corollary 1, ‘conditional belief with probability one’ asdefined by Battigalli and Siniscalchi (2002) and Ben-Porath (1997) isisomorphic to the conditional belief operator B(φ) derived from an epis-temic model satisfying the assumptions of Section 4.1 of the presentchapter.

Given that the conditional belief operator of Battigalli and Siniscalchi(2002) thus coincides with the B(φ) operator of the present paper, we



Belief operators 51

can define their ‘strong belief’ operator as follows: Let ΦH (⊆ Φ) be somenon-empty subcollection of the collection of subsets that are subjectivelypossible at any state; e.g., in an extensive game ΦH may consist of thesubsets that correspond to subgames. Then ΦH ∩ ΦE is the collection of subsets φ satisfying φ ∈ ΦH and having the property that E is subjec-tively possible conditional on φ whenever E is subjectively possible.

Definition 8 (Battigalli and Siniscalchi, 2002) At d the deci-sion maker strongly believes E if d ∈

ΦH ∩ΦE

B(φ).

Hence, at d an event E is strongly believed if E is robustly believed in

the following sense: E is believed conditional on any subset φ in ΦH thatdoes not make E subjectively impossible. Since ΦE ⊇ ΦH ∩ ΦE ⊇ {F },it follows that the ‘strong belief’ operator is bounded by the robust belief and (unconditional) belief operators.

Proposition 19 If d ∈ B0(E ), then E is strongly believed at d. If E is strongly believed at d, then d ∈ B(F )E .

As suggested by Battigalli and Bonanno (1999), the ‘strong belief’operator may also be defined w.r.t. other subcollections of Φ than thecollection of subsets that correspond to subgames, and may be seen asa generalization of robust belief by not necessarily requiring belief to be

“absolutely robust” in the sense of Stalnaker (1998). However, providedthat F is included, Proposition 19 still holds.

In any case, the ‘strong belief’ operator shares the properties of robustbelief: Also ‘strong belief’ satisfies the properties of Proposition 17, butis not monotonic.





Chapter 5

BASIC CHARACTERIZATIONS

In this chapter we present characterizations of basic game-theoreticconcepts. After presenting the concept of an epistemic model of a strate-gic game form in Section 5.1, we turn to the characterizations of Nashequilibrium and rationalizability in Section 5.2 and characterizations of (strategic form) perfect equilibrium and permissibility in the final Sec-tion 5.3.

The characterizations of Nash equilibrium and rationalizability will

be done by means of the event that each player has preferences that areconsistent with the game and the preferences of the opponent. Like-wise, the characterizations of (strategic form) perfect equilibrium andpermissibility will be done by means of the event that each player haspreferences that are admissibly consistent with the game and the pref-erences of the opponent. Hence, the chapter illustrates the ‘consistentpreferences’ approach and set the stage for the analysis of subsequentchapters.

Note that the results of this chapter are variants of results that can befound in the literature. In particular, the characterizations of Nash equi-librium and (strategic form) perfect equilibrium are variants of Propo-

sitions 3 and 4 of Blume et al. (1991b).

5.1 Epistemic modeling of strategic games

The purpose of this section is to present a framework for strategicgames where each player is modeled as a decision maker under uncer-tainty. The analysis builds on the two previous chapters and introduces




the concept of an epistemic model for a strategic game form. In thischapter preferences are assumed to be complete, an assumption thatwill be relaxed in Chapter 6.

A strategic game form. Denote by S i player i’s finite set of pure strategies , and let z : S → Z map strategy profiles into outcomes, whereS = S 1 × S 2 is the set of strategy profiles and Z is the finite set of outcomes. Then (S 1, S 2, z) is a finite strategic two-player game form .

An epistemic model. For each player i, any of i’s strategies isan act from strategy choices of his opponent j to outcomes. The un-certainty faced by a player i in a strategic game form concerns (a) j’sstrategy choice, (b) j’s preferences over acts from i’s strategy choicesto outcomes, and so on (cf. the discussion in Section 1.3). A type of aplayer i corresponds to (a) preferences over acts from j’s strategy choices,(b) preferences over acts from j ’s preferences over acts from i’s strategychoices, and so on.

For any player i, i’s decision is to choose one of his own strategies. Asthe player is not uncertain of his own choice, the player’s preferences overacts from his own strategy choices is not relevant and can be ignored.

Hence, in line with the discussion in Section 1.3, consider an implicitmodel—with a finite type set T i for each player i—where the preferencesof a player corresponds to the player’s type, and where the preferences

of the player are over acts from the opponent’s strategy-type pairs tooutcomes.

If we let each player be aware of his own type (as we will assumethroughout), this leads to an epistemic model where the state space of player i is T i × S j × T j, and where, for each ti ∈ T i,

{ti} × S j × T j

constitutes an equivalence class, being the set of states that are indis-tinguishable for player i at ti, and a non-empty subset of {ti} × S j × T j,κti, is the set of states that player i deems subjectively possible at ti.

Definition 9 An epistemic model for the finite strategic two-playergame form (S 1, S 2, z) is a structure

(S 1, T 1, S 2, T 2) ,

where, for each type ti of any player i, ti corresponds to a system of conditional preferences on the collection of sets of acts from elements of

Φti := {φ ⊆ T i × S j × T j | κti ∩ φ = ∅}



Basic characterizations 55

to ∆(Z ), where κti is a non-empty subset of {ti} × S j × T j.

An implicit model with a finite set of types for each player, as consid-ered throughout this book, does not allow for ‘preference-completeness’,where, for each player i, there exists some type of i for any feasible pref-erences that i may have.1 Still, even a finite implicit model gives rise toinfinite hierarchies of preferences, and – in effect – we assume that eachplayer as a decision maker is able to represent his subjective hierarchyof preferences by means of a finite implicit model. Then, at the trueprofile of types, the two players’ subjective hierarchies can be embeddedin a single implicit model that includes the types of the two players that

are needed to represent each player’s hierarchy. Such a construction canfruitfully be used to analyze a wide range of game-theoretic concepts, aswill be demonstrated throughout this book.

However, when embedding the two player’s subjective hierarchies intoa single implicit model, it is illegitimate to require that player i deemsthe true type of his opponent j subjectively possible. Rather, we can-not rule out that, at the true type profile, player j’s true type is notneeded to represent player i’s subjective hierarchy of preferences; this isparticularly relevant for the analysis of non-equilibrium game-theoreticconcepts. Hence, when applying finite implicit models for interactiveanalysis of games, it is important to allow—as we do in the framework

of the present text—the decision maker to hold objectively possible op-ponent preferences as subjectively impossible.

Throughout this book we will consider two different kinds of epistemicmodels that differ according to the kind of assumption imposed on theset of conditional preferences that ti determines. For the present chapter,as well Chapters 8, 9, and 10, we will make the following assumption.

Assumption 1 For each ti of any player i, (a) tiφ satisfies Axioms 1,

2, and 4 if ∅ = φ ⊆ T i ×S j × T j, and Axiom 3 if and only if φ ∈ Φti, (b)the system of conditional preferences {ti

φ | φ ∈ Φti} satisfies Axioms 5,6 , and 16, and (c) there exists a non-empty subset of opponent types,T j

ti, such that κti = {ti} × S j × T jti.

1‘Preference-completeness’ is needed for the interactive epistemic analyses of, e.g., Branden-burger and Keisler (2002) and Battigalli and Siniscalchi (2002), but not for the analysispresented in this book. Brandenburger and Keisler (1999) show that there need not exist

a ‘preference-complete’ interactive epistemic model when preferences are not representableby subjective probabilities, implying that ‘preference-completeness’ may be inconsistent withthe analysis of Chapters 6, 7, 11, and 12, where Axiom 1 is not imposed.




In this assumption, T jti is the non-empty set of opponent types that

player i deems subjectively possible at ti. The assumption explicitlyallows for preferences over acts from subsets of T i × S j × T j, φ, whereprojS j

φ may be a strict subset of S j. This accommodates the analysisof extensive game concepts in Chapters 8 and 9 and will permit theconcepts in Tables 2.1 and 2.2 to be treated in a common framework.

Write ti for player i’s preferences conditional on being of type ti;i.e., for ti

φ when φ = {ti} × S j × T j. We will refer to ti as player i’sunconditional preferences at ti.

Under Assumption 1 it follows from Proposition 5 that, for each typeti of any player i, i’s system of conditional preferences at ti can be

represented by a vNM utility function υtii : ∆(Z ) → R and an SCLP

(λti , ti) on T i × S j × T j with support κti = {ti} × S j × T jti . Throughout,

we will adopt an interim perspective, where player i has already becomeaware of his own type. This entails that we can w.l.o.g. assume that,for any φ ∈ Φti , (φ) =

φ ∩ ({ti} × S j × T j)

. The interpretation is

player i’s preferences at ti are not changed by ruling out states that ican distinguish from the true state at ti. Consequently, for expositionalsimplicity we choose to let the SCLP (λti , ti) be defined on S j × T j withsupport S j × T j

ti

Preferences over strategies. It follows from the above assumptions

that, for each type ti of any player i, player i’s unconditional preferencesat ti, ti , are a complete and transitive binary relation on the set of actsfrom S j × T j to ∆(Z ) that is represented by a vNM utility function υti

i

and an LPS λti = (µti

1 , . . . , µti ), where = (S j × T j). Since each pure

strategy si ∈ S i is a function that assigns the deterministic outcomez(si, s j) to any (s j , t j ) ∈ S j × T j and is thus an act from S j × T j to∆(Z ), we have that ti determines complete and transitive preferenceson i’s set of pure strategies, S i.

Player i’s choice set at ti, S tii , is player i’s set of rational pure strate-gies at ti:

S tii := {si ∈ S i| ∀si ∈ S i, si ti s

i} .

Since ti is complete and transitive and satisfies objective independence,and S i is finite, it follows that the choice set S tii is non-empty, and thatthe set of rational mixed strategies equals ∆(S tii ).

A strategic game. Let, for each i, ui : S → R be a vNM utility func-tion that assigns payoff to any strategy profile. Then G = (S 1, S 2, u1, u2)




is a finite strategic two-player game . Assume that, for each i, there exists = (s1, s2), s = (s

1, s2) ∈ S such that ui(s) > ui(s). The event that i

plays the game G is given by

[ui] := {(t1, t2) ∈ T 1 ×T 2| υtii ◦z is a positive affine transformation of ui} ,

while [u] := [u1] ∩ [u2] is the event that both players play G.Denote by pi, q i ∈ ∆(S i) mixed strategies for player i, and let S j

(⊆ S j) be a non-empty set of opponent strategies. Say that pi strongly dominates q i on S j if, ∀s j ∈ S j, ui( pi, s j) > ui(q i, s j). Say that q i isstrongly dominated on S j if there exists pi ∈ ∆(S i) such that pi strongly

dominated q i on S

j . Say that pi weakly dominates q i on S

j if, ∀s j ∈ S

j,ui( pi, s j) ≥ ui(q i, s j) with strict inequality for some s

j ∈ S j. Say that q iis weakly dominated on S j if there exists pi ∈ ∆(S i) such that pi weaklydominated q i on S j.

The following two results will be helpful for some of the proofs.

Lemma 4 Let G = (S 1, S 2, u1, u2) be a finite strategic two-player game.For each i, pi ∈ ∆(S i) is strongly dominated on S j if and only if there does not exist µ ∈ ∆(S j) with suppµ ⊆ S j such that, ∀s

i ∈ S i,

sj∈S j

µ(s j)ui( pi, s j) ≥

sj∈S j

µ(s j)ui(si, s j) .

Proof. Lemma 3 of Pearce (1984).

Lemma 5 Let G = (S 1, S 2, u1, u2) be a finite strategic two-player game.For each i, pi ∈ ∆(S i) is weakly dominated on S j if and only if there does not exist µ ∈ ∆(S j) with suppµ = S j such that, ∀s

i ∈ S i,

sj∈S j

µ(s j)ui( pi, s j) ≥

sj∈S j

µ(s j)ui(si, s j) .

Proof. Lemma 4 of Pearce (1984).

Certain belief. In the present chapter, as well in Chapters 8, 9, and10, we will apply the certain belief operator (cf. Definition 4 of Chapter4) for events that are subsets of the set of type profiles, T 1 × T 2. InAssumption 1 we allow for the possibility that each player deems someopponent types subjectively impossible, corresponding to an SCLP thatdoes not have full support along the type dimension. Therefore, certainbelief (meaning that the complement is subjectively impossible) can be




derived from the epistemic model and defined for events that are subsetsof T 1 × T 2. For any E ⊆ T 1 × T 2, say that player i certainly believes theevent E at ti if ti ∈ projT i

KiE , where

KiE := {(t1, t2) ∈ T 1 × T 2| projT 1×T 2κti = {ti} × T j

ti ⊆ E } .

Say that there is mutual certain belief of E at (t1, t2) if (t1, t2) ∈ KE ,where KE := K1E ∩ K2E . Say that there is common certain belief of E at (t1, t2) if (t1, t2) ∈ CKE , where CKE := KE ∩ KKE ∩ KKKE ∩ . . . .

As established in Proposition 14, Ki corresponds to a KD45 system.Moreover, the mutual certain belief operator, K, has the following prop-erties, where we write K0E := E , and for each g ≥ 1, KgE := KKg−1E .

Proposition 20 (i) For any E ⊆ T 1×T 2 and all g > 1, KgE ⊆ Kg−1E .If E = E 1 ∩ E 2, where, for each i, E i = projT i

E i × T j, then KE ⊆ E .(ii) For any E ⊆ T 1 × T 2, there exists g ≥ 0 such that KgE = CKE

for g ≥ g , implying that CKE = KCKE .

Proof. Part (i). If E = E 1∩E 2, where, for each i, E i = projT iE i×T j,

then KE = K1E ∩ K2E ⊆ K1E 1 ∩ K2E 2 = E 1 ∩ E 2 = E , establishingthe second half of part (i).

Since, for any E ⊆ T 1 × T 2, KE = K1E ∩ K2E , where, for each i,KiE = projT i

KiE × T j, the first half of part (i) follows from the resultof the second half.

Part (ii) is a consequence of part (i) and T 1 × T 2 being finite.

5.2 Consistency of preferences

In the present section we define the event of consistency of preferencesand show how this event can be used to provide characterizations of mixed-strategy Nash equilibrium and mixed rationalizable strategies.

Inducing rational choice. In line with the discussion in Section 1.1,and following a tradition from Harsanyi (1973) to Blume et al. (1991b),a mixed strategy will interpreted, not as an object of choice, but as anexpression of the beliefs for the other player. Say that the mixed strategy

p j

ti|tj

is induced for t j by ti if t j ∈ T jti

and, for all s j ∈ S j,

p jti|tj(s j ) =

µti (s j, t j)

µti (S j , t j)

,

where µti (S j, t j) :=

sj∈S j

µti (s j, t j), and where denotes the first level

of λti for which µti (S j , t j ) > 0. Furthermore, define the set of type pro-

files for which ti i nduces a r ational mixed strategy for any subjectively






remains to show that (t1, t2) ∈ [ir], i.e., for each i, pi ∈ ∆(S tii ). Since,by Definition 10, it holds for each i that, ∀s

i ∈ S i, ui( pi, p j) ≥ ui(si, p j),

it follows from the construction of (λti , ti) that pi ∈ ∆(S tii ).(If.) Suppose that there exists an epistemic model with (t1, t2) ∈

[u] ∩ [ir] such that there is mutual certain belief of {(t1, t2)} at (t1, t2),and, for each i, pi is induced for ti by t j. Then, for each i, ti isrepresented by υti

i satisfying that υtii ◦z is a positive affine transformation

of ui and an LPS λti = (µti

1 , . . . , µti ), where ∀s j ∈ S j, µti

1 (s j, t j ) = p j (s j),and where = (S j × T j) ≥ 1. Suppose, for some i and p

i ∈ ∆(S i),ui( pi, p j ) < ui( p

i, p j). Then there is some si ∈ S i with pi(si) > 0 andsome s

i ∈ S i such that ui(si, p j ) < ui(si, p j), or equivalently

sjµti

1 (s j , t j )ui(si, s j) <

sjµti

1 (s j, t j)ui(si, s j) .

This means that si /∈ S tii , which, since pi(si) > 0, contradicts (t1, t2) ∈[ir j]. Hence, by Definition 10, ( p1, p2) is a Nash equilibrium.

For the “if” part of Proposition 21, it is sufficient that there is mutualcertain belief of the beliefs that each player has about the strategy choice.We do not need the stronger condition that (1) entails. Hence, higherorder certain belief plays no role in the characterization, in line with thefundamental insights of Aumann and Brandenburger (1995).

Characterizing rationalizability. We now turn to analysis of de-ductive reasoning in games and present a characterization of (ordinary)rationalizability. Since we are only concerned with two-player games,there is no difference between rationalizability, as defined by Bernheim(1984) and Pearce (1984), and correlated rationalizability, where conjec-tures are allowed to be correlated. As it follows that rationalizability intwo-player games corresponds to IESDS, we use the latter procedure asthe primitive definition. For any (∅ =) X = X 1 × X 2 ⊆ S 1 × S 2, writec(X ) := c1(X 2) × c2(X 1), where

ci(X j) := S i \ {si ∈ S i| ∃ pi ∈ ∆(S i) s.t. pi strongly dominates si on X j} .

Definition 11 Let G = (S 1, S 2, u1, u2) be a finite strategic two-playergame. Consider the sequence defined by X (0) = S 1 × S 2 and, ∀g ≥ 1,X (g) = c(X (g − 1)). A pure strategy si is said to be rationalizable if

si ∈ Ri :=∞

g=0X i(g) .

A mixed strategy pi is said to be rationalizable if pi is not stronglydominated on R j.




While any pure strategy in the support of a rationalizable mixed strat-egy is itself rationalizable (due to what Pearce calls the pure strategyproperty), the mixture on a set of rationalizable pure strategies need notbe rationalizable.

The following lemma is a straightforward implication of Definition 11.

Lemma 6 (i) For each i, Ri = ∅. (ii) R = c(R). (iii) For each i,si ∈ Ri if and only if there exists X = X 1 × X 2 with si ∈ Ri such that X ⊆ c(X ).

We next characterize the concept of rationalizable mixed strategies asinduced mixed strategies under common certain belief of [u] ∩ [ir].

Proposition 22 A mixed strategy pi for i is rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with (t1, t2) ∈ CK([u] ∩ [ir]) such that pi is induced for ti by t j.

Proof. Part 1: If p∗i is rationalizable, then there exists an epistemic

model with (t∗1, t∗

2) ∈ CK([u] ∩ [ir]) such that p∗i is induced for t∗

i by t∗ j .

Step 1: Construct an epistemic model with T 1 × T 2 ⊆ CK([u] ∩ [ir])such that for each si ∈ Ri of any player i, there exists ti ∈ T i with,si ∈ S tii . Construct an epistemic model with, for each i, a bijectionsi : T i → Ri from the set of types to the the set of rationalizable purestrategies. Assume that, for each ti ∈ T i of any player i, υti

i satisfies that

(a) υtii ◦ z = ui (so that T 1 × T 2 ⊆ [u]),

and the SCLP (λti , ti) on S j × T j has the properties that

(b) λti = (µti1 , . . . , µti

L ) with support S j × T jti satisfies that suppµt1

1 ∩(S j × {t j}) = {(s j (t j), t j)} for all t j ∈ T j

ti (so that, ∀t j ∈ T jti ,

piti|tj(s j(t j)) = 1),

(c) ti satisfies ti(S j × T j) = 1.

Property (b) entails that the support of the marginal of µti1 on S j is

included in R j. By properties (a) and (c) and Lemmas 4 and 6(ii),

we can still choose µti1 (and T iti) so that si(ti) ∈ S tii . This combinedwith property (b) means that T 1 × T 2 ⊆ [ir]. Furthermore, T 1 × T 2 ⊆CK([u] ∩ [ir]) since T j

ti ⊆ T j for each ti ∈ T i of any player i. Since, foreach player i, si is onto Ri, it follows that, for each si ∈ Ri of any playeri, there exists ti ∈ T i with si ∈ S tii .

Step 2: Add type t∗i to T i. Assume that υi

t∗i satisfies (a) and (λt∗i , t∗i )satisfies (b) and (c). Then µ1

t∗i can be chosen so that p∗i ∈ ∆(S i

t∗i ).




Furthermore, (T i ∪ {t∗i }) × T j ⊆ [u] ∩ [ir], and since T jt∗i ⊆ T j, (T i ∪

{t∗i }) × T j ⊆ CK([u] ∩ [ir]).Step 3: Add type t∗

j to T j . Assume that υ jt∗j satisfies (a) and the SCLP

(λt∗j , t∗j ) on S i × (T i ∪ {t∗i }) has the property that λt∗j = (µ1

t∗j , . . . , µLt∗j )

with support S i×{t∗i } satisfies that, ∀si ∈ S i, µ1

t∗j (si, t∗i ) = p∗

i (si), so that p∗

i is induced for t∗i by t∗

j . Furthermore, (T i∪{t∗i })×(T j ∪{t∗

j }) ⊆ [u]∩[ir],and since T i

t∗j ⊆ T i ∪{t∗i }, (T i ∪{t∗

i })×(T j ∪{t∗ j }) ⊆ CK([u]∩[ir]). Hence,

(t∗1, t∗

2) ∈ CK([u] ∩ [ir]) and p∗i is induced for t∗

i by t∗ j .

Part 2: If there exists an epistemic model with (t∗1, t∗

2) ∈ CK([u] ∩ [ir])such that p∗


j , then p∗i is rationalizable.

Assume that there exists an epistemic model with (t∗1, t∗

2) ∈ CK([u] ∩

[ir]) such that p∗i is induced for t∗i by t∗ j . In particular, CK([u]∩ [ir]) = ∅.Let, for each i, T i := projT i

CK([u] ∩ [ir]) and X i :=

ti∈T iS tii . ByProposition 20(ii), for each ti ∈ T i of any player i, ti deems (s j, t j)subjectively impossible if t j ∈ T j \T j since CK([u] ∩ [ir]) = KCK([u] ∩[ir]) ⊆ KiCK([u] ∩ [ir]), implying T j

ti ⊆ T j. By the definitions of [u] and[ir], it follows that, for each ti ∈ T i of any player i, ti is representedby υti

i satisfying that υtii ◦ z is a positive affine transformation of ui

and an LPS λti = (µti

1 , . . . , µti ), where = (S j × T j) ≥ 1, and where

suppµti1 ⊆ X j ×T j . Hence, by Lemma 4, for each ti ∈ T i of any player i, if

pi ∈ ∆(S tii ), then no strategy in the support of pi is strongly dominatedon X j, since it follows from pi ∈ ∆(S tii ) and suppµti

1 ⊆ X j × T j that,

∀si ∈ supp pi and ∀si ∈ S i,sj∈X j

tj∈T j

µti1 (s j , t j )ui(si, s j) ≥

sj∈X j

tj∈T j

µti1 (s j, t j)ui(s

i, s j) .

This implies X ⊆ c(X ), entailing by Lemma 6(iii) that, for each i,X i ⊆ Ri. Furthermore, since (t∗

1, t∗2) ∈ CK([u] ∩ [ir]) and the mixed

strategy induced for t∗i by t∗

j , p∗i , satisfies p∗

i ∈ ∆(S it∗i ), it follows that

p∗i is not strongly dominated on X j ⊆ R j. By Definition 11 this implies

that p∗i is a rationalizable mixed strategy.

5.3 Admissible consistency of preferences

We next refine the event of consistency of preferences and show howthis leads to characterizations of (strategic form) perfect equilibrium andmixed permissible strategies.

Caution. Player i has preference for cautious behavior at ti if hetakes into account all opponent strategies for any opponent type that isdeemed subjectively possible.




Throughout this chapter as well as chapters 8, 9, and 10, we assumethat Assumption 1 is satisfied so that κti = {ti} × S j × T j

ti . Under As-sumption 1 player i is cautious at ti if {ti

φ | ∅ = φ ⊆ Φti} satisfies Axiom6. Because then it follows from Proposition 2 that player i’s uncondi-tional preferences at ti, ti , are represented by υ ti

i and an LPS λti withsupport S j × T j

ti . Since thus (s j , t j ) ∈ suppλti for any (s j, t j) satisfyingt j ∈ T j

ti , player i at ti takes into account all opponent strategies forany opponent type that is deemed subjectively possible. Hence, underAssumption 1, we can define the event

[caui] := {(t1, t2) ∈ T 1 × T 2| {tiφ | ∅ = φ ⊆ Φti} satisfies Axiom 6 } .

In terms of the representation of the system of conditional preferences,{ti

φ | ∅ = φ ⊆ Φti}, by means of a vNM utility function and an SCLP(cf. Proposition 5), caution imposes the additional requirement that foreach type ti of any player i the full LPS λti is used to form the conditionalbeliefs over opponent strategy-type pairs. Formally, if L denotes thenumber of levels in the LPS λti , then

[caui] = {(t1, t2) ∈ T 1 × T 2| ti(S j × T j) = L} .

Since ti is non-increasing w.r.t. set inclusion, ti ∈ projT i[caui] implies

that ti(projS j×T jφ) = L for all subsets φ of {ti} × S j × T j with well-

defined conditional beliefs. Since it follows from Assumption 1 that λti

has full support on S j, ti ∈ projT i [caui] means that i’s choice set at ti

never admits a weakly dominated strategy, thereby inducing preferencefor cautious behavior.

Write [cau] := [cau1] ∩ [cau2].Say that at ti player i’s preferences over his strategies are admissibly

consistent with the game G = (S 1, S 2, u1, u2) and the preferences of his opponent , if ti ∈ projT i

([ui] ∩ [iri] ∩ [caui]). Refer to [u] ∩ [ir] ∩ [cau] asthe event of admissible consistency .

Characterizing perfect equilibrium. We now characterize theconcept of a strategic form (or “trembling-hand”) perfect equilibrium as

profiles of induced mixed strategies at a type profile in [u] ∩ [ir] ∩ [cau]where there is mutual certain belief of the type profile (i.e., for eachplayer, only the true opponent type is deemed subjectively possible).Before doing so, we define a (strategic form) perfect equilibrium.

Definition 12 Let G = (S 1, S 2, u1, u2) be a finite strategic two-playergame. A mixed strategy profile p = ( p1, p2) is a (strategic form) perfect equilibrium if there is a sequence ( p(n))n∈

of completely mixed strategy




profiles converging to p such that for each i and every n ∈ N,

ui( pi, p j (n)) = max pi

ui( pi, p j(n)) .

The following holds in two-player games.

Lemma 7 Let G = (S 1, S 2, u1, u2) be a finite strategic two-player game.A mixed strategy profile p = ( p1, p2) is a (strategic form) perfect equilib-rium if and only if p is a mixed-strategy Nash equilibrium and, for each i, pi is not weakly dominated.

Proof. Proposition 248.2 of Osborne and Rubinstein (1994).

The characterization result—which is a variant of Proposition 4 of Blume et al. (1991b)—can now be stated.

Proposition 23 Consider a finite strategic two-player game G. A profile of mixed strategies p = ( p1, p2) is a (strategic form) perfect equi-librium if and only if there exists an epistemic model with (t1, t2) ∈[u] ∩ [ir] ∩ [cau] such that (1) there is mutual certain belief of {(t1, t2)}at (t1, t2), and (2) for each i, pi is induced for ti by t j.

Proof. (Only if.) Let ( p1, p2) be a (strategic form) perfect equilib-rium. Then, by Lemma 7, ( p1, p2) be a mixed-strategy Nash equilibrium

and, for each i, pi is not weakly dominated. Construct the followingepistemic model. Let T 1 = {t1} and T 2 = {t2}. Assume that, for each i,

υtii satisfies that υti

i ◦ z = ui,

the SCLP (λti , ti) has the properties that λti = (µti1 , µti

2 ) with sup-port S j × {t j} has two levels, with the first level chosen so that,∀s j ∈ S j, µti

1 (s j, t j) = p j(s j ), and the second level chosen so thatsuppµti

2 = S j × {t j} and, ∀si ∈ S i,

sjµti

2 (s j, t j)ui( pi, s j ) ≥

sjµti

2 (s j , t j )ui(si, s j)

(which is possible by Lemma 5 since pi is not weakly dominated),and ti satisfies that ti(S j × T j ) = 2.

Then, it is clear that (t1, t2) ∈ [u] ∩ [cau], that there is mutual certainbelief of {(t1, t2)} at (t1, t2), and that, for each i, pi is induced for ti by t j.It remains to show that (t1, t2) ∈ [ir], i.e., for each i, pi ∈ ∆(S tii ). Since,by Lemma 7, it holds for each i that, ∀s

i ∈ S i, ui( pi, p j) ≥ ui(si, p j ), it

follows from the construction of (λti , ti) that pi ∈ ∆(S tii ).







Lemma 8 (i) For each i, P i = ∅. (ii) P = a(P ). (iii) For each i, si ∈ P iif and only if there exists X = X 1 ×X 2 with si ∈ P i such that X ⊆ a(X ).

We next characterize the concept of permissible mixed strategies asinduced mixed strategies under common certain belief of [u] ∩ [ir]∩ [cau].

Proposition 24 A mixed strategy pi for i is permissible in a finite strategic two-player game G if and only if there exists an epistemic model with (t1, t2) ∈ CK([u] ∩ [ir] ∩ [cau]) such that pi is induced for ti by t j.

Proof. Part 1: If p∗

i is permissible, then there exists an epistemic model with (t∗1, t∗

2) ∈ CK([u] ∩ [ir] ∩ [cau]) such that pi is induced for t∗i

by t∗ j .

Step 1: Construct an epistemic model with T 1 × T 2 ⊆ CK([u] ∩ [ir] ∩[cau]) such that for each si ∈ P i of any player i, there exists ti ∈ T i with,si ∈ S tii . Construct an epistemic model with, for each i, a bijectionsi : T i → P i from the set of types to the the set of permissible purestrategies. Assume that, for each ti ∈ T i of any player i, υti

i satisfies that



(b) λti = (µti1 , µti2 ) with support S j × T jti has two levels and satisfiesthat suppµt1

1 ∩ (S j × {t j}) = {(s j(t j), t j)} for all t j ∈ T jti (so that,

∀t j ∈ T jti , pi

ti|tj (s j (t j)) = 1),

(c) ti satisfies ti(S j × T j) = 2 (so that T 1 × T 2 ⊆ [cau]).


included in P j. By properties (a) and (c) and Lemmas 4, 5 and 8(ii),we can still choose µti

1 and µti2 (and T i

ti) so that si(ti) ∈ S tii . Thiscombined with property (b) means that T 1 × T 2 ⊆ [ir]. Furthermore,T 1 × T 2 ⊆ CK([u] ∩ [ir] ∩ [cau]) since T j

ti ⊆ T j for each ti ∈ T i of anyplayer i. Since, for each player i, si is onto P i, it follows that, for each

si ∈ P i of any player i, there exists ti ∈ T i with si ∈ S tii .Step 2: Add type t∗

i to T i. Assume that υit∗i satisfies (a) and (λt∗i , t∗i )

satisfies (b) and (c). Then µ1t∗i and µ2

t∗i can be chosen so that p∗i ∈

∆(S it∗i ). Furthermore, (T i ∪ {t∗

i }) × T j ⊆ [u] ∩ [ir] ∩ [cau], and sinceT j

t∗i ⊆ T j, (T i ∪ {t∗i }) × T j ⊆ CK([u] ∩ [ir] ∩ [cau]).

Step 3: Add type t∗ j to T j. Assume that υ j

t∗j satisfies (a) and the SCLP(λt∗j , t∗j ) on S i × (T i ∪ {t∗

i }) has the property that λt∗j = (µ1t∗j , . . . , µL

t∗j )




with support S i × {t∗i } satisfies that, ∀si ∈ S i, µ1

t∗j (si, t∗i ) = p∗

i (si), sothat p∗


j , and t∗j satisfies that t∗j (T i ∪ {t∗i }) = L.

Furthermore, (T i ∪ {t∗i }) × (T j ∪ {t∗

j }) ⊆ [u] ∩ [ir] ∩ [cau], and sinceT i

t∗j ⊆ T i ∪ {t∗i }, (T i ∪ {t∗

i }) × (T j ∪ {t∗ j }) ⊆ CK([u] ∩ [ir] ∩ [cau]). Hence,

(t∗1, t∗

2) ∈ CK([u] ∩ [ir] ∩ [cau]) and p∗i is induced for t∗

i by t∗ j .

Part 2: If there exists an epistemic model with (t∗1, t∗

2) ∈ CK([u] ∩[ir] ∩ [cau]) such that p∗


j , then p∗i is permissible.


2) ∈ CK([u] ∩[ir] ∩ [cau]) such that p∗


j . In particular, CK([u] ∩[ir] ∩ [cau]) = ∅. Let, for each i, T i := projT i

CK([u] ∩ [ir] ∩ [cau]) andX i := ti∈T iS tii . By Proposition 20(ii), for each ti ∈ T i of any player

i, ti deems (s j , t j ) subjectively impossible if t j ∈ T j\T j since CK([u] ∩[ir] ∩ [cau]) = KCK([u] ∩ [ir] ∩ [cau]) ⊆ KiCK([u] ∩ [ir] ∩ [cau]), implyingT j

ti ⊆ T j . By the definitions of [u], [ir], and [cau], it follows that, foreach ti ∈ T i of any player i, ti is represented by υti

i satisfying that υtii ◦z

is a positive affine transformation of ui and an LPS λti = (µti1 , . . . , µti

L ),and where suppµti

1 ⊆ X j × T j , and where suppλti = S j × T jti . Hence,

by Lemma 4, for each ti ∈ T i of any player i, if pi ∈ ∆(S tii ), then nostrategy in the support of pi is strongly dominated on X j, since it followsfrom pi ∈ ∆(S tii ) and suppµti

1 ⊆ X j × T j that, ∀si ∈ supp pi and ∀si ∈ S i,

sj∈X j

tj∈T j


sj∈X j

tj∈T j


i, s j) .

Furthermore, since the projection of λti on S j has full support, nostrategy in the support in pi is weakly dominated on S j. This impliesX ⊆ a(X ), entailing by Lemma 8(iii) that, for each i, X i ⊆ P i. Finally,since (t∗

1, t∗2) ∈ CK([u] ∩ [ir] ∩ [cau]) and the mixed strategy induced for

t∗i by t∗

j , p∗i , satisfies p∗

i ∈ ∆(S it∗i ), it follows that p∗

i is not strongly dom-inated on X j ⊆ P j and p∗

i is not weakly dominated on S j. By Definition13 this implies that p∗

i is a permissible mixed strategy.





Chapter 6

RELAXING COMPLETENESS

In the previous chapter, we have presented epistemic characteriza-tions of rationalizability and permissibility. For these non-equilibriumdeductive concepts, we have used, respectively, IESDS and the Dekel-Fudenberg procedure (one round of weak elimination followed by iteratedstrong domination) as the primitive definitions. Neither of these proce-dures rely on players having subjective probabilities over the strategychoice of the opponent. In contrast, the epistemic characterizations—byrelying on Assumption 1—require that players have complete preferencesthat are representable by means of subjective probabilities.

In this chapter we show how rationalizability and permissibility can beepistemically characterized without requiring that players have completepreferences that are representable by means of subjective probabilities.The resulting structure will also be used for the epistemic analysis of backward induction in Chapter 7 and forward induction in Chapter 11.Hence, even though the results of the present chapter may have limitedinterest in their own right, they set the stage for later analysis.

6.1 Epistemic modeling of strategic games (cont.)

The purpose of this section is to present a framework for strategicgames where each player is modeled as a decision maker under uncer-tainty with preferences that are allowed to be incomplete.

An epistemic model. Consider an epistemic model for a finitestrategic game form (S 1, S 2, z) as formalized in Definition 9, with a finitetype set T i for each player i, and where the preferences of a player corres-ponds to the player’s type. Hence, for each type ti of any player i,




{ti} × S j × T j is the set of states that are indistinguishable for playeri at ti,

a non-empty subset of {ti} × S j × T j, κti , is the set of states thatplayer i deems subjectively possible at ti, and

ti corresponds to a system of conditional preferences on the collectionof sets of acts from subsets of T i × S j × T j whose intersection withκti is non-empty to ∆(Z ).

However, instead Assumption 1, impose the following assumption,where Φti still denotes {φ ⊆ T i × S j × T j | κti ∩ φ = ∅}.

Assumption 2 For each ti of any player i, (a) tiφ satisfies Axioms 1 ,2, and 4 if ∅ = φ ⊆ T i × S j × T j, and Axiom 3 if and only if φ ∈ Φti,and (b) the system of conditional preferences {ti

φ | φ ∈ Φti} satisfies Axioms 5, 6, and 11.

As before, write ti for player i’s unconditional preferences at ti;i.e., for ti

φ when φ = {ti} × S j × T j. W.l.o.g. we may consider ti

to be preferences over acts from S j × T j to ∆(Z ) (instead of acts from{ti}×S j ×T j to ∆(Z )). Under Assumption 2 it follows from Proposition4 that, for each ti of any player i, i’s unconditional preferences at ti canbe conditionally represented by a vNM utility function υ ti

i : ∆(Z ) → R.Conditional representation implies that strong and weak dominance

are is well-defined: Let E j ⊆ S j × T j. Say that one act pE j strongly dominates another act qE j at ti if,

∀(s j, t j) ∈ E j, υtii (pE j (s j, t j))) > υti

i (qE j(s j, t j))) .

Say that pE j weakly dominates qE j at ti if,

∀(s j, t j) ∈ E j, υtii (pE j (s j, t j))) ≥ υti

i (qE j(s j, t j))) ,

with strict inequality for some (s j, t

j) ∈ E j. Say that ti is admissible on{ti}× E j if E j is non-empty and p ti q whenever pE j weakly dominatesqE j at ti. Assumption 2 entails that ti is admissible on κti . Indeed, asshown in Section 4.1, there exists a vector of nested sets, ( ρti

1 , . . . , ρtiL ),

on which ti is admissible, satisfying:

∅ = ρti1 ⊂ · · · ⊂ ρti

⊂ · · · ⊂ ρtiL = κti ⊆ {ti} × S j × T j

(where ⊂ denotes ⊆ and =).

Preferences over strategies. It follows from the above assumptionsthat, for each type ti of any player i, player i’s unconditional preferences






i plays the game G is given by

[ui] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

υtii ◦ z is a positive affine transformation of ui} ,

while [u] := [u1] ∩ [u2] is the event that both players play G.

Belief operators. Since Assumptions 2 is compatible with the frame-work of Chapter 4, we can in line with Section 4.2 define belief operatorsas follows. For these definitions, say that E ⊆ S 1 × T 1 × S 2 × T 2 does not concern player i’s strategy choice if E = S i × projT i×S j×T j

E .If E does not concern player i’s strategy choice, say that player i

certainly believes the event E at ti if ti ∈ projT iKiE , where

KiE := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2| κti ⊆ projT 1×S 2×T 2E } .

If E does not concern the strategy choice of either player, say thatthere is mutual certain belief of E at (t1, t2) if (t1, t2) ∈ projT i×T 2

KE ,where KE := K1E ∩ K2E . If E does not concern the strategy choice of either player, say that there is common certain belief of E at (t1, t2) if (t1, t2) ∈ projT i×T 2

CKE , where CKE := KE ∩ KKE ∩ KKKE ∩ . . . .If E does not concern player i’s strategy choice, say that player i

(unconditionally) believes the event E at ti if ti ∈ projT iBiE , where

BiE := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2| β ti ⊆ projT 1×S 2×T 2E } ,

and where β ti := ρti1 denotes the smallest set on which ti is admissible.

If E does not concern the strategy choice of either player, say that thereis mutual belief of E at (t1, t2) if (t1, t2) ∈ projT i×T 2BE , where BE :=B1E ∩ B2E . If E does not concern the strategy choice of either player,say that there is common belief of E at (t1, t2) if (t1, t2) ∈ projT i×T 2

CBE ,where CBE := BE ∩ BBE ∩ BBBE ∩ . . . .

As established in Proposition 14, Ki and Bi correspond to KD45 sys-tems. Moreover, the mutual certain belief and mutual belief operators,K and B, have the following properties, where we write K0E := E andB0E := E , and for each g ≥ 1, KgE := KKg−1E and BgE := BBg−1E .

Proposition 25 (i) For any E ⊆ T 1 ×T 2 and all g > 1, Kg

E ⊆ Kg−1

E and BgE ⊆ Bg−1E . If E = E 1 ∩ E 2, where, for each i, E i = S i ×projT i

E i × S j × T j , then KE ⊆ E and BE ⊆ E .(ii) For any E ⊆ T 1 × T 2, there exist g and g ≥ 0 such that KgE =

CKE for g ≥ g and BgE = CBE for g ≥ g, implying that CKE =KCKE and CBE = BCBE .

Proof. See the proof of Proposition 20.



Relaxing completeness 73

6.2 Consistency of preferences (cont.)

In the present section we define the event of consistency of preferencesin the case described by Assumption 2, where preferences need not becomplete, and use this event to characterize the concept of rationalizablepure strategies.

Belief of opponent rationality. In the context of the present chap-ter, define as follows the event that player i’s preferences over his strate-gies are consistent with the game G = (S 1, S 2, u1, u2) and the preferences of his opponent :

C i := [ui] ∩ Bi[rat j ] .

Write C := C 1 ∩ C 2 for the event of consistency .

Characterizing rationalizability. We now characterize the con-cept of rationalizable pure strategies (cf. Definition 11 of Chapter 5) asmaximal pure strategies under common certain belief of consistency.

Proposition 26 A pure strategy si for i is rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CKC .

To prove Proposition 26, it is helpful to establish a variant of Lemma

6. Write, for any (∅ =) X = X 1 ×X 2 ⊆ S 1 ×S 2, c(X ) := c1(X 2)×c2(X 1),where

ci(X j) := {si ∈ S i| ∃(∅ =) Y j ⊆ X j such that, ∀ pi ∈ ∆(S i),

pi does not weakly dominate si on Y j} .

Lemma 9 (i) R = c(R). (ii) For each i, si ∈ Ri if and only if there exists X = X 1 × X 2 with si ∈ Ri such that X ⊆ c(X ).

Proof. In view of Lemma 6, it is sufficient to show that, for any ( ∅ =)X j ⊆ S j, ci(X j) = ci(X j ).

Part 1: ci(X j ) ⊆ ci(X j). If si /∈ ci(X j), then ∃ pi ∈ ∆(S i) s.t. pi

strongly dominates si on X j. From this it follows that ∀ (∅ =) Y j ⊆ X j,∃ pi ∈ ∆(S i) s.t. pi weakly dominates si on Y j, implying that si /∈ ci(X j).

Part 2: ci(X j ) ⊇ ci(X j ). If si ∈ ci(X j), then there does not exist pi ∈ ∆(S i) s.t. pi strongly dominates si on X j. Hence, by Lemma 4, thereexists a subjective probability distribution µ ∈ ∆(S j ) with suppµ ⊆ X jsuch that si is maximal in ∆(S i) w.r.t. the preferences represented bythe vNM utility function ui and the subjective probability distribution




µ. Then there does not exist pi ∈ ∆(S i) s.t. pi weakly dominates si onsuppµ (⊆ X j), implying that si ∈ ci(X j).

Proof of Proposition 26. Part 1: If si is rationalizable, then there exists an epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2

CKC .It is sufficient to construct a belief system with S 1 × T 1 × S 2 × T 2 ⊆ CKC such that, for each si ∈ Ri of any player i, there exists ti ∈ T i with si ∈S tii . Construct a belief system with, for each i, a bijection si : T i → Ri

from the set of types to the the set of rationalizable pure strategies. ByLemma 9(i) we have that, for each ti ∈ T i of any player i, there existsY j

ti ⊆ Ri such that there does not exist pi ∈ ∆(S i) such that pi weakly

dominates si(ti) on Y jti

. Determine the set of opponent types that tideems subjectively possible as follows: T j

ti = {t j ∈ T j| s j(t j) ∈ Y jti}.

Let, for each ti ∈ T i of any player i, ti satisfy

1. υtii ◦ z = ui (so that S 1 × T 1 × S 2 × T 2 ⊆ [u]), and

2. p ti q iff pE j weakly dominates qE j for E j = E jti := {(s j , t j )|s j =

s j (t j) and t j ∈ T ti j }, which implies that β ti = κti = {ti} × E jti .

By the construction of E jti , this means that S tii si(ti) since, for any

acts p and q on S j × T j satisfying that there exist mixed strategies pi, q i ∈ ∆(S i) such that, ∀(s j, t j ) ∈ S j × T j , p(s j, t j) = z( pi, s j ) andq(s

j, t

j) = z(q

i, s

j), p ti q iff p

E j weakly dominates q

E j for E

j = Y

j

ti ×T j. This in turn implies, for each ti ∈ T i any player i,

3. β ti ⊆ projT i×S j×T j[rat j ] (so that S 1 × T 1 × S 2 × T 2 ⊆ Bi[rat j] ∩

B j[rati]).

Furthermore, S 1 × T 1 × S 2 × T 2 ⊆ CKC since T jti ⊆ T j for each ti ∈ T i

of any player i. Since, for each player i, si is onto Ri, it follows that, foreach si ∈ Ri of any player i, there exists ti ∈ T i with si ∈ S tii .

Part 2: If there exists an epistemic model with s∗i ∈ S i

t∗i for some (t∗

1, t∗2) ∈ projT 1×T 2

CKC , then s∗i is rationalizable.

Assume that there exists an epistemic model with s∗i ∈ S i

t∗i for some(t∗

1, t∗

2) ∈ proj

T 1×T 2CKC . In particular, CKC = ∅. Let, for each i,

T i := projT iCKC and X i :=

ti∈T iS tii . It is sufficient to show that, for

each i, X i ⊆ Ri. By Proposition 25(ii), for each ti ∈ T i of any playeri, β ti ⊆ κti ⊆ {ti} × S j × T j since CKC = KCKC ⊆ KiCKC . By thedefinition of C , it follows that, for each ti ∈ T i of any player i,

1. ti is conditionally represented by υtii satisfying that υti

i ◦ z is apositive affine transformation of ui, and



Relaxing completeness 75

2. p ti q if pE j weakly dominates qE j for E j = E jti := projS j×T j

β ti ,where β ti ⊆ projT i×S j×T j

[rat j].

Write Y jti := projS j

E jti = projS j

β ti , and note that β ti ⊆ ({ti}×S j ×T j)∩projT i×S j×T j

[rat j] implies Y jti ⊆ X i. It follows that, for any acts p and

q on S j × T j satisfying that there exist mixed strategies pi, q i ∈ ∆(S i)such that, ∀(s j , t j ) ∈ S j ×T j, p(s j, t j) = z( pi, s j) and q(s j, t j) = z(q i, s j),p ti q if pE j weakly dominates qE j for E j = Y j

ti×T j. Hence, if si ∈ S tii ,then there does not exist pi ∈ ∆(S i) such that pi weakly dominates si

on Y jti . Since this holds for each ti ∈ T i of any player i, we have that

X ⊆ c(X ). Hence, Lemma 9(ii) entails that, for each i, X i ⊆ Ri.

Proposition 26 is obtained also if CBC is used instead of CKC .

6.3 Admissible consistency of preferences (cont.)

In the present section we define the event of admissible consistency of preferences in the case considered by Assumption 2, where preferencesneed not be complete, and use this event to characterize the concept of permissible pure strategies.

Caution. As in Section 5.3, player i has preference for cautiousbehavior at ti if he takes into account all opponent strategies for anyopponent type that is deemed subjectively possible. Throughout this

chapter, as well as Chapter 7 and 11, we assume that Assumption 2 issatisfied, so that the system of conditional preferences {ti

φ | φ ∈ Φti}satisfies Axiom 6, where Φti denotes {φ ⊆ T i × S j × T j | κti ∩ φ = ∅}, andwhere κti—the set of states that player i deems subjectively possible atti—satisfies ∅ = κti ⊆ {ti} × S j × T j. Hence, T j

ti := projT jκti is the set

of opponent types that player i deems subjectively possible.Under Assumption 2, player i is cautious at ti if κti = {ti} × S j × T j

ti .Because then, player i at ti takes into account all opponent strategies forany opponent type that is deemed subjectively possible. This means thati’s choice set at ti never admits a weakly dominated strategy, therebyinducing preference for cautious behavior. Hence, under Assumption 2

we can define the event[caui] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

∃T jti such that κti = {ti} × S j × T j

ti } .

Write [cau] := [cau1] ∩ [cau2].In the context of the present chapter, define as follows the event that

player i’s preferences over his strategies are admissibly consistent with




the game G = (S 1, S 2, u1, u2) and the preferences of his opponent :

Ai := [ui] ∩ Bi[rat j] ∩ [caui] .

Write A := A1 ∩ A2 for the event of admissible consistency .

Characterizing permissibility. We now characterize the conceptof permissible pure strategies (cf. Definition 13 of Chapter 5) as maximalpure strategies under common certain belief of admissible consistency.

Proposition 27 A pure strategy si for i is permissible in a finite strate-gic two-player game G if and only if there exists an epistemic model with

si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CKA.

To prove Proposition 27, it is helpful to establish a variant of Lemma8. Define, for any (∅ =) Y j ⊆ S j ,

Di(Y j ) := {si ∈ S i| ∃ pi ∈ ∆(S i) such that

pi weakly dominates si on Y j or S j} ,

and write, for any (∅ =) X = X 1 × X 2 ⊆ S 1 × S 2, a(X ) := a1(X 2) ×a2(X 1), where

ai(X j ) := {si ∈ S i| ∃(∅ =) Y j ⊆ X j such that si ∈ S i\Di(Y i)} .

Lemma 10 (i) P = a(P ). (ii) For each i, si ∈ P i if and only if there exists X = X 1 × X 2 with si ∈ P i such that X ⊆ a(X ).

Proof. In view of Lemma 8, it is sufficient to show that, for any ( ∅ =)X j ⊆ S j, ai(X j) = ai(X j).

Part 1: ai(X j) ⊆ ai(X j). If si /∈ ai(X j), then ∃ pi ∈ ∆(S i) s.t. pi

strongly dominates si on X j or pi weakly dominates si on S j. From thisit follows that ∀ (∅ =) Y j ⊆ X j , ∃ pi ∈ ∆(S i) s.t. pi weakly dominates si

on Y j or S j, implying that ∀ (∅ =) Y j ⊆ X j, si ∈ Di(Y j). This meansthat si /∈ ai(X j).

Part 2: ai(X j) ⊇ ai(X j). If si ∈ ai(X j), then there does not exist pi ∈

∆(S i) s.t. pi strongly dominates si on X j or pi weakly dominates si on S j.Hence, by Lemmas 4 and 5, there exists an LPS λ = (µ1, µ2) ∈ L∆(S j)with suppµ1 ⊆ X j and suppµ2 = S j such that si is maximal in ∆(S i)w.r.t. the preferences represented by the vNM utility function ui and theLPS λ. Then there does not exist pi ∈ ∆(S i) s.t. pi weakly dominatessi on suppµ1 (⊆ X j) or suppµ2 (= S j), implying that si /∈ Di(Y j) forY j = suppµ1 ⊆ X j. This means that si ∈ ai(X j).






2. p ti q if pE j weakly dominates qE j for E j = E jti := projS j×T j

β ti

or E j = S j × T jti , where β ti ⊆ projT i×S j×T j

[rat j].

Write Y jti := projS j

E jti = projS j

β ti , and note that β ti ⊆ ({ti}×S j ×T j)∩projT i×S j×T j

[rat j] implies Y jti ⊆ X i. It follows that, for any acts p and

q on S j × T j satisfying that there exist mixed strategies pi, q i ∈ ∆(S i)such that, ∀(s j, t j ) ∈ S j ×T j, p(s j, t j) = z( pi, s j) and q(s j , t j) = z(q i, s j),p ti q if pE j weakly dominates qE j for E j = Y j

ti × T j or E j = S j × T j.Hence, S tii ⊆ S i\Di(Y j

ti). Since this holds for each ti ∈ T i of any playeri, we have that X ⊆ a(X ). Hence, Lemma 10(ii) entails that, for eachi, X i ⊆ P i.

Proposition 27 is obtained also if CBA is used instead of CKA; thisis essentially the corresponding result by Brandenburger (1992). Onemay argue that the result above is more complicated as it involves twodifferent epistemic operators. Still, it yields the insight that the essentialfeature in a characterization of the Dekel-Fudenberg procedure is to letirrational opponent choice be deemed subjectively possible. It also turnsout to be a useful benchmark for the analysis of backward induction inSection 7.3 where the certain belief operator Ki – rather than the belief operator Bi – must be used for the interactive epistemology (cf. theanalysis of Γ5 illustrated in Figure 7.1).






ter, this is purely a matter of convenience as everything can directly begeneralized to n-player games (with n > 2).

Among the large literature on backward induction during the lastcouple of decades,1 Reny’s (1993) impossibility result is of special im-portance. Reny associates a player’s ‘rationality’ in an extensive gamewith perfect (or almost perfect) information with what is called ‘weaksequential rationality’; i.e., that a player chooses rationally in all sub-games that are not precluded from being reached by the player’s ownstrategy. He shows that there exist perfect information games wherethe event that both players satisfy weak sequential rationality cannot be commonly believed in all subgames . E.g., in the “centipede” game that

is illustrated in Figure 2.4, common belief of weak sequential rational-ity cannot be held in the subgame defined by 2’s decision node. Thereason is that if 1 believes that 2 is rational in the subgame, and if 1believes that 2 believes that 1 will be rational in the subgame defined by1’s second decision node, then 1 believes that 2 will choose , implyingthat only Out is a best response for 1. Then the fact that the subgamedefined by 2’s decision node has been reached, contradicts 2’s belief that1 is rational in the whole game.

As a response, Ben-Porath (1997) imposes that common belief of weak sequential rationality is held initially, in the whole game , only. However,backward induction is not implied if weak sequential rationality is com-

monly believed initially, in the whole game, only. In the “centipede”game of Figure 2.4, the strategies Out and InL for player 1 and andr for player 2 are consistent with such common belief, while backwardinduction implies that down is played at any decision node.

In order to obtain an epistemic characterization of backward induc-tion, Aumann (1995) considers ‘sequential rationality’ in the sense thata player chooses rationally in all subgames (see also footnote 3 of thischapter). However, the event that players satisfy sequential rationalityis somewhat problematic. If—in the “centipede” game of Figure 2.4—1believes or knows that 2 chooses , then only by choosing the strategyOutL will 1 satisfy sequential rationality. However, what does it mean

that 1 chooses OutL in the counterfactual event that player 2’s decisionnode were reached? It is perhaps more natural—as suggested by Stal-

1Among contributions that are not otherwise referred to in this chapter are Basu (1990),

Bicchieri (1989), Binmore (1987, 1995), Bonanno (1991, 2001), Clausing and Wilks (2000),Dufwenberg and Linden (1996) Feinberg (2004a), Gul (1997), Kaneko (1999), Rabinowicz(1997), and Rosenthal (1981).



Backward induction 81

naker (1998)—to consider 2’s belief about 1’s subsequent action if 2’sdecision node were reached. Since Aumann (1995) assumes knowledgeof rational choice in an S5 partition structure, such a question of belief revision cannot be asked within Aumann’s model.

By imposing a full support restriction by considering players of typesin projT 1×T 2

[cau] (cf. the definition of [cau] in Section 6.3), the presentchapter ensures that each player takes all opponent strategies into ac-count, having the structural implication that conditional beliefs are well-defined and the behavioral implication that a rational choice in the wholegame is a rational choice in all subgames that are not precluded from

being reached by the player’s own strategy. Hence, by this restriction,we may consider ‘rationality’ instead of ‘weak sequential rationality’ (asshown by Lemma 11 and the subsequent text).

The main distinguishing feature of the present analysis is, however, toconsider the event that a player believes in opponent rationality ratherthan the event that the player himself chooses rationally. This is of course in line with the ‘consistent preferences’ approach that is the basisfor this book. As is shown by Proposition 27 of Chapter 6, permis-sible pure strategies—strategies surviving the Dekel-Fudenberg proce-dure, where one round of weak elimination is followed by iterated strongelimination—can be characterized as maximal strategies when there is

common certain belief that each player believes initially, in the whole game, that the opponent chooses rationally (‘belief of opponent ratio-nality’). For generic perfect information games, Ben-Porath shows thatthe set of outcomes consistent with common belief of weak sequentialrationality corresponds to the set of outcomes that survives the Dekel-Fudenberg procedure. Hence, maximal strategies when there is commoncertain belief of ‘belief of opponent rationality’ correspond to outcomesthat are promoted by Ben-Porath’s analysis.

An extensive game offers choice situations, not only initially, in thewhole game, but also in proper subgames. In perfect information games(and, more generally, in multi-stage games) the subgames constitute an

exhaustive set of such choice situations. Hence, in perfect informationgames one can replace ‘belief of opponent rationality’ by ‘belief in eachsubgame of opponent rationality’: Each player believes in each subgame that his opponent chooses rationally in the subgame . The main results of the present chapter (Propositions 28 and 29 of Section 7.3) show how,for generic perfect information games, common certain belief of ‘belief in each subgame of opponent rationality’ is possible and uniquely deter-




mines the backward induction outcome. Hence, by substituting ‘belief in each subgame of opponent rationality’ for ‘belief of opponent ratio-nality’, the present analysis provides an alternative route to Aumann’sconclusion, namely that common knowledge (or certain belief) of anappropriate form of (belief of) rationality implies backward induction.

This epistemic foundation for backward induction requires commoncertain belief of ‘belief in each subgame of opponent rationality’, wherethe term ‘certain belief’ is being used in the sense that an event is cer-tainly believed if the complement is subjectively impossible. As shownby a counterexample in Section 7.3, the characterization does not ob-tain if instead common belief is applied.2 Furthermore, the event of

which there is common certain belief — namely ‘belief in each subgameof opponent rationality’ — cannot be further restricted by taking theintersection with the event of ‘rationality’. The reason is that the fullsupport restriction (i.e., that players are of types in projT 1×T 2

[cau]) isinconsistent with certain belief of opponent ‘rationality’, as the latterprevents a player from taking into account irrational opponent choicesand rules out a well-defined theory of belief revision.

7.1 Epistemic modeling of extensive games

The purpose of this section is to present a framework for extensivegames of almost perfect information where each player is modeled as a

decision maker under uncertainty, with preferences that are allowed tobe incomplete.

An extensive game form. Inspired by Dubey and Kaneko (1984)and Chapter 6 of Osborne and Rubinstein (1994), a finite extensive two-person game form of almost perfect information with M − 1 stages canbe described as follows. The set of histories is determined inductively:The set of histories at the beginning of the first stage 1 is H 1 = {∅}.Let H m denote the set of histories at the beginning of stage m. Ath ∈ H m let, for each player i, i’s action set be denoted Ai(h), where iis inactive at h if Ai(h) is a singleton. Write A(h) := A1(h) × A2(h).Define the set of histories at the beginning of stage m + 1 by H m+1 :={(h, a) |h ∈ H m and a ∈ A(h)}. This concludes the induction. Denoteby H :=

M −1m=1 H m the set of subgames and denote by Z := H M the set

of outcomes.

2For definitions of the certain belief operator Ki and the belief operator Bi in the currentcontext, see Section 6.1.




A pure strategy for player i is a function si that assigns an action inAi(h) to any h ∈ H . Denote by S i player i’s finite set of pure strategies,and let z : S → Z map strategy profiles into outcomes, where S := S 1 ×S 2 is the set of strategy profiles.3 Then (S 1, S 2, z) is the correspondingfinite strategic two-person game form. For any h ∈ H ∪ Z , let S (h) =S 1(h) × S 2(h) denote the set of strategy profiles that are consistent withh being reached. Note that S (∅) = S . For any h, h ∈ H ∪ Z , h (weakly)precedes h if and only if S (h) ⊇ S (h). If si ∈ S i and h ∈ H , letsi|h denote the strategy in S i(h) having the following properties: (1) atsubgames preceding h, si|h determines the unique action leading to h,and (2) at all other subgames, si|h coincides with si.

Epistemic modeling. Since the extensive game form determines afinite strategic game form, we may represent the strategic interactionby means of an epistemic model as defined by Definition 9 of Chapter5. Since backward induction is a procedure—like IESDS and the Dekel-Fudenberg procedure—that does not rely on subjective probabilities,the analysis will allow for incomplete preferences. Hence, the epistemicmodel is combined with Assumption 2 of Chapter 6. In this respect thepresent analysis follows Aumann (1995) who presents a characterizationof backward induction where subjective probabilities play no role.

Conditional preferences over strategies. Write tih for player

i’s preferences at ti conditional on subgame h ∈ H being reached; i.e.,for ti

φ when φ = {ti} × S j(h) × T j. W.l.o.g. we may consider tih to

be preferences over acts from S j(h) × T j to ∆(Z ) (instead of acts from{ti} × S j(h) × T j to ∆(Z )). Denote by

H ti := {h ∈ H | κti ∩ ({ti} × S j(h) × T j) = ∅}

the set of subgames that i deems subjectively possible at ti. UnderAssumption 2 it follows from Proposition 4 that, for each ti of any playeri and all h ∈ H ti , i’s conditional preferences at ti in subgame h can beconditionally represented by a vNM utility function υti

i : ∆(Z ) → R that

does not depend on h.

3A pure strategy si ∈ S i can be viewed as an act on S j that assigns z(si, sj) ∈ Z to any

sj ∈ S j . The set of pure strategies S i is partitioned into equivalent classes of acts since apure strategy si also determines actions in subgames which si prevents from being reached.

Each such equivalent class corresponds to a plan of action , in the sense of Rubinstein (1991).As there is no need here to differentiate between identical acts in the present analysis, theconcept of a plan of action would have sufficed.




Hence, for each type ti of any player i, player i’s conditional prefer-ences at ti in subgame h, ti

h , is a reflexive and transitive binary rela-tion on acts from S j(h) × T j to ∆(Z ) that is conditionally representedby a vNM utility function υti

i if h ∈ H ti . Since each mixed strategy pi ∈ ∆(S i(h)) is a function that assigns the randomized outcome z( pi, s j)to any (s j, t j) ∈ S j(h) × T j and is thus an act from S j(h) × T j to ∆(Z ),we have that ti

h determines reflexive and transitive preferences on i’sset of mixed strategies, ∆(S i).

Player i’s choice function at ti is a function S tii (·) that assigns to everyh ∈ H player i’s set of maximal pure strategies at ti in subgame h:

S tii (h) := {si ∈ S i(h)| pi ∈ ∆(S i(h)), pi

tih si} .

Hence, a pure strategy, si, is in the set determined by i’s choice functionat ti in subgame h if there is no mixed strategy in ∆(S i(h)) that is strictlypreferred to si given i’s (possibly incomplete) conditional preferences atti in subgame h. Refer to S tii (h) as player i’s choice set at ti in subgameh, and write S tii = S tii (∅), thereby following the notation of Chapter 6.

Since tih is reflexive and transitive and satisfies objective indepen-

dence, and S i(h) is finite, it follows that the choice set S tii (h) is non-empty and supports any maximal mixed strategies: If q i ∈ ∆(S i(h)) and pi ∈ ∆(S i(h)) such that pi ti

h q i, then q i ∈ ∆(S tii (h)).By the following lemma, if si is maximal at ti in subgame h, then si

is maximal at ti in any later subgame that si is consistent with.

Lemma 11 If si ∈ S tii (h), then si ∈ S tii (h) for any h ∈ H with si ∈S i(h) ⊆ S i(h).

Proof. The proof of this lemma is based on the concept of a ‘strate-gically independent set’ due to Mailath et al. (1993). The set S ⊆S is strategically independent for player i in a strategic game G =(S 1, S 2, u1, u2) if S = S 1 × S 2 and, ∀si, s

i ∈ S i, ∃si ∈ S i such that

ui(si , s j ) = ui(s

i, s j) for all s j ∈ S j and ui(si , s j) = ui(si, s j ) for all

s j ∈ S j\S j. It follows from Mailath et al. (Definitions 2 and 3 and the‘if’ part of Theorem 1) that S (h) is strategically independent for i for any

subgame h in a finite extensive game of almost perfect information, andthis does not depend on the vNM utility function that assigns payoff toany outcome. The argument is based on the property that ∃s

i ∈ S i(h)such that z(s

i , s j) = z (si, s j) for all s j ∈ S j(h) and z(s

i , s j) = z (si, s j)for all s j ∈ S j \S j(h). The point is that i’s decision conditional on jchoosing a strategy consistent with h and i’s decision conditional on jchoosing a strategy in consistent with h can be made independently.




Suppose that si is not a maximal strategy at ti in the subgame h.Then there exists s

i ∈ S i(h) such that si ti

h si. As noted above, S (h) isstrategically independent for i. Hence, ∃s

i ∈ S i(h) such that z(si , s j) =

z(si, s j) for all s j ∈ S j(h) and z(s

i , s j) = z(si, s j) for all s j ∈ S j\S j(h).By Assumption 2 this implies that s

i tih si, which contradicts that si is

most preferred at ti in the subgame h.

The event that player i is rational in subgame h is defined by

[rati(h)] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2| si ∈ S tii (h)} .

Write [rati] = [rati(∅)], thereby following the notation of Chapter 6.

The imposition of a full support restriction by considering players of types in projT 1×T 2

[cau] (cf. the definition of [cau] in Section 6.3) has thestructural implication that, for all h, the conditional preferences, ti

h , arenontrivial. Moreover, by Lemma 11 it has the behavioral implication thatany choice si that is rational in h is also rational in any later subgamethat si is consistent with. This means that ‘rationality’ implies ‘weaksequential rationality’. In fact, ti

h is admissible on {ti} × S j(h) × T jti

(cf. Section 6.1), implying that any strategy that is weakly dominatedin h cannot be rational in h. Thus, preference for cautious behavior isinduced. However, in the context of generic perfect information games(cf. Section 7.2 of the present chapter) such admissibility has no cut-

ting power beyond ensuring that ‘rationality’ implies ‘weak sequentialrationality’; see, e.g., Lemmas 1.1 and 1.2 of Ben-Porath (1997). Hence,in the class of games considered in our main results it is of no conse-quence to use ‘rationality’ combined with full support rather than ‘weaksequential rationality’.

An extensive game. Consider an extensive game form, and let, foreach i, υi : Z → R be a vNM utility function that assigns a payoff to anyoutcome. Then the pair of the extensive game form and the vNM utilityfunctions (υ1, υ2) is a finite extensive two-player game of almost perfectinformation, Γ. Let G = (S 1, S 2, u1, u2) be the corresponding finitestrategic game , where for each i, the vNM utility function ui : S → R is

defined by ui = υi ◦ z (i.e., ui(s) = υi(z(s)) for any s = (s1, s2) ∈ S ).Assume that, for each i, there exist s = (s1, s2), s = (s

1, s2) ∈ S such

that ui(s) > ui(s).As before, the event that i plays the game G is given by

[ui] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

υtii ◦ z is a positive affine transformation of ui} ,






7.2 Initial belief of opponent rationality

A finite extensive game is

... of perfect information if, at any h ∈ H , there exists at most oneplayer that has a non-singleton action set.

... generic if, for each i, υi(z) = υi(z) whenever z and z are differentoutcomes.

Generic extensive games of perfect information have a unique subgame-perfect equilibrium. Moreover, in such games the procedure of backwardinduction yields in any subgame the unique subgame-perfect equilibriumoutcome. If s∗ denotes the unique subgame-perfect equilibrium, then,for any subgame h, z(s∗|h) is the backward induction outcome in thesubgame h, and S (z(s∗|h)) is the set of strategy vectors consistent withthe backward induction outcome in the subgame h.

Both Aumann (1995) and Ben-Porath (1997) analyze generic exten-sive games of perfect information. As already pointed out, while Au-mann establishes that common (true) knowledge of (sequential) rational-ity5 implies that the backward induction outcome is reached, Ben-Porathshows that the backward induction outcome is not the only outcome thatis consistent with common belief (in the whole game) of (weak sequen-tial) rationality. The purpose of the present section is to interpret the

analysis of Ben-Porath by applying Proposition 27 to the class of genericperfect information games.

Applying admissible consistency to extensive games. Recallthat the event of admissible consistency is defined as A := A1 ∩ A2,where

Ai := [ui] ∩ Bi[rat j] ∩ [caui] .

Again note that a full support restriction is imposed by consideringplayers of types in projT 1×T 2

[cau], ensuring that each player takes allopponent strategies into account.

In Proposition 27 of Chapter 6 we have established that the concept of

permissible pure strategies can be characterized as maximal pure strate-gies under common certain belief of admissible consistency. Recall alsothat permissible strategies (cf. Definition 13 of Chapter 5) correspond to

5Aumann (1995) uses the term substantive rationality, meaning that for all histories h, if aplayer were to reach h, then the player would choose rationally at h. See Aumann (1995, pp.14–16) and Aumann (1998) as well as Halpern (2001) and Stalnaker (1998, Section 5).




strategies surviving the Dekel-Fudenberg procedure, where one round of weak elimination is followed by iterated strong elimination. In the con-text of generic perfect information games, Ben-Porath (1997) establishesthrough his Theorem 1 that the set of outcomes consistent with com-mon belief (initially, in the whole game) of (weak sequential) rationalitycorresponds to the set of outcomes that survive the Dekel-Fudenbergprocedure. Hence, by Proposition 27, maximal strategies when thereis common certain belief of admissible consistency correspond to theoutcomes promoted by Ben-Porath’s analysis.

An example. To illustrate how common certain belief of admissible

consistency is consistent with outcomes other than the unique backwardinduction outcome, consider the strategic game G

3, with correspond-ing extensive form Γ

3; i.e., the “centipede” game illustrated in Figure2.4. Here, backward induction implies that down is being played at anydecision node. Let T 1 = {t

1, t1} and T 2 = {t

2, t2}. Assume that the pref-

erences of each type ti of any player i are represented by a vNM utilityfunction υti

i satisfying υ tii ◦ z = ui and a 2-level LPS on S j × T j . In Ta-

ble 7.1, the first numbers in the parentheses express primary probabilitydistributions, while the second numbers express secondary probabilitydistributions. The strategies OutL and OutR are merged as their rel-ative likelihood does not matter; see footnote 3. Note that all types

are in projT 1×T 2[cau], implying that players take all opponent strategiesinto account. With these 2-level LPSs each type’s preferences over theplayer’s own strategies are given by

Out t1 InL t1 InR

InL t1 Out t1 InR

t2 r

r t2

It is easy to check that both players satisfy ‘belief of opponent rationality’at each of their types; e.g., both t

2 and t2 assign positive (primary)

probability to an opponent strategy-type pair only if it is a maximalstrategy for the opponent type (i.e., Out in the case of t

1 and InL in the

case of t1). Thus, S 1 × T 1 × S 2 × T 2 ⊆ A. Since, for each ti ∈ T i of anyplayer i, κti ⊆ {ti} × S j × T j, it follows that S 1 × T 1 × S 2 × T 2 ⊆ C KA.Hence, preferences consistent with common certain belief of admissibleconsistency need not reflect backward induction since InL and r aremaximal strategies.

Note that, conditional on player 2’s decision node being reached (i.e.1 choosing InL or InR), player 2 at t

2 updates her beliefs about the type




Table 7.1. An epistemic model for G3 with corresponding extensive form Γ3.

t1 : t

2 t2 t

1 : t2 t

2

45 , 7

10

0, 1

10

35 , 5

10

0, 1

10

r

0, 1

10

15 , 1

10

r

0, 1

10

25 , 3

10

t

2 : t1 t

1 t2 : t

1 t1

Out

12 , 1

4

0, 1

8

Out

1, 1

2

(0, 0)

InL

0, 1

8 12 , 1

4 InL

0, 1

4 (0, 0)

InR 0, 1

8 0, 1

8 InR 0, 1

4 (0, 0)

of player 1 and assigns (primary) probability one to player 1 being of type t

1. Consequently, the conditional belief of player 2 at t2 assigns

(primary) probability one to player 1 choosing InL. Player 2 at t2, on

the other hand, does not admit the possibility that 1 is of another typethan t

1. Since the choice of In at 1’s first decision node is not rationalfor player 1 at t

1, there is no restriction concerning the conditional belief of player 2 at t

2 about the choice at 1’s second decision node. In theterminology of Ben-Porath, a “surprise” has occurred. Subsequent tosuch a surprise, a player need not believe that the opponent chooses

rationally among his remaining strategies.

7.3 Belief in each subgame of opponentrationality

A simultaneous game offers only one choice situation. Hence, for agame in this class, it seems reasonable that belief of opponent rationalityis held in the whole game only, as formalized by the requirement ‘belief of opponent rationality’. An extensive game with a nontrivial dynamicstructure, however, offers such choice situations, not only initially, inthe whole game, but also in proper subgames. Moreover, for extensivegames of almost perfect information, the subgames constitute an ex-

haustive set of such choice situations. This motivates imposing belief in each subgame of opponent rationality . Hence, consider the event that ibelieves conditional on subgame h ∈ H ti that j is rational in h:

Bi(h)[rat j(h)] = {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

(s j, t

j) ∈ projS j×T jβ ti(h) implies s

j ∈ S jtj (h)} ,




Since H ti = H whenever ti ∈ projT i[caui], it follows that, if ti ∈

projT i

h∈H ti Bi(h)[rat j(h)]

∩ [caui]

, then at ti player i believes con-

ditional on any subgame h that j is rational in h. In other words,h∈H ti

Bi(h)[rat j (h)]

∩ [caui]

is the event that player i believes in each subgame h that the opponent j is rational in h.6

Consider a finite extensive two-player game Γ of almost perfect in-formation with corresponding strategic game G. Say that at ti playeri’s preferences over his strategies are admissibly subgame consistent with

the game Γ and the preferences of his opponent if ti ∈ projT iA∗i , where

A∗i := [ui] ∩

h∈H ti

Bi(h)[rat j (h)]

∩ [caui] .

Refer to A∗ := A∗1 ∩ A∗

2 as the event of admissible subgame consistency .This definition of admissible subgame consistency can be applied to anyfinite extensive game of almost perfect information. However, in orderto relate to Aumann’s (1995) Theorems A and B, the following analysisis concerned with generic perfect information games.

The example revisited. In the belief system of Table 7.1, player 2at type t

2 does not satisfy ‘belief in each subgame of opponent rational-

ity’. By ‘belief in each subgame of opponent rationality’, player 2 mustbelieve, conditional on the subgame defined by 2’s decision node, that1 chooses his maximal strategy, InL, in the subgame. This means thatplayer 2 prefers to r, implying that player 1 must prefer Out to InLif he satisfies ‘belief in each subgame of opponent rationality’. Thus,common certain belief of admissible subgame consistency entails thatany types of players 1 and 2 have the preferences

Out t1 InL t1 InR t2 r

6Note that the requirement of such ‘belief in each subgame of opponent rationality’ allowsa player to update his belief about the type of his opponent. Hence, there is no assumptionof ‘epistemic independence’ between different agents in the sense of Stalnaker (1998); cf.the remark after the proof of Proposition 28 as well as Section 7.4. Still, the requirementcan be considered a non-inductive analog to ‘forward knowledge of rationality’ as defined by

Balkenborg and Winter (1997), and it is related to the requirement in Section 5 of Samet(1996) that each player hypothesizes that if h were reached, then the opponent would behaverationally at h.




respectively, meaning that if a player chooses a maximal strategy in asubgame, then his choice is made in accordance with backward induction.Demonstrating that this conclusion holds in general for generic perfectinformation games constitutes the main results of the present chapter.

Main results. In analogy with Aumann’s (1995) Theorems A andB, it is established that

... any vector of maximal strategies in a subgame of a generic perfectinformation game, in a state where there is common certain belief of admissible subgame consistency, leads to the backward inductionoutcome in the subgame (Proposition 28). Hence, by substituting

h∈H ti Bi(h)[rat j (h)] for Bi[rat j], the present analysis yields supportto Aumann’s conclusion, namely that if there is common knowledge(or certain belief) of an appropriate form of (belief of) rationality,then backward induction results.

... for any generic perfect information game, common certain belief of admissible subgame consistency is possible (Proposition 29). Hence,the result of Proposition 28 is not empty.

Proposition 28 Consider a finite generic extensive two-player game of perfect information Γ with corresponding strategic game G. If, for

some epistemic model, (t1, t2) ∈ projT 1×T 2CKA

∗

, then, for each h ∈ H ,S t11 (h) × S t22 (h) ⊆ S (z(s∗|h)), where s∗ denotes the unique subgame-perfect equilibrium.

Proof. In view of properties of the certain belief operator (cf. Propo-sition 25(ii)), it suffices to show for any g = 0, . . . , M − 2 that S t11 (h) ×S t22 (h) ⊆ S (z(s∗|h)) for any h ∈ H M −1−g if there exists an epistemicmodel with (t1, t2) ∈ projT 1×T 2

KgA∗. This is established by induction.(g = 0) Let h ∈ H M −1. First, consider j with a singleton action set

at h. Then trivially S jtj (h) = S j(h) = S j(z(s∗|h)). Now, consider i with

a non-singleton action set at h; since Γ has perfect information, there isat most one such i. Let ti ∈ projT i

K0A∗ = projT iA∗. Then it follows

that S tii (h) = S i(z(s∗|h)) since Γ is generic and A∗ ⊆ [ui] ∩ [caui].(g = 1, . . . , M − 2) Suppose that it has been established for g =

0, . . . , g − 1 that S t11 (h) × S t22 (h) ⊆ S (z(s∗|h)) for any h ∈ H M −1−g

if there exists an epistemic model with (t1, t2) ∈ projT 1×T 2K gA∗. Let

h ∈ H M −1−g. Part 1. Consider j with a singleton action set at h. Lett j ∈ projT j

Kg−1A∗. Then S j(h) = S j(h, a) and, by Lemma 11 and thepremise, S j

tj(h) ⊆ S jtj (h, a) ⊆ S j (z(s∗|(h,a))) if a is a feasible action




vector at h. This implies that

S jtj (h) ⊆

a

S j(z(s∗|(h,a))) ⊆ S j (z(s∗|h)) .

Hence, if s j ∈ S jtj(h), then s j is consistent with the backward induc-

tion outcome in any subgame (h, a) immediately succeeding h. Part 2. Consider i with a non-singleton action set at h; since Γ has perfectinformation, there is at most one such i. Let ti ∈ projT i

KgA∗. Thepreceding argument implies that S j

tj(h) ⊆

a S j(z(s∗|(h,a))) whenevert j ∈ T j

ti since ti ∈ projT iKgA∗ ⊆ projT i

KiKg−1A∗. Let si ∈ S i(h) be astrategy that differs from s∗

i |h by assigning a different action at h (i.e.,z(si, s∗

j |h) = z(s∗|h) and si(h) = s∗i |h(h) whenever S i(h) ⊃ S i(h)).

Let p and q be acts on S j × T j satisfying that, ∀(s j, t j) ∈ S j × T j,p(s j, t j) = z(s∗

i , s j) and q(s j , t j ) = z(si, s j ). Then,

p∩aS j(z( p|(h,a)))×T j strongly dominates q∩aS j(z( p|(h,a)))×T j

by backward induction since Γ is generic and ti ∈ projT iKgA∗ ⊆ [ui].

Since S jtj (h) ⊆

a S j (z(s∗|(h,a))) whenever t j ∈ T j

ti , it follows that,∀t j ∈ T j

ti ,

pS j

tj (h)×{tj} strongly dominates qS j

tj (h)×{tj} ,

and, thus, ti ∈ projT iKgA∗ ⊆ Bi(h)[rat j(h)] ∩ [caui] implies that

p tih q .

It has thereby been established that si ∈ S i(h)\S tii (h) if si differs frombackward induction only by the action taken at h. However, the premisethat S tii (h, a) ⊆ S i(z(s∗|(h,a))) if a is a feasible action vector at h, itfollows that any si ∈ S tii (h) is consistent with the backward inductionoutcome in the subgame (h, (si(h), a j)) immediately succeeding h wheni plays the action si(h) at h (since si ∈ S i(h, (s j (h), a j)) and, by Lemma11, si ∈ S tii (h, (si(h), a j)). Hence, S tii (h) ⊆ S i(z( p|h)).

It follows from the proof of Proposition 28 that, for a generic perfectinformation game with M −1 stages, it is sufficient with M −2 order mu-

tual certain belief of admissible subgame consistency in order to obtainbackward induction. Hence, KM −2A∗ can be substituted for CKA∗.

Backward induction will not be obtained, however, if CBA∗ is substi-tuted for CKA∗. This can be shown by considering a counter-examplethat builds on the four-legged centipede game of Figure 7.1 and the epis-temic model of Table 7.2. In the table the preferences of each type ti

of any player i are represented by a vNM utility function υtii satisfying




1 2 1 2 6

In r R r 4Out L

2 1 4 30 3 2 5

Figure 7.1. Γ5 (a four-legged “centipede” game).

Table 7.2. An epistemic model for Γ5.

t1 : t1 : t1 :t2 t2 t2 t2 t2 t2

45

, 710

, 712

0, 110

, 112

35

, 510

, 512

0, 110

, 112

110

110

r

0, 110

, 112

15

, 110

, 112

r

0, 110

, 112

25

, 310

, 312

r

110

110

rr

0, 0, 112

0, 0, 112

rr

0, 0, 112

0, 0, 112

rr

310

310

t2 : t2 :

t1 t1 t1 t1 t1 t1Out

12

, 13

, 14

0, 16

, 18

(0, 0, 0) Out

1, 12

, 13

(0, 0, 0)

0, 0, 112

InL

0, 16

, 18

12

, 13

, 14

(0, 0, 0) InL

0, 14

, 16

(0, 0, 0)

0, 0, 112

InR

0, 0, 18

0, 0, 18

(0, 0, 0) InR

0, 0, 16

(0, 0, 0)

0, 14

, 16

υtii ◦ z = ui and a 1 or 3-level LPS on S j × T j, where T 1 = {t

1, t1, t

1 }and T 2 = {t

2, t2}. While all types are in projT 1×T 2

[cau], implying thatplayers take all opponent strategies into account, inspection shows thatA∗ = S 1 ×{t

1, t1}×S 2 ×{t

2, t2}, since player 1 at t

1 does not satisfy ‘be-lief in each subgame of opponent rationality’. Furthermore, each player ibelieves at t

i or ti that the opponent is of a type in {t

j , t j }. This implies

that CBA∗ = A∗. Since InL is the maximal strategy for 1 at t1 and r

is the maximal strategy for 2 at t2, it follows that preferences consistent

with common belief of admissible subgame consistency need not reflectbackward induction. However, 2 does not certainly believe at t

2 that theopponent is not of type t

1 . Therefore, KA∗ = A∗ = S × {t1, t

1} × {t2},

while KKA∗ = ∅. Hence, preferences that yield maximal strategies incontradiction with backward induction are not consistent with commoncertain belief of admissible subgame consistency.

The example shows that ti ∈ projT iA∗

i is consistent with player i at ti

updating his beliefs about the preferences of his opponent conditional ona subgame being reached. I.e., 1 at t

1 assigns initially, in the whole, (pri-mary) probability 4

5 to 2 being of type t2 with preferences r rr,




while in the subgame defined by 1’s second decision node 1 at t1 as-

signs (primary) probability one to 2 being of type t2 with preferences

r ∼ rr. This shows that Stalnaker’s (1998) assumption of ‘epis-temic independence’ is not made; a player is in principle allowed to learnabout the type of his opponent on the basis of previous play. However,in an epistemic model with CKA∗ = ∅, ti ∈ projT i

CKA∗ implies that 1certainly believes at ti that 2 is of a type with preferences r rr.In other words, if there is common certain belief of admissible subgameconsistency, there is essentially nothing to learn about the opponent.

Proposition 29 For any finite generic two-player extensive game of perfect information Γ with corresponding strategic game G, there exists a belief system for G with CKA∗ = ∅.

Proof. Construct an epistemic model with one type of each player:T 1 = {t1} and T 2 = {t2}. Write, for each player j , ∀m ∈ {1, . . . , M − 1},S ∗ m

j := {s∗ j |h | h ∈ H m} and, S ∗ M

j := S j. Let, for each player i,λti = (µti

1 , . . . , µtiM ) ∈ L∆(S j × {t j}) satisfy the following requirement:

∀m ∈ {1, . . . , M }, suppµtim = S ∗ m

j × {t j }. By letting ti be repre-sented by a vNM utility function υti

i satisfying υtii ◦ z = ui and the

LPS λti , then (1) [ui] ∩ [caui] = S 1 × T 1 × S 2 × T 2. Let, ∀h ∈ H ,λti |

h = (µ ti

1 , . . . µ ti

M |h) denote the conditional of λti on S

j(h)×T

j. By the

properties of a subgame-perfect equilibrium, ∀h ∈ H , µ ti1 (s∗

j |h, t j) = 1and s∗

i |h ∈ S tii (h). Hence, since likewise s∗ j |h ∈ S j

tj(h), we have that (2)h∈H ti Bi(h)[rat j(h)] = S 1 × T 1 × S 2 × T 2. As (1) and (2) hold for both

players, it follows that C KA∗ = A∗ = S 1 × T 1 × S 2 × T 2 = ∅.

7.4 Discussion

In this section we first interpret our analysis in view of Aumann (1995)and then present a discussion of the relationship to Battigalli (1996a).

Adding belief revision to Aumann’s analysis. Consider a genericperfect information game. Say that a player’s preferences (at a given

type) are in accordance with backward induction if, in any subgame, astrategy is a rational choice only if it is consistent with the backwardinduction outcome. Using this terminology, Proposition 28 can be re-stated as follows: Under common certain belief of admissible subgameconsistency, players are of types with preferences that are in accordancewith backward induction. Furthermore, common certain belief of admis-sible subgame consistency implies that each player deems it subjectively






her strategy, he believes that the opponent will follow her strategy inthe remaining subgame.

There is no assumption of ‘epistemic independence’ in the current in-terpretation of Aumann’s result. Instead, we have changed statements‘about opponents’ from being concerned with strategy choice to beingrelated to preferences. While it is desirable when modeling backwardinduction to have an explicit theory of revision of beliefs about oppo-nent choice, a theory of revision of beliefs about opponent preferences isinconsistent with maintaining both (a) that preferences are necessarilyrevealed from choice, and (b) that there is common certain belief of thegame being played (i.e., consider the case where Ai(∅) is non-singleton,

and ai ∈ Ai(∅) ends the game and leads to an outcome that is pre-ferred by i to any other outcome). Here we have kept the assumptionthat there is common certain belief of the game, meaning that the gameis of ‘complete information’, while requiring only conditional belief ineach subgame of opponent rationality, meaning that irrational opponentchoices—although being probability zero events—are not subjectivelyimpossible.

We have shown how common certain belief of admissible subgameconsistency implies that each player deems it impossible that the op-ponent has preferences not in accordance with backward induction andthus interprets any deviation from the backward induction path as the

opponent not having made a rational choice. In this way we present amodel that combines a result that resembles Aumann (1995) by associ-ating backward induction with certainty about opponent type, with ananalysis that unlike Aumann’s yields a theory of belief revision aboutopponent choice.

Rationality orderings. The constructive proof of Proposition 29shows how common certain belief of admissible subgame rationality maylead player i at ti to have preferences over i’s strategies that are rep-resented by a vNM utility function υti

i satisfying υtii ◦ z = ui and an

LPS λti = (µti1 ,...,µti

L) ∈ L∆(S j × T j) with more than two levels of subjective probability distributions (i.e., L > 2). E.g., in the “centipede”

game of Figure 2.4, common certain belief of admissible subgame con-sistency implies that player 2 at any type t2 has preferences that canbe represented by υt2

2 satisfying υt22 ◦ z = u2 and λt2 = (µt2

1 , µt22 , µt2

3 )satisfying projS 1

suppµt21 = {Out}, projS 1

suppµt22 = {Out, InL}, and

projS 1suppµt2

3 = S 1. One may interpret

projS jsuppµti

1 to be j’s “most rational” strategies,




projS jsuppµti

L\

k<L projS jsuppµti

k to be j’s “completely irrational”strategies, and

projS jsuppµti

\

k< projS jsuppµti

k , for = 2, . . . , L − 1, to consist of strategies for j that are at “intermediate degrees of rationality”.

This illustrates thatprojS j

suppµti1 , . . . , projS j

suppµtiL \

k<L

projS jsuppµti

k

corresponds closely to what Battigalli (1996a) calls a rationality ordering for j.

However, the present construction of such a rationality ordering dif-

fers from the one proposed by Battigalli. This difference is along twodimensions:

1 Battigalli considers best responses in reachable subgames only (seehis Definition 2.1), while here belief of opponent rationality is held inall subgames (cf., ‘belief in each subgame of opponent rationality’).

2 Battigalli considers best responses given beliefs where opponent stra-tegies that are less than “most rational” are given positive probability,while here each player always believes that the opponent choosesrationally.

This difference has the following consequences:

Although Battigalli’s construction of rationality orderings also yieldsthe backward induction outcome in any generic perfect informationgame, his proof (cf. Battigalli, 1997) is not tied to the backwardinduction procedure .

Battigalli’s construction promotes the forward induction outcome(InL, ) in the “battle-of-the-sexes with an outside option” game il-lustrated in Figure 2.6. This conclusion is not reached in the presentanalysis since there is no choice situation in which 1 under all cir-cumstances will have a particular preference between his “battle-of-the-sexes” strategies.8

This also indicates how the epistemic foundation for the backward in-

duction procedure offered here differs from the epistemic foundation forbackward (and forward) induction outcomes provided by Battigalli andSiniscalchi (2002).

8Chapter 11 will, following Asheim and Dufwenberg (2003a), demonstrate how the conceptof admissible consistency can be strengthened so that the forward induction outcome ispromoted in the “battle-of-the-sexes with an outside option” game.





Chapter 8

SEQUENTIALITY

One major problem in the theory of extensive games is the following:How should a player react when he finds himself at an information setthat contradicts his previous belief about the opponent’s strategy choice?Different approaches have been proposed to this problem. As mentionedin Chapter 2, Ben-Porath (1997) and Reny (1992) have formulated ra-tionalizability and equilibrium notions based on weak sequentiality , inwhich a player is allowed to believe, in this situation, that his opponent

will no longer choose rationally. Battigalli and Siniscalchi (2002) haveshown that Pearce’s (1984) extensive form rationalizability can be char-acterized by assuming that a player, in such a situation, should look forthe highest degree of “strategic sophistication” that is compatible withthe event of reaching this information set, and stick to this degree untilit is contradicted later on in the game. Perea (2002, 2003) suggests thatthe player, in such a situation, may revise his conjecture about the op-ponent’s utility function in order to rationalize her “surprising” move,while maintaining common belief of rational choice at all informationsets. The most prominent position, however, is that the player shouldstill believe that his opponent will choose rationally in the remainder

of the game; this underlies concepts that promote backward induction.Such concepts will be presented in this and the next chapter, whichreproduce joint work with Andres Perea, cf. Asheim and Perea (2004).

We define sequential rationalizability by imposing common certainbelief of the event that each player believes that the opponent choosesrationally at all her information sets. Since this is a non-equilibriumconcepts, each player need not be certain of the beliefs that the oppo-




nent has about the player’s own action choice. However, by assuming that each player is certain of the beliefs that the opponent has aboutthe player’s own action choice, we obtain an epistemic characterizationof the corresponding equilibrium concept: sequential equilibrium . Whenapplied to generic games with perfect information, sequential rationaliz-ability yields the backward induction procedure. As elsewhere, to avoidthe issue of whether (and if so, how) each player’s beliefs about theaction choice of his opponents are stochastically independent, all anal-ysis is limited to two-player games. The assumption is essential in thepresent context where a behavior strategy of a player will be interpretedas an expression of the belief of his opponent.

For the above mentioned definitions and characterizations, we mustdescribe what a player believes both conditional on reaching his owninformation sets (to evaluate his rationality) and conditional on his op-ponent reaching her information sets (to determine his beliefs about herchoices). Hence, we must specify a system of conditional beliefs for eachplayer. For reasons given in Section 3.1, this will be done by means of ourconcept of a system of conditional lexicographic probabilities (SCLP) asdefined in Definition 1 and characterized in Proposition 5.

We embed the notion of an SCLP in an epistemic model, as definedby Definition 9 of Chapter 5, by invoking Assumption 1. For each type ti

of any player i, ti is described by an SCLP, inducing a behavior strategyfor each opponent type t j that is deemed subjectively possible by ti. Theevent that ‘player i believes that the opponent j chooses rationally ateach information set’ can then be defined as the event where player i isof a type ti that, for each subjectively possible opponent type t j, inducesa behavior strategy which is sequentially rational given t j’s own SCLP.

The characterization of sequential equilibrium reported in Proposition30 is included in order to motivate the analogous non-equilibrium con-cept, namely sequential rationalizability. The result may, however, beof interest in its own right and in comparison with other such epistemiccharacterizations; see, e.g., Theorem 2 of Feinberg (2004b).

The concept of sequential rationalizability as stated in Definition 15is related to various other concepts proposed in the literature. Alreadyin Bernheim (1984) there are suggestions concerning how to define non-equilibrium concepts that involve rational choice at all information sets.By requiring rationalizability in every subgame, Bernheim defines theconcept of subgame rationalizability —which coincides with our definitionof sequential rationalizability for games of almost perfect information—



Sequentiality 101

but no epistemic characterization is offered. On p. 1022 Bernheim claimsthat it is possible to define a concept of sequential rationalizability, butdoes not indicate how this can be done. After related work by Green-berg (1996), sequential rationalizability was finally defined by Dekel etal. (1999, 2002), whose concept coincides with ours in our two-playersetting. Our definition of quasi-perfect rationalizability is new. Dekel etal. (1999) and Greenberg et al. (2003) consider also extensive game con-cepts that lie between equilibrium and rationalizability; such conceptswill not be considered here.

8.1 Epistemic modeling of extensive games

(cont.)The purpose of this section is to present a framework for a general

class of extensive games where each player is modeled as a decision makerunder uncertainty with complete preferences.

An extensive game form. Consider a finite extensive two-playergame form without chance moves. Assume that the extensive game formsatisfies perfect recall. Denote by H i the finite collection of informationsets controlled by player i. For every information set h ∈ H i, let A(h)be the set of actions available at h. A pure strategy for player i is afunction si which assigns to every information set h ∈ H i some action

si(h) ∈ A(h). Denote by S i the set of pure strategies for player i, where,in the subsequent analysis, there is no need to differentiate between purestrategies in S i that differ only at non-reachable information sets. WriteS = S 1 × S 2, denote by Z the set of outcomes (or terminal nodes), andlet z : S → Z map strategy profiles into terminal nodes. Then (S 1, S 2, z)is the corresponding finite strategic two-player game form.

For any h ∈ H 1 ∪ H 2, let S i(h) be the set of strategies si for whichthere is some strategy s j such that (si, s j ) reaches h. For any h and anynode x ∈ h, denote by S (x) = S 1(x) × S 2(x) the set of pure strategyprofiles for which x is reached, and write S (h) :=

x∈h S (x). By perfect

recall, it holds that S (h) = S 1(h) × S 2(h) for all information sets h. For

any h, h ∈ H i, h (weakly) precedes h if and only if S (h) ⊇ S (h). Forany h ∈ H i and a ∈ A(h), write S i(h, a) := {si ∈ S i(h)|si(h) = a}.

A behavior strategy for player i is a function σi that assigns to everyh ∈ H i some randomization σi(h) ∈ ∆(A(h)) on the set of available ac-tions. If h ∈ H i, denote by σi|h the behavior strategy with the followingproperties: (1) at player i information sets preceding h, σi|h determineswith probability one the unique action leading to h, and (2) at all other




player i information sets, σi|h coincides with σi. Say that σi is outcome-equivalent to a mixed strategy pi (∈ ∆(S i)) if, for any s j ∈ S j , σi and pi

induce the same probability distribution over terminal nodes. For anyh ∈ H i, σi|h is outcome-equivalent to some pi ∈ ∆(S i(h)).

Epistemic modeling. Since the extensive game form determines afinite strategic game form, we may represent the strategic interaction bymeans of an epistemic model as defined by Definition 9 of Chapter 5.Since a behavior strategy of a player will be interpreted as an expressionof the belief of his opponent, it is essential that the analysis assumescomplete preferences. Hence, the epistemic model is combined with

Assumption 1 of Chapter 5.Under Assumption 1 it follows from Proposition 5 that, for each type

ti of any player i, i’s system of conditional preferences at ti can berepresented by a vNM utility function υti

i : ∆(Z ) → R and an SCLP(λti , ti), which for expositional simplicity is defined on S j × T j withsupport S j × T j

ti (instead of being defined on T i × S j × T j with supportκti = {ti} × S j × T j

ti). Hence, writing tih for player i’s preferences

at ti conditional on player i information set h ∈ H i being reached, weconsider w.l.o.g. ti

h to be preferences over acts from S j(h) × T j to ∆(Z )(instead of acts from {ti} × S j (h) × T j to ∆(Z )).

Conditional preferences over strategies. It follows that, for each

ti of any player i and all h ∈ H i, i’s conditional preferences at ti insubgame h can be represented by the vNM utility function υti

i : ∆(Z ) →R that does not depend on h, and an LPS

λti(S j(h)×T j)|S j(h)×T j = (µ

1, . . . µ(S j(h)×T j)|S j(h)×T j

)

derived from the SCLP (λti , ti) on S j × T j with support S j × T jti .

Recall from Assumption 1 that player i deems an opponent strategy-type pair (s j, t j) subjectively possible at ti if and only if s j ∈ S j andt j ∈ T j

ti . This means that conditional preferences are non-trivial foran event E j (⊆ S j × T j) if and only if E j ∩ (S j × T ti j ) = ∅. Note that{S j(h)×T j | h ∈ H i} is the set of events that are objectively observable by

i. Hence, conditional preferences are always non-trivial for such eventssince, for any h ∈ H i, (S j(h) × T j ) ∩ (S j × T j

ti) = (S j(h) × T jti) = ∅.

Since, for all h ∈ H i, each pure strategy si ∈ S i(h) is a function thatassigns the deterministic outcome z(si, s j) to any (s j, t j) ∈ S j(h) × T j,it follows that si ∈ S i(h) is an act from S j (h) × T j to ∆(Z ), and wehave that ti

h determines complete and transitive preferences on i’s setof pure strategies, S i(h), conditional on h ∈ H i being reached.



Sequentiality 103

Player i’s choice function at ti is a function S tii (·) that assigns toevery h ∈ H i player i’s set of rational pure strategies at ti conditionalon h ∈ H i:

S tii (h) := {si ∈ S i(h)| ∀si ∈ S i(h), si ti

h si} .

Refer to S tii (h) as player i’s choice set at ti conditional on player iinformation set h, and write S tii = S tii (∅), thereby following the notationof Chapter 5.

Since tih is complete and transitive and satisfies objective indepen-

dence, and S i(h) is finite, it follows that the choice set S tii (h) is non-empty, and that the set of rational mixed strategies equals ∆(S ti

i (h)).

Note that Lemma 11 does not hold under Assumption 1, unless alsocaution is imposed (cf. Section 9.1). The assumption that player i’s sys-tem of conditional preferences at ti is representable by means of an SCLPwhere the set of subjectively possible opponent types equals S j × T j

ti

has the structural implication that, for all h ∈ H i, the conditional pref-erences, ti

h , are nontrivial, even without imposing caution. However,representation by means of an SCLP does not have the behavioral impli-cation that any choice si that is rational conditional on h is also rationalin any later player i information set that si is consistent with. Thismeans that ‘rationality’ does not imply ‘weak sequential rationality’ if caution is not imposed.

An extensive game. As in Chapter 7, a finite extensive two-playergame consists of the pair of the extensive game form and the vNM utilityfunctions (υ1, υ2), with G = (S 1, S 2, u1, u2) denoting the correspondingfinite strategic game , where for each i, the vNM utility function ui :S → R is defined by ui = υi ◦ z. As before—but transferred to T 1 × T 2space—the event that i plays the game G is given by

[ui] := {(t1, t2) ∈ T 1×T 2|υtii ◦z is a positive affine transformation of ui},

while [u] := [u1] ∩ [u2] is the event that both players play G.

Certain belief. As in Chapter 5, say for any E ⊆ T 1 × T 2 that playeri certainly believes the event E at ti if ti ∈ projT iKiE , where

KiE := {(t1, t2) ∈ T 1 × T 2| projT 1×T 2κti = {ti} × T j

ti ⊆ E } .

Say that there is mutual certain belief of E at (t1, t2) if (t1, t2) ∈ KE ,where KE := K1E ∩ K2E . Say that there is common certain belief of E at (t1, t2) if (t1, t2) ∈ CKE , where CKE := KE ∩ KKE ∩ KKKE ∩ . . . .





Sequentiality 105

distributions also at player j information sets that are unreachable giveni’s initial belief at ti about t

j’s behavior. Hence, if ti ∈ projT i[isri], then

player i believes at ti that each subjectively possible opponent type t j

chooses rationally also at player j information sets that contradict ti’sinitial belief about the behavior of t

j. The above observation explainswhy we can characterize a sequential equilibrium as a profile of inducedbehavior strategies at a type profile in [isr] where there is mutual certainbelief of the type profile (i.e., for each player, only the true opponenttype is deemed subjectively possible).

Characterizing sequential equilibrium. We first define sequential

equilibrium. Player i’s beliefs over past opponent actions at i’s informa-tion sets is a function β i that to any h ∈ H i assigns a probability distri-bution over the nodes in h. An assessment (σ, β ) = ((σ1, σ2), (β 1, β 2)),consisting of a pair of behavior strategies and a pair of beliefs, is consis-tent if there is a sequence (σ(n), β (n))n∈

of assessments converging to(σ, β ) such that for every n, σ(n) is completely mixed and β (n) is inducedby σ(n) using Bayes’ rule. If σi and σ j are any behavior strategies for iand j, and β i are the beliefs of i, then let, for each h ∈ H i, ui(σi, σ j; β i)|h

denote i’s expected payoff conditional on h, given the belief β i(h), andgiven that future behavior is determined by σi and σ j.

Definition 14 An assessment (σ, β ) = ((σ1, σ2), (β 1, β 2)) is a sequen-

tial equilibrium if it is consistent and it satisfies that for each i and everyh ∈ H i,

ui(σi, σ j; β i)|h = maxσi

ui(σi, σ j; β i)|h .

The characterization result can now be stated; it is proven in Ap-pendix B.

Proposition 30 Consider a finite extensive two-player game Γ. A profile of behavior strategies σ = (σ1, σ2) can be extended to a sequential equilibrium if and only if there exists an epistemic model with (t1, t2) ∈[u] ∩ [isr] such that (1) there is mutual certain belief of {(t1, t2)} at (t

1, t

2), and (2) for each i, σi is induced for ti by t j.

For the “if” part, it is sufficient that there is mutual certain belief of the beliefs that each player has about the action choice of his opponentat each of her information sets. We do not need the stronger conditionthat (1) entails. Hence, higher order certain belief plays no role in thecharacterization, in line with the fundamental insights of Aumann andBrandenburger (1995).




Defining sequential rationalizability. We next define the conceptof sequentially rationalizable behavior strategies as induced behaviorstrategies under common certain belief of [isr].

Definition 15 A behavior strategy σi for i is sequentially rationalizable in a finite extensive two-player game Γ if there exists an epistemic modelwith (t1, t2) ∈ CK([u] ∩ [isr]) such that σi is induced for ti by t j .

It follows from Proposition 30 that a behavior strategy is sequentiallyrationalizable if it is part of a profile of behavior strategies that canbe extended to a sequential equilibrium. Since a sequential equilibrium

always exists, we obtain as an immediate consequence that sequentiallyrationalizable behavior strategies always exist.For the concept of sequential rationalizability—as indeed, throughout

the book—we restrict our attention to games with two players. A naturalquestion which arises is whether, and if so how, the present analysis canbe extended to the case of three or more players. In order to illustratethe potential difficulties of such an extension, consider a three playergame in which player 3 has an information set h with two nodes, xand y, where x is preceded by the player 1 action a and the player 2action c, and y is preceded by the player 1 action b and the player 2action d. Suppose that player 3 views b and c as suboptimal choices,and hence player 3 deems a infinitely more likely than b, and deems dinfinitely more likely than c. Then, player 3’s LPS at h over player 1’sstrategy choice and player 3’s LPS at h over player 2’s strategy choice donot provide sufficient information to derive player 3’s relative likelihoodsattached to nodes x and y, and these relative likelihoods are crucial toassess player 3’s rational behavior at h. Hence, in addition to the twoLPSs mentioned above, we need another aggregated LPS for player 3 ath over his opponents’ collective strategy profiles.

The key problem would then be what restrictions to impose upon theconnection between the LPSs over individual strategies on the one handand the aggregate LPS over strategy profiles on the other hand. Bothclasses of LPSs are needed, since the former are crucial in order to evalu-

ate the beliefs about rationality of individual players, and the latter areneeded in order to determine the conditional preferences of each player,as shown above. This issue is closely related to the problem of howto characterize consistency of assessments in algebraic terms, withoutthe use of sequences; cf. McLennan (1989a, 1989b), Battigalli (1996b),Kohlberg and Reny (1997), and Perea et al. (1997). In these papers,the consistency requirement for assessments has been characterized by



Sequentiality 107

means of conditional probability systems, relative probability systemsand lexicographic probability systems, satisfying some appropriate ad-ditional conditions. Perea et al. (1997), for instance, use a refinementof LPS in which, at every information set, not only an LPS over theavailable actions is defined, but moreover the relative likelihood levelbetween actions is “quantified” by an additional parameter, wheneverone action is deemed infinitely more likely than the other. This addi-tional parameter makes it possible to derive a unique aggregate LPS overaction profiles (and hence also over strategy profiles). A similar approachcan be found in Govindan and Klumpp (2002). Such an approach couldpossibly be useful when extending our analysis of, e.g., sequentiality to

the case of more than two players. For the moment, we leave this issuefor future research.

8.3 Weak sequential consistency

In the previous section we have shown how imposing that each playerbelieves that the opponent chooses rationally at all her information setscan be used to characterize sequential equilibrium and define sequentialrationalizability. Table 2.2 suggests the following claim: Imposing thateach player believes that the opponent chooses rationally only at herreachable information sets can be used to characterize the notion of weak sequential rationalizability , due to Ben-Porath (1997) and coined ‘weak

extensive form rationalizablity’ by Battigalli and Bonanno (1999). Inthis section we verify this claim and shed light on the difference betweensequentiality and weak sequentiality.

Inducing weak sequential rationality. Recall from Chapter 5 thatthe mixed strategy p j

ti|tj is induced for t j by ti if t j ∈ T jti and, for all

s j ∈ S j ,

p jti|tj (s j ) =

µti (s j, t j)

µti (S j, t j)

,

where is the first level of λti for which µti (S j , t j ) > 0.

Say that a mixed strategy pi is weak sequentially rational for i at ti

if, ∀h ∈ H i s.t. supp pi ∩ S i(h) = ∅, supp pi ∩ S i(h) ⊆ S tii (h), and define

the event that player i is of a type that i nduces a w eakly sequentiallyr ational mixed strategy for any opponent type that is deemed subjec-tively possible:

[iwri] := {(t1, t2) ∈ T 1 × T 2| ∀t j ∈ T j

ti ,

p jti|tj is weak sequentially rational for t

j} .




Write [iwr] := [iwr1] ∩ [iwr2].Say that at ti player i’s preferences over his strategies are weak sequen-

tially consistent with the game Γ and the preferences of his opponent , if ti ∈ projT i

([ui] ∩ [iwri]). Refer to [u] ∩ [iwr] as the event of weak sequen-tial consistency .

Note that the mixed strategy induced for t j by ti may be interpreted

as i’s initial belief at ti about the behavior of t j. In contrast to the

behavior strategy induced for t j by ti, as defined in the previous section,

the induced mixed strategy gives no information about how i at ti reviseshis belief about the behavior of t

j at player j information sets thatare unreachable given i’s initial belief at ti about t

j’s behavior. Hence,

if ti ∈ projT i [iwri], then player i believes at ti that each subjectivelypossible opponent type t

j chooses rationally at player j information setsthat do not contradict i’s initial belief at ti about the behavior of t

j.However, and this is the crucial difference when compared to the casewhere ti ∈ projT i

[isri]: ti ∈ projT i[iwri] entails no restriction on how

i at ti revises his beliefs about t j’s behavior conditional on t

j reaching“surprising” information sets. The above observation explains why weaksequentially rationalizable mixed strategies can be shown to correspondto induced mixed strategies under common certain belief of [u] ∩ [iwr].

Characterizing weak sequential rationalizability. We first de-

fine weak sequential rationalizability. Since weak sequential rationaliz-ability in two-player games corresponds to iterated elimination of strate-gies that are strongly dominated at some reachable information set, weuse the latter procedure as the primitive definition. For any (∅ =)X = X 1 × X 2 ⊆ S , write b(X ) := b1(X 2) × b2(X 1), where

bi(X j) := S i \ {si ∈ S i| ∃ pi ∈ ∆(S i) s.t. pi strongly dominates si on X j

or ∃h ∈ H i with S i(h) si and q i ∈ ∆(S i(h))

s.t. q i strongly dominates si on S j(h)} .

If pi is a mixed strategy and h ∈ H i satisfies that supp pi ∩ S i(h) = ∅,then let pi|h be defined by

pi|h(si) =

pi(si) pi(S i(h)) if si ∈ S i(h)

0 otherwise .

Definition 16 Let Γ be a finite extensive two-player game. Con-sider the sequence defined by X (0) = S 1 × S 2 and, ∀g ≥ 1, X (g) =b(X (g − 1)). A pure strategy si is said to be weak sequentially rational-



Sequentiality 109

izable if si ∈ W i :=

∞

g=0X i(g) .

A mixed strategy pi is said to be weak sequentially rationalizable if pi

is not strongly dominated on W j and there does not exist h ∈ H i withsupp pi ∩ S i(h) = ∅ such that pi|h is strongly dominated on S j (h).

While any pure strategy in the support of a weak sequentially rationaliz-able mixed strategy is itself weak sequentially rationalizable, the mixtureover a set of weak sequentially rationalizable pure strategies need not beweak sequentially rationalizable.


Lemma 12 (i) For each i, W i = ∅. (ii) W = b(W ). (iii) For each i,si ∈ W i if and only if there exists X = X 1 × X 2 with si ∈ W i such that X ⊆ b(X ).

We next characterize the concept of weak sequentially rationalizablemixed strategies as induced mixed strategies under common certain be-lief of [u] ∩ [iwr].

Proposition 31 A mixed strategy pi for i is weak sequentially rational-izable in a finite extensive two-player game Γ if and only if there exists an epistemic model with (t1, t2) ∈ CK([u] ∩ [iwr]) such that pi is induced for ti by t j.

Proof. Part 1: If p∗i is weak sequentially rationalizable, then there

exists an epistemic model with (t∗1, t∗

2) ∈ CK([u] ∩ [iwr]) such that p∗i is

induced for t∗i by t∗

j .Step 1: Construct an epistemic model with T 1 × T 2 ⊆ CK([u] ∩ [iwr])

such that for each si ∈ W i of any player i, there exists ti ∈ T i with,si ∈ S tii . Construct an epistemic model with, for each i, a bijectionsi : T i → W i from the set of types to the the set of weak sequentiallyrationalizable pure strategies. Assume that, for each ti ∈ T i of any playeri, υ ti

i satisfies that



(b) λti = (µti1 , . . . , µti

L ) with support S j × T jti satisfies that suppµt1

1 ∩(S j × {t j}) = {(s j (t j), t j)} for all t j ∈ T j

ti (so that, ∀t j ∈ T jti ,

piti|tj(s j(t j)) = 1),

(c) ∀E j ∈ S j×T j such that E j∩(S j × T jti) = ∅, ti(E j) = min{| suppλti

=∅} (so that, by Corollary 1, the SCLP corresponds to a CPS).





included in W j. By properties (a) and (c) and Lemmas 4 and 12(ii),we can still choose µti

1 (and T iti) so that si(ti) ∈ S tii . Since information

sets correspond to strategically independent sets (cf. the discussion inconnection with Lemmas 11 and 13) we have that, ∀h ∈ H i s.t. S i(h) si(ti) and suppµti

1 ∩ S j(h) = ∅, si(ti) ∈ S tii (h), while, ∀h ∈ H i s.t. S i(h) si(ti) and suppµti

1 ∩S j (h) = ∅, si(ti) ∈ S tii (h) by choosing the lower levelsof λti appropriately (again invoking properties (a) and (c) and Lemmas 4and 12(ii)). This combined with property (b) means that T 1×T 2 ⊆ [iwr].Furthermore, T 1 × T 2 ⊆ CK([u] ∩ [iwr]) since T j

ti ⊆ T j for each ti ∈ T iof any player i. Since, for each player i, si is onto W i, it follows that,

for each si ∈ W i of any player i, there exists ti ∈ T i with si ∈ S tii .Step 2: Add type t∗

i to T i. Assume that υit∗i satisfies (a) and (λt∗i , t∗i )

satisfies (b) and (c). Then µ1t∗i can be chosen so that p∗

i ∈ ∆(S it∗i ), and

consequently, ∀h ∈ H i s.t. supp p∗i ∩ S i(h) = ∅ and suppµ1

t∗i ∩ S j(h) = ∅,supp p∗

i ∩ S i(h) ⊆ S it∗i (h), while, ∀h ∈ H i s.t. supp p∗

i ∩ S i(h) = ∅ andsuppµ1

t∗i ∩ S j(h) = ∅, supp p∗i ∩ S i(h) ⊆ S i

t∗i (h) by choosing the lowerlevels of λt∗i appropriately. Furthermore, (T i ∪ {t∗

i }) × T j ⊆ [u] ∩ [iwr],and since T j

t∗i ⊆ T j, (T i ∪ {t∗i }) × T j ⊆ CK([u] ∩ [iwr]).

Step 3: Add type t∗ j to T j . Assume that υ j

t∗j satisfies (a) and the SCLP(λt∗j , t∗j ) on S i × (T i ∪ {t∗

i }) has the property that λt∗j = (µ1t∗j , . . . , µL

t∗j )with support S i×{t∗

i } satisfies that, ∀si ∈ S i, µ1t∗j (si, t∗

i ) = p∗i (si), so that

p∗i is induced for t∗i by t∗ j . Furthermore, (T i ∪ {t∗i }) × (T j ∪ {t∗ j }) ⊆ [u] ∩[iwr], and since T i

t∗j ⊆ T i ∪{t∗i }, (T i ∪{t∗

i })×(T j ∪{t∗ j }) ⊆ CK([u]∩[iwr]).

Hence, (t∗1, t∗

2) ∈ CK([u] ∩ [iwr]) and pi is induced for t∗i by t∗

j .Part 2: If there exists an epistemic model with (t∗

1, t∗2) ∈ CK([u] ∩

[iwr]) such that p∗i is induced for t∗

i by t∗ j , then p∗

i is weak sequentially rationalizable.


2) ∈ CK([u] ∩[iwr]) such that p∗


j . In particular, CK([u]∩[iwr]) =∅. Let, for each i, T i := projT i

CK([u] ∩ [iwr]) and

X i :=

ti∈T i{si ∈ S i|∀h ∈ H i s.t. S i(h) si, si ∈ S tii (h)} .

By Proposition 20(ii), for each ti ∈ T i of any player i, ti deems (s j, t j)subjectively impossible if t j ∈ T j\T j since CK([u] ∩ [iwr]) = KCK([u] ∩[iwr]) ⊆ KiCK([u] ∩ [iwr]), implying T j

ti ⊆ T j. By the definitions of [u] and [iwr], it follows that, for each ti ∈ T i of any player i, ti isrepresented by υti

i satisfying that υtii ◦z is a positive affine transformation

of ui and an LPS λti = (µti

1 , . . . , µti ), where = (S j × T j) ≥ 1, and

where suppµti1 ⊆ X j × T j. Hence, by Lemma 4, for each ti ∈ T i of any



Sequentiality 111

2 1

d D f F

33

00

11

d f

D

F

1, 1 0, 0

1, 1 3, 3

Figure 8.1. Γ6 and its strategic form.

player i, if pi ∈ ∆(S i) satisfies that, ∀h ∈ H i s.t. supp pi ∩ S i(h) = ∅,supp pi ∩ S i(h) ⊆ S tii (h), then

no strategy in the support of pi is strongly dominated on X j , sincethen pi ∈ ∆(S tii ), and it follows from pi ∈ ∆(S tii ) and suppµti

1 ⊆X j × T j that, ∀si ∈ supp pi and ∀s

i ∈ S i,sj∈X j

tj∈T j


sj∈X j

tj∈T j


i, s j) ,

∀h ∈ H i s.t. supp pi ∩ S i(h) = ∅, no strategy in supp pi ∩ S i(h) isstrongly dominated on S j (h) since supp pi ∩ S i(h) ⊆ S tii (h).

This implies X ⊆ b(X ), entailing by Lemma 12(iii) that, for each i, X i ⊆W i. Furthermore, since (t∗

1, t∗2) ∈ CK([u] ∩ [iwr]) and the mixed strategy

induced for t∗

i

by t∗

j

, p∗

i

, satisfies that, ∀h ∈ H i s.t. supp p∗

i

∩ S i(h) = ∅,supp p∗

i ∩ S i(h) ⊆ S iti(h), it follows that p∗

i is not strongly dominated onX j ⊆ W j and there does not exist h ∈ H i with supp p∗

i ∩ S i(h) = ∅ suchthat pi|h is strongly dominated on S j(h). By Definition 16 this impliesthat p∗

i is a weak sequentially rationalizable mixed strategy.

The following observation (which is stated without proof) can now beused to establish the relationships between the rationalizability conceptson the lower row of Table 2.2.

Proposition 32 For any epistemic model and for each player i,

[isri] ⊆ [iwri] ⊆ [iri] .

Since [iwri] ⊆ [iri], Propositions 31 and 22 entail that weak sequentialrationalizability refines (ordinary) rationalizability, and since [isri] ⊆[iwri], Definition 15 and Proposition 31 entail that sequential rational-izability refines weak sequential rationalizability. That the two latter in-clusions can be strict, is illustrated by Γ6 and Γ

6 of Figures 8.1 and 8.2,respectively. In Γ6 rationalizability does not have any bite, while weak




1 2 1 3F f F 3

D d D

2 1 02 1 0

d f

D

FD

FF

2, 2 2, 2

1, 1 0, 0

1, 1 3, 3

Figure 8.2. Γ6 and its pure strategy reduced strategic form.

sequential rationalizability promotes that player 1 plays F and player 2plays f . In Γ

6—introduced by Reny (1992, Figure 1)—weak sequentialrationalizability only precludes the play of D at 1’s second decision node.This can be established by applying the Dekel-Fudenberg procedure (i.e.,one round of weak elimination followed by iterated strong elimination)which eliminates a strategy if and only if it is not permissible. Since allterminal nodes yield different payoffs, weak sequential rationalizabilityleads to the same conclusion.1 However, only the play of F at bothof 1’s decision nodes and the play of f at 2’s single decision node aresequentially rationalizable. This follows from Proposition 33 of the nextsection, showing that the latter concepts imply the backward induction procedure .

Extensive form rationalizability (EFR), cf. Pearce (1984) as well

as Battigalli (1997) and Battigalli and Siniscalchi (2002), is an iterativedeletion procedure where, at any information set reached by a remainingstrategy, any deleted strategy is deemed infinitely less likely than someremaining strategy. Even though EFR only requires players to chooserationally at reachable information sets and preference for cautious be-havior is not imposed, EFR is different from weak sequential rationaliz-ability. Unlike all concepts in Tables 2, EFR yields forward induction incommon examples like the “battle-of-the-sexes with an outside option”game, see Figure 2.6.2 EFR also leads to the backward induction out-

1To see how the characterization in Proposition 31 of weak sequential rationalizability isconsistent with (D, d) in Γ6, let T 1 = {t1} with λt1 = ((1, 0), (0, 1)) (assigning probabilities

to (d, t2) and (f, t2) respectively), and T 2 = {t2} with λt2 = ((1, 0, 0), (0, 1, 0), (0, 0, 1))(assigning probabilities to (D, t1), (F D , t1), and (F F , t1) respectively). Then, independentlyof how t1 and t2 are specified, (t1, t2) ∈ CK([u] ∩ [iwr]), and, for each i, pi is induced for

ti by tj , where p1(D) = 1 and p2(d) = 1.2By strengthening permissibility, Asheim and Dufwenberg (2003a) define a rationalizability

concept, fully permissible sets, which is different from those of Table 2.2 as well as EFR, asit yields forward induction, but does not always promote backward induction. This conceptwill be presented in Chapters 11 and 12.



Sequentiality 113

come . However, unlike sequential rationalibility, EFR need not promotethe backward induction procedure.

8.4 Relation to backward induction

The following result shows how sequential rationalizability implies thebackward induction procedure in perfect information games. A finiteextensive game Γ, as introduced in Section 8.1, is of perfect information if, at any information set h ∈ H 1 ∪ H 2, h = {x}; i.e., h contains onlyone node. It is generic if, for each i, υi(z) = υi(z) whenever z and z

are different outcomes. A generic extensive game of perfect informationhas a unique subgame-perfect equilibrium in pure strategies. Moreover,

in such games the backward induction procedure yields in any subgamethe unique subgame-perfect equilibrium outcome.

Proposition 33 Consider a finite generic extensive two-player game of perfect information Γ. If there exists an epistemic model with (t1, t2) ∈CK([u]∩[isr]) and, for each i, σi is induced for ti by t j, then σ = (σ1, σ2)is the subgame-perfect equilibrium.

Proof. In a perfect information game, the action a ∈ A(h) taken atthe information set h determines the immediate succeeding informationset, which can thus be denoted (h, a). Also, any information set h ∈H 1 ∪ H 2 determines a subgame. Set H −1 = Z (i.e. the set of terminalnodes) and determine H g for g ≥ 0 by induction: h ∈ H g if and only if h satisfies

max{g| ∃h ∈ H g

and a ∈ A(h) such that h = (h, a)} = g − 1 .

In words, h ∈ H g if and only if g is the maximal number of decisionnodes between h and a terminal node in the subgame determined byh. If σ is a profile of behavior strategies and h ∈ H 1 ∪ H 2, denote byσ|h the strategy profile with the following properties: (1) at informationsets preceding h, σ |h determines with probability one the unique action

leading to h, and (2) at all other information sets, σ|h coincides withσ. Say that σ is outcome-equivalent to σ if σ and σ induce the sameprobability distribution over terminal nodes.

In view of properties of the certain belief operator (cf. Proposition 20of Chapter 5), it is sufficient to show for any g = 0, . . . , max{g|H g

= ∅}

that if there exists an epistemic model with (t1, t2) ∈ Kg([u] ∩ [isr]) and,for each i, σi is induced for ti by t j, then, ∀h ∈ H g, σ|h is outcome-




equivalent to σ∗|h, where σ∗ = (σ∗1, σ∗

2) denotes the subgame-perfectequilibrium. This is established by induction.

(g = 0) Let (t1, t2) ∈ K0([u] ∩ [isr]) = [u] ∩ [isr] and, for each i, σi

be induced for ti by t j. Let h ∈ H 0 and assume w.l.o.g. that h ∈ H i.Since (t1, t2) ∈ [ui] ∩ [isr j ] and j takes no action at h, σ|h is outcomeequivalent to σ∗|h.

(g = 1, . . . , max{g|H g

= ∅}) Suppose that it has been established forg = 0, . . . , g − 1 that if there exists an epistemic model with (t1, t2) ∈Kg([u] ∩ [isr]) and, for each i, σi is induced for ti by t j , then, ∀h ∈ H g

,

σ|h is outcome-equivalent to σ∗|h . Let (t1, t2) ∈ Kg([u] ∩ [isr]) and,for each i, σi be induced for ti by t j . Let h ∈ H g and assume w.l.o.g.

that h ∈ H i. Since (t1, t2) ∈ KiKg−1[isr], it follows from the premise of the inductive step that ti’s SCLP (λti , ti) satisfies, ∀t

j ∈ T ti j , ∀h ∈ H jsucceeding h, and ∀a ∈ A(h),

µti (S j(h, a), t

j )

µti (S j (h), t

j)= σ∗

j (h)(a) ,

where is the first level of λti for which µti (S j(h), t

j ) > 0. Since Γis generic, σi is sequentially rational for ti only if σi(h) = σ ∗

i (h). Since(t1, t2) ∈ [ui] ∩ [isr j] and j takes no action at h, it follows from thepremise that σ |h is outcome-equivalent to σ ∗|h.

Since sequentially rationalizable strategies always exist, there is anepistemic model with (t1, t2) ∈ CK([u] ∩ [isr]), implying that the resultof Proposition 33 is not empty.



Chapter 9

QUASI-PERFECTNESS

In Chapter 5 we saw how the characterizations of Nash equilibriumand rationalizability lead to characterizations of (strategic form) perfectequilibrium and permissibility by adding preference for cautious behav-ior. In this chapter we show that the characterization of sequentialequilibrium leads to a characterization of quasi-perfect equilibrium byadding caution. The concept of a quasi-perfect equilibrium, proposed byvan Damme (1984), differs from Selten’s (1975) extensive form perfect

equilibrium by the property that, at each information set, the playertaking an action ignores the possibility of his own future mistakes.

So, parallelling Chapter 8, we define quasi-perfect rationalizability byimposing common certain belief of the event that each player has pref-erence for cautious behavior (i.e., at every information set, one strat-egy is preferred to another if the former weakly dominates the latter)and believes that the opponent chooses rationally at all her informationsets. Moreover, by assuming that each player is certain of the beliefsthat the opponent has about the player’s own action choice, we obtainan epistemic characterization of the corresponding equilibrium concept:quasi-perfect equilibrium . Since quasi-perfect rationalizability refines se-

quential rationalizability, it follows from Proposition 33 that also theformer concept yields the backward induction procedure.

By embedding the notion of an SCLP in an epistemic model with a setof epistemic types for each player, we are able to model quasi-perfectnessas a special case of sequentiality. For each type ti of any player i, ti isdescribed by an SCLP, which under the event that “player i believesthat the opponent j chooses rationally at each information set” induces,




for each opponent type t j that is deemed subjectively possible by ti, abehavior strategy which is sequentially rational given t j ’s own SCLP.An SCLP ensures well-defined conditional beliefs representing nontriv-ial conditional preferences, while allowing for flexibility w.r.t. whether toassume preference for cautious behavior. Preference for cautious behav-ior, as needed for quasi-perfect rationalizability, is obtained by imposingthe following additional requirement on ti’s SCLP for each conditioningevent: If an opponent strategy-type pair (s j , t j) is compatible with theevent and t j is deemed subjectively possible by ti, then (s j , t j ) is in thesupport the LPS that represents type ti’s conditional preferences.

This chapter’s definition of quasi-perfect rationalizability was pro-

posed by Asheim and Perea (2004).

9.1 Quasi-perfect consistency

In this section, we add preference for cautious behavior to the analysisof Chapter 8. This enables us to

characterize quasi-perfect equilibrium (van Damme, 1984), and

define quasi-perfect rationalizability as a non-equilibrium analog tothe concept of van Damme (1984).

The epistemic modeling is identical to the one given in Section 8.1; hence,this will not be recapitulated here.

Caution. Under Assumption 1 it follows from Proposition 5 that,for each type ti of any player i, i’s system of conditional preferences atti can be represented by a vNM utility function υti

i : ∆(Z ) → R and anSCLP (λti , ti) on S j × T j with support S j × T j

ti . Recall from Section5.3 that caution imposes the additional requirement that for each typeti of any player i the full LPS λti is used to form the conditional beliefsover opponent strategy-type pairs. Formally, if L denotes the number of levels in the LPS λti , then

[caui] = {(t1, t2) ∈ T 1 × T 2| ti(S j × T j) = L} .

Since ti is non-increasing w.r.t. set inclusion, ti ∈ projT i [caui] impliesthat ti(projS j×T j

φ) = L for all subsets φ of {ti} × S j × T j with well-defined conditional beliefs. Since it follows from Assumption 1 that λti

has full support on S j, ti ∈ projT i[caui] means that i’s choice function at

ti never admits a weakly dominated strategy, thereby inducing preferencefor cautious behavior.

As before, write [cau] := [cau1] ∩ [cau2].



Quasi-perfectness 117

Say that at ti player i’s preferences over his strategies are quasi-perfectly consistent with the game Γ and the preferences of his opponent ,if ti ∈ projT i

([ui] ∩ [isri] ∩ [caui]). Refer to [u] ∩ [isr] ∩ [cau] as the eventof quasi-perfect consistency .

Characterizing quasi-perfect equilibrium. We now characterize the concept of a quasi-perfect equilibrium as profiles of induced behaviorstrategies at a type profile in [u] ∩ [isr] ∩ [cau] where there is mutualcertain belief of the type profile (i.e., for each player, only the trueopponent type is deemed subjectively possible). To state the definition of quasi-perfect equilibrium, we need some preliminary definitions. Define

the concepts of a behavior representation of a mixed strategy and themixed representation of a behavior strategy in the standard way, cf., e.g.,p. 159 of Myerson (1991). If a behavior strategy σ j and a mixed strategy p j are both completely mixed, and σ j is a behavior representation of p j

or p j is the mixed representation of σ j, then, ∀h ∈ H j, ∀a ∈ A(h),

σ j (h)(a) = p j(S j(h, a))

p j(S j(h)) .

If σi is any behavior strategy for i and σ j is a completely mixed behaviorstrategy for j, then abuse notation slightly by writing, for each h ∈ H i,

ui(σi, σ j)|h := ui( pi, p j|h) ,where pi is outcome-equivalent to σi|h and p j is the mixed representationof σ j.

Definition 17 A behavior strategy profile σ = (σ1, σ2) is a quasi-perfect equilibrium if there is a sequence (σ(n))n∈

of completely mixedbehavior strategy profiles converging to σ such that for each i and everyn ∈ N and h ∈ H i,

ui(σi, σ j (n))|h = maxσi

ui(σi, σ j(n))|h .

The characterization result can now be stated; it is proven in AppendixB.

Proposition 34 Consider a finite extensive two-player game Γ. A profile of behavior strategies σ = (σ1, σ2) is a quasi-perfect equilibrium if and only if there exists an epistemic model with (t1, t2) ∈ [u] ∩ [isr] ∩ [cau]such that (1) there is mutual certain belief of {(t1, t2)} at (t1, t2), and (2) for each i, σi is induced for ti by t j.




As for Proposition 31, higher order certain belief plays no role in thischaracterization.

Defining quasi-perfect rationalizability. We next define the con-cept of quasi-perfectly rationalizable behavior strategies as induced be-havior strategies under common certain belief of [u] ∩ [isr] ∩ [cau].

Definition 18 A behavior strategy σi for i is quasi-perfectly rational-izable in a finite extensive two-player game Γ if there exists an epistemicmodel with (t1, t2) ∈ CK([u] ∩ [isr] ∩ [cau]) such that σi is induced forti by t j.

It follows from Proposition 34 that a behavior strategy is quasi-perfectlyrationalizable if it is part of a quasi-perfect equilibrium. Since a quasi-perfect equilibrium always exists, we obtain as an immediate conse-quence that quasi-perfectly rationalizable behavior strategies always ex-ist.

Propositions 30 and 34 imply the well-known result that every quasi-perfect equilibrium can be extended to a sequential equilibrium, whileDefinitions 15 and 18 imply that the set of quasi-perfectly rationalizablestrategies is included in the set of sequentially rationalizable strategies.To illustrate that this inclusion can be strict, consider Γ4 of Figure 3.1.Both concepts predict that player 2 plays d with probability one. How-

ever, only quasi-perfect rationalizability predicts that player 1 plays Dwith probability one. Preferring D to F amounts to preference for cau-tious behavior since by choosing D player 1 avoids the risk that player2 may choose f .

Since quasi-perfect rationalizability is thus a refinement of sequen-tial rationalizability, it follows from Proposition 33 that quasi-perfectrationalizability implies the backward induction procedure in perfect in-formation games.

9.2 Relating rationalizability concepts

The following result helps establishing some of the remaining relation-

ships between the rationalizability concepts of Table 2.2.

Proposition 35 For any epistemic model and for each player i,

[iri] ∩ K i[cau j] ⊆ [iwri] .

To prove Proposition 35 we need the following lemma.



Quasi-perfectness 119

Lemma 13 If ti ∈ projT iKi[cau j ], then, for each t j ∈ T j

ti and any h ∈ H j, s j ∈ S j(h)\S j

tj(h) implies that there exists s j ∈ S j(h) such that

s j tj s j .

Proof. As for Lemma 11 the proof of this lemma is based on the conceptof a strategically independent set due to Mailath et al. (1993). It followsfrom Mailath et al. (Definitions 2 and 3 and the ‘if’ part of Thm. 1)that S (h) is strategically independent for j at any player j informationset h in a finite extensive game, and this does not depend on the vNMutility function that assigns payoff to any outcome.

If ti ∈ projT iKi[cau j], then the following holds for each t j ∈ T j

ti :

Player j ’s system of conditional preferences at t j satisfies Axiom 6 (Con-ditionality). Suppose s j ∈ S j (h)\S j

tj(h). Then there exists s j ∈ S j (h)

such that s j

tjh s j. As noted above, S (h) is a strategically independent

set for j. Hence, s j can be chosen such that z(s

j, si) = z(s j, si) for allsi ∈ S i\S i(h). By Axiom 6 (Conditionality), this implies s

j tj s j.

Proof of Proposition 35. Consider any epistemic model with

ti ∈ projT i([iri] ∩ Ki[cau j ]) .

Suppose ti /∈ projT i[iwri]; i.e., there exist t j ∈ T j

ti and h ∈ H j such that p j

ti|tj(s j ) > 0 for some s j ∈ S j(h)\S jtj(h). Since ti ∈ projT i

Ki[cau j ], it

follows from Lemma 13 that ∃s j ∈ S j (h) s.t. s

j

tj

s j . Hence,

p jti|tj /∈ ∆(S j

tj ) ,

contradicting ti ∈ projT i[iri]. This shows that ti ∈ projT i

[iwri].

Since [iri] ∩ K i[cau j ] ⊆ [iwri], the cell in Table 2.2 to the left of ‘permissibility’ is not applicable, and permissibility refines weak sequen-tial rationalizability. Figure 3.1 shows that the inclusion can be strict:Permissibility, but not weak sequential rationalizability, precludes thatplayer 1 plays F in Γ4.

Since [isri] ⊆ [iri], Definition 18 and Proposition 24 entail that quasi-

perfect rationalizability refines permissibility. That the latter inclusioncan be strict is illustrated by Γ6 of Figure 8.2. Since this is a generic ex-

tensive game, imposing preference for cautious behavior has no bite, andthe difference between permissibility and quasi-perfect rationalizabilitycorresponds to the difference between weak sequential rationalizabilityand sequential rationalizability, as discussed in Section 8.3.





Chapter 10

PROPERNESS

Most contributions on the relation between common knowledge/belief of rationality and backward induction in perfect information games per-form the analysis in the extensive form of the game. Indeed, the ana-lyses in Chapters 7 and 8 of this book are examples of this. An ex-ception to this rule is Schuhmacher (1999) who—based on Myerson’s(1978) concept of a proper equilibrium , but without making equilib-rium assumptions—defines the concept of proper rationalizability in the

strategic form and shows that proper rationalizable play leads to back-ward induction.

Schuhmacher defines the concept of ε-proper rationalizability by as-suming that players make mistakes, but where more costly mistakesare made with a much smaller probability than less costly ones. Aproperly rationalizable strategy can then be defined as the limit of asequence of ε-properly rationalizable strategies as ε goes to zero. Fora given ε, Schuhmacher offers an epistemic foundation for ε-proper ra-tionalizability. However, this does not provide an epistemic foundationfor the limiting concept, i.e. proper rationalizability. It is one purposeof the present chapter, which reproduces Asheim (2001), to establishhow proper rationalizability can be given an epistemic characterizationin strategic two-player games, within an epistemic model where prefer-ences are represented by a vNM utility function and an SCLP (i.e., anepistemic model satisfying Assumption 1 of Chapter 5).

Blume et al. (1991b) characterize proper equilibrium as a propertyof preferences. When doing so they represent a player’s preferences




by a vNM utility function and an LPS, whereby the player may deemone opponent strategy to be infinitely more likely than another whilestill taking the latter strategy into account. In two-player games, theircharacterization of proper equilibrium can be described by the followingtwo properties.

1 Each player is certain of the preferences of his opponent,

2 Each player’s preferences satisfies that the player takes all opponentstrategies into account (‘caution’) and that the player deems one op-ponent strategy to be infinitely more likely than another if the oppo-nent prefers the one to the other (‘respect of opponent preferences’).

The present characterization of proper rationalizability in two-playergames drops property 1, which is an equilibrium assumption; instead itwill be assumed that there is common certain belief of property 2, whichwill be referred to as proper consistency.

Since, in the present framework, a player is not certain of the prefer-ences of his opponent, player i’s preferences must be defined on acts fromS j × T j, where S j denotes the set of opponent strategies and T j denotesthe set of opponent types. Under Assumption 1, each type of player icorresponds to a vNM utility function and an SCLP on S j × T j . As be-fore, a player i has preference for cautious behavior at ti if he takes intoaccount all strategies of any opponent type that is deemed subjectivelypossible. Moreover, a player i is said to respect opponent preferences at ti if, for any opponent type that is deemed subjectively possible, hedeems one strategy of the opponent type to be infinitely more likely thananother if the opponent type prefers the one to the other. At ti playeri’s preferences are said to be properly consistent with the game and thepreferences of his opponent if at ti i both has preference for cautious be-havior and respects opponent preferences. Hence, the present analysisfollows the ‘consistent preferences’ approach by imposing requirementson the preferences of players rather than their choice.

In this chapter it is first shown (in Proposition 36) how the eventof proper consistency combined with mutual certain belief of the type

profile can be used to characterize the concept of proper equilibrium.It is then established (in Proposition 37) that common certain belief of proper consistency corresponds to Schuhmacher’s (1999) concept of proper rationalizability. Furthermore, by relating ‘respect of preferences’to ‘inducement of sequential rationality’ in Proposition 38, it follows bycomparing Proposition 37 with Proposition 33 of Chapter 8 that onlystrategies leading to the backward induction outcome are properly ra-



Properness 123

c r

U

M

D

1, 1 1, 1 1, 0

1, 1 2, 2 2, 2

0, 1 2, 2 3, 3

Figure 10.1. G7, illustrating common certain belief of proper consistency.

tionalizable in the strategic form of a generic perfect information game.Thus, Schuhmacher’s Theorem 2 (which shows that the backward induc-tion outcome obtains with “high” probability for any given “small” ε) is

strengthened, and an epistemic foundation for the backward inductionprocedure, as an alternative to Aumann’s (1995) and others, is provided.Lastly, it is illustrated through an example how proper rationalizabilitycan be used to test the robustness of inductive procedures.

10.1 An illustration

The symmetric game of Figure 10.1 is an example where commoncertain belief of proper consistency is sufficient to determine completelyeach player’s preferences over his or her own strategies. The game is dueto Blume et al. (1991b, Figure 1).

In this game, caution implies that player 1 prefers M to U since

M weakly dominates U . Likewise, player 2 prefers c to . Since 1respects the preferences of 2 and, in addition, certainly believes that 2has preference for cautious behavior, it follows that 1 deems c infinitelymore likely than . This in turn implies that 1 prefers D to U . Likewise,since 2 respects the preferences of 1 and, in addition, certainly believesthat 1 has preference for cautious behavior, it follows that 2 prefers rto . As a consequence, since 1 respects the preferences of 2, certainlybelieves that 2 respects the preferences of 1, and certainly believes that2 certainly believes that 1 has preference for cautious behavior, it followsthat 1 deems r infinitely more likely than . Consequently, 1 prefers Dto M . A symmetric reasoning entails that 2 prefers r to c. Hence, if

there is common certain belief of proper consistency, it follows that theplayers’ preferences over their own strategies are given by

1’s preferences: D M U 2’s preferences: r c .

The facts that D is the unique most preferred strategy for 1 and r is theunique most preferred strategy for 2 mean that only D and r are properly




rationalizable; cf. Proposition 37 of Section 10.2. By Proposition 36 of the same section, it then follows that the pure strategy profile (D, r) isthe unique proper equilibrium, which can easily be checked. However,note that in the argument above, each player obtains certainty aboutthe preferences of his opponent through deductive reasoning; i.e. suchcertainty is not assumed as in the concept of proper equilibrium.

The concept of proper rationalizability yields a strict refinement of (ordinary) rationalizability (cf. Definition 11 of Chapter 5). All strate-gies for both players are rationalizable, which is implied by the factthat, in addition to (D, r), the pure strategy profiles (U, ) and (M, c)are also Nash equilibria. The concept of proper rationalizability yields

even a strict refinement when compared permissibility (cf. Definition 13of Chapter 5), corresponding to the Dekel-Fudenberg procedure, whereone round of weak elimination followed by iterated strong elimination.When the Dekel-Fudenberg procedure is employed, only U is eliminatedfor 1, and only is eliminated for 2, reflecting that also the pure strat-egy profile (M, c) is a strategic form perfect equilibrium. It is a generalresult that proper rationalizability refines the Dekel-Fudenberg proce-dure; this follows from Section 10.3 as well as Theorem 4 of Herings andVannetelbosch (1999).

10.2 Proper consistency

In this section, we add respect for opponent preferences to the analysisof Chapter 5. This enables us to characterize

proper equilibrium (Myerson, 1978), and

proper rationalizability (Schuhmacher, 1999).

The epistemic modeling is identical to the one given in Section 5.1; hence,this will not be recapitulated here.

Respect of opponent preferences. Player i respects the preferen-ces of his opponent at ti if the following holds for any opponent type thatis deemed subjectively possible: Player i deems one opponent strategy of

the opponent type to be infinitely more likely than another if the oppo-nent type prefers the one to the other. To capture this, define the event

[respi] := {(t1, t2) ∈ T 1 × T 2| (s j, t j) ti (s

j, t j)

whenever t j ∈ T i

ti and s j tj s j } ,

where the notation ti means “infinitely more likely at ti”, as definedin Section 3.2.



Properness 125

Write [resp] := [resp1] ∩ [resp2].Say that at ti player i’s preferences over his strategies are properly

consistent with the game G = (S 1, S 2, u1, u2) and the preferences of his opponent , if ti ∈ projT i

([ui] ∩ [respi] ∩ [caui]). Refer to [u]∩[resp]∩ [cau]as the event of proper consistency .

Characterizing proper equilibrium. We now characterize theconcept of a proper equilibrium as profiles of induced mixed strategiesat a type profile in [u] ∩ [resp] ∩ [cau] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent typeis deemed subjectively possible). Before doing so, we define a proper

equilibrium.Definition 19 Let G = (S 1, S 2, u1, u2) be a finite strategic two-playergame. A completely mixed strategy profile p = ( p1, p2) is a ε-proper equilibrium if, for each i,

εpi(si) ≥ pi(si) whenever ui(si, p j) > ui(s

i, p j ) .

A mixed strategy profile p = ( p1, p2) is a proper equilibrium if there isa sequence ( p(n))n∈

of ε(n)-proper equilibria converging to p, whereε(n) → 0 as n → ∞.

The characterization result—which is a variant of Proposition 5 of Blume et al. (1991b)—can now be stated. For this result, recall from

Sections 5.2 and 8.3 that the mixed strategy p jti|tj is induced for t j by ti if t j ∈ T j

ti and, for all s j ∈ S j ,

p jti|tj (s j ) =

µti (s j, t j)

µti (S j, t j)

,

where is the first level of λti for which µti (S j , t j ) > 0.

Proposition 36 Consider a finite strategic two-player game G. A profile of mixed strategies p = ( p1, p2) is a proper equilibrium if and only if there exists an epistemic model with (t1, t2) ∈ [u] ∩ [resp] ∩ [cau] such that (1) there is mutual certain belief of {(t1, t2)} at (t1, t2), and (2) for each i, pi is induced for ti by t j.

The proof is contained in Appendix B. As for similar earlier results,higher order certain belief plays no role in this characterization.

Characterizing proper rationalizability. We now turn to thenon-equilibrium analog to proper equilibrium, namely the concept of proper rationalizability; cf. Schuhmacher (1999). To define the conceptof properly rationalizable strategies, we must introduce the following




variant of an epistemic model, with a mixed strategy piti being associated

to each type ti of any player i, where piti is completely mixed.

Definition 20 An ∗-epistemic model for the finite strategic two-playergame form (S 1, S 2, z) is a structure

(S 1, T 1, S 2, T 2) ,

where, for each type ti of any player i, ti corresponds to (1) mixed strat-egy pi

ti , where supp piti = S i, and (2) a system of conditional preferences

on the collection of sets of acts from elements of

Φti := {φ ⊆ T i × S j × T j | κti ∩ φ = ∅}

to ∆(Z ), where κti is a non-empty subset of {ti} × S j × T j.

Moreover, Schuhmacher (1999) in effect makes the following assumption.

Assumption 3 For each ti of any player i, (a) tiφ satisfies Axioms 1,

2, and 4 if ∅ = φ ⊆ T i × S j × T j, and Axiom 3 if and only if φ ∈ Φti, (b)the system of conditional preferences {ti

φ | φ ∈ Φti} satisfies Axioms 5 and 6, and (c) there exists a non-empty subset of opponent types, T j

ti,such that κti = {ti} × S j × T j

ti.

Under Assumption 3 it follows from Proposition 1 that, for each typeti of any player i, i’s system of conditional preferences at ti can be

represented by a vNM utility function υtii : ∆(Z ) → R and a subjective

probability distribution µti which for expositional simplicity is defined onS j ×T j with support S j ×T j

ti (instead of being defined on T i×S j ×T j withsupport κti = {ti} × S j × T j

ti). Hence, as before we consider w.l.o.g. i’sunconditional preferences at ti, ti , to be preferences over acts fromS j × T j to ∆(Z ) (instead of acts from {ti} × S j × T j to ∆(Z )).

The combination of κti having full support on S i and Axiom 6 (Con-ditionality) being satisfied means that all opponent strategies are takeninto account for any opponent type that is deemed subjectively possi-ble, something that is reflected by µti having full support on S j . Hence,preference for cautious behavior need not be explicitly imposed. Rather,

following Schuhmacher (1999) we consider the following events. First,define the set of type profiles for which ti, for any subjectively possibleopponent type, ind uces that type’s mixed strategy:

[indi] :=

(t1, t2) ∈ T 1 × T 2

∀t j ∈ T j

ti , p jti|tj = p j

tj

.

Write [ind] := [ind1] ∩ [ind2]. Furthermore, define the set of type profilesfor which ti, according to his mixed strategy pi

ti , plays a pure strategy






Proof. Consider any epistemic model with

ti ∈ projT i([respi] ∩ Ki[cau j ]) .

Suppose ti /∈ projT i[isri]; i.e., there exist t j ∈ T j

ti and h ∈ H j suchthat σ j

ti|tj |h is outcome equivalent to p j , where p j (s j) > 0 for somes j ∈ S j(h)\S j

tj(h). Since ti ∈ projT iKi[cau j], it follows from Lemma

13 that ∃s j ∈ S j(h) s.t. s

j tj s j . Since ti ∈ projT i[respi], this means

that ∃s j ∈ S j (h) s.t. (s

j, t j) ti (s j, t j). Furthermore, p j(s j ) > 0 impliesµti

(s j, t j) > 0, where is the first level of λti for which µti (S j (h), t j) >

0. Since then is also the first level of λti for which µti ({s j , s

j}, t j ) > 0,this contradicts (s

j, t

j) ti (s

j, t

j) and shows that t

i ∈ proj

T i[isr

i].

Since proper rationalizability is thus a refinement of quasi-perfect ra-tionalizability, which in turn is a refinement of sequential rationalizabil-ity, it follows from Proposition 33 that proper rationalizability impliesthe backward induction procedure in perfect information games. E.g.,in the “centipede” game illustrated in Γ

3 of Figure 2.4, common certainbelief of proper consistency implies that the players’ preferences overtheir own strategies are given by

1’s preferences: Out InL OutR2’s preferences: r .

This property of proper rationalizability has been discussed by bothSchuhmacher (1999) and Asheim (2001).

From the proof of Proposition 1 in Mailath et al. (1997) one canconjecture that quasi-perfect rationalizability in every extensive formcorresponding to a given strategic game coincides with proper rational-izability in that game. However, for any given extensive form the setof proper rationalizable strategies can be a strict subset of the set of quasi-perfect rationalizable strategies, as illustrated by Γ

2 of Figure 2.5.Here, quasi-perfect rationalizability only precludes the play of InR withpositive probability. However, since InL strongly dominates InR, it fol-lows that 2 prefers to r if she respects 1’s preferences. Hence, only

with probability one is properly rationalizable for 2, which implies thatonly InL with probability one is properly rationalizable for 1.

10.4 Induction in a betting game

The games G7 (of Figure 10.1), Γ3 (of Figure 2.4), and Γ

2 (of Figure2.5) have in common that the properly rationalizable strategies coincidewith those surviving iterated (maximal) elimination of weakly domi-



Properness 129

a b cPlayer 1

Player 2

-9 6 -3

9 -6 31/3 1/3 1/3

Figure 10.2. A betting game.

nated strategies (IEWDS). In the present section it will be shown thatthis conclusion does not hold in general. Rather, the concept of proper

rationalizability can be used to test the robustness of IEWDS and otherinductive procedures.

Figure 10.2 illustrates a simplified version of a betting game intro-duced by Sonsino et al. (2000) for the purpose of experimental study;Søvik (2001) has subsequently repeated their experiment in alternativedesigns. The two players consider to bet and have a common and uni-form prior over the states that determine the outcome of the bet. If thestate is a, then 1 looses 9 and 2 wins 9 if betting occurs. If the state isb, then 1 wins 6 and 2 looses 6 if betting occurs. Finally, if the state isc, then 1 looses 3 and 2 wins 3 if betting occurs. Player 1 is informed of whether the state of the bet is equal to a or in the set {b, c}. Player 2 is

informed of whether the state of the bet is in the set {a, b} or equal to c.As a function of their information, each player can announce to acceptthe bet or not. For player 1 the strategy YN means to accept the bet if informed of a and not to accept the bet if informed of {b, c}, etc. Forplayer 2 the strategy yn means to accept the bet if informed of {a, b} andnot to accept the bet if informed of c, etc. Betting occurs if and onlyif both players have accepted the bet. This yields the strategic game of Figure 10.3.

An inductive procedure. If player 2 naively believes that player 1is equally likely to accept the bet when informed of a as when informedof {b, c}, then 2 will wish to accept the bet when informed of {a, b}.

However, the following, seemingly intuitive, inductive procedure appearsto indicate that 2 should never accept the bet if informed of {a, b}: Player1 should not accept the bet when informed of a since he cannot win bydoing so. This eliminates his strategies YY and YN . Player 2, realizingthis, should never accept the bet when informed of {a, b}, since—aslong as 1 never accepts the bet when informed of a—she cannot win bydoing so. This eliminates her strategies yy and yn . This in turn means




yy yn ny nn

YY

YN

NY

NN

-2, 2 -1, 1 -1, 1 0, 0

-3, 3 -3, 3 0, 0 0, 0

1, -1 2, -2 -1, 1 0, 0

0, 0 0, 0 0, 0 0, 0

Figure 10.3. The strategic form of the betting game.

that player 1, realizing this, should never accept the bet when informed

of {b, c}, since—as long as 2 never accepts the bet when informed of {a, b}—he cannot win by doing so. This eliminates his strategy NY .This inductive argument corresponds to IEWDS, except that the latterprocedure eliminates 2’s strategies yn and nn in the first round. Theargument seems to imply that player 2 should never accept the betif informed of {a, b} and that player 1 should never accept the bet if informed of {b, c}. Is this a robust conclusion?

Proper rationalizability in the betting game. The strategicgame of Figure 10.3 has a set of Nash equilibria that includes the purestrategy profiles (NN , ny ) and (NN , nn ), and a set of (strategic form)perfect equilibria that includes the pure strategy profile (NN,ny). How-ever, there is a unique proper equilibrium where player 1 plays NN withprobability one, and where player 2 mixes between yy with probability1/5 and ny with probability 4/5. It is instructive to see why the purestrategy profile (NN , ny ) is not a proper equilibrium. If 1 assigns prob-ability one to 2 playing ny , then he prefers YN to NY (since the moreserious mistake to avoid is to accept the bet when being informed of {b, c}). However, if 2 respects 1’s preferences and certainly believes that1 prefers YN to NY , then she will herself prefer yy to ny , undermining(NN , ny ) as a proper equilibrium. The mixture between yy and ny inthe proper equilibrium is constructed so that 1 is indifferent between YN and NY .

Since any mixed strategy is properly rationalizable if it is part of aproper equilibrium, it follows that both yy and yn are properly ratio-nalizable pure strategies for 2. Moreover, if 1 certainly believes that2 is of a type with only yy as a most preferred strategy, then NY isa most preferred strategy for 1, implying that NY in addition to NN is a properly rationalizable strategy for 1. That these strategies arein fact properly rationalizable is verified by the epistemic model of Ta-



Properness 131

Table 10.1. An epistemic model for the betting game.

t1 t

2 t2 t

1 t2 t

2

yy (0, 0, 1, 0) (0, 0, 0, 0) yy (0, 0, 0, 0) (1, 0, 0, 0)yn (0, 0, 0, 1) (0, 0, 0, 0) yn (0, 0, 0, 0) (0, 1, 0, 0)ny (1, 0, 0, 0) (0, 0, 0, 0) ny (0, 0, 0, 0) (0, 0, 1, 0)nn (0, 1, 0, 0) (0, 0, 0, 0) nn (0, 0, 0, 0) (0, 0, 0, 1)

t2 t

1 t1 t

2 t1 t

1

Y Y (0, 0, 0, 0) (0, 0, 1, 0) Y Y (0, 0, 0, 1) (0, 0, 0, 0)Y N (0, 0, 0, 0) (0, 0, 0, 1) Y N (0, 1, 0, 0) (0, 0, 0, 0)NY (0, 0, 0, 0) (1, 0, 0, 0) NY (0, 0, 1, 0) (0, 0, 0, 0)NN (0, 0, 0, 0) (0, 1, 0, 0) N N (1, 0, 0, 0) (0, 0, 0, 0)

ble 10.1. In the table the preferences of any player i at each type ti

are represented by a vNM utility function υtii satisfying υti

i ◦ z = ui

and a 4-level LPS on S j × {t j , t

j }, with the first numbers in the paran-theses expressing primary probability distributions, the second numbersexpressing secondary probability distributions, etc. It can be checkedthat {t

1, t1} × {t

2, t2} ⊆ [u] ∩ [resp] ∩ [cau], which in turn implies

{t1, t

1} × {t

2, t

2} ⊆ CK([u] ∩ [resp] ∩ [cau]) since, for each ti ∈ T i of

any player i, T jti ⊆ {t

j , t j }. Since each type’s preferences over his/her

own strategies are given by

N N t1 Y N t1 NY t1 Y Y

N Y t1 NN t1 Y Y t1 Y N

ny t2 nn t2 yy t2 yn

yy t2 yn t2 ny t2 nn ,

it follows that NY and NN are properly rationalizable for player 1 andyy and ny are properly rationalizable for player 2. Note that YY and YN for player 1 and yn and nn for player 2 cannot be properly rationalizable

since these strategies are weakly dominated and, thus, cannot be mostpreferred strategies for cautious players.

The lesson to be learned from this analysis is that is not obvious thatdeductive reasoning should lead players to refrain from accepting thebet in the betting game. The experiments by Sonsino et al. (2000)and Søvik (2001) show that some subjects do in fact accept the betin a slightly more complicated version of this game. By comparison to




Propositions 33 and 38, the analysis can be used to support the argumentthat backward induction in generic perfect information games is moreconvincing than the inductive procedure for the betting game discussedabove.



Chapter 11

CAPTURING FORWARD INDUCTION

THROUGH FULL PERMISSIBILITY

The procedure of iterated (maximal) elimination of weakly dominatedstrategies (IEWDS) has a long history and some intuitive appeal, yet itis not as easy to interpret as iterated elimination of strongly dominatedstrategies (IESDS). IESDS is known to be equivalent to common belief of rational choice; cf. Tan and Werlang (1988) as well as Propositions 22and 26 of this book. IEWDS would appear simply to add a requirementof admissibility , i.e., that one strategy should be preferred to another if the former weakly dominates the latter on a set of strategies that theopponent “may choose”. However, numerous authors—in particular,Samuelson (1992)—have noted that it is not clear that we can interpretIEWDS this way. To see this, consider the following two examples.

The left-hand side of Figure 2.6 shows G1, the pure strategy reduced

strategic form of the “battle-of-the-sexes with an outside option” game.Here IEWDS works by eliminating InR, r, and Out, leading to the for-ward induction outcome (InL, ). This prediction appears consistent: if 2 believes that 1 will choose InL, then she will prefer to r as 2’s pref-erence over her strategies depends only on the relative likelihood of InLand InR.

The situation is different in G8 of Figure 11.1, where IEWDS works

by eliminating D, r, and M , leading to (U, ). Since 2 is indifferentat the predicted outcome, we must here appeal to admissibility on asuperset of {U }, namely {U, M }, to justify the statement that 2 must play L. However, it is not clear that this is reasonable. Admissibility on{U, M } means that 2’s preferences respect weak dominance on this setand implies that M is deemed infinitely more likely than D (in the senseof Blume et al., 1991a, Definition 5.1; see also Chapter 3). However, why




r

U

M

D

1, 1 1, 1

0, 1 2, 0

1, 0 0, 1

Figure 11.1. G8, illustrating that IEWDS may be problematic.

should 2 deem M more likely than D? If 2 believes that 1 believes inthe prediction that 2 plays (as IEWDS suggests), then it seems odd to

assume that 2 believes that 1 considers D to be a less attractive choicethan M .A sense in which D is “less rational” than M is simply that it was

eliminated first. This hardly seems a justification for insisting on thebelief that D is much less likely than M . Still, Stahl (1995) has shownthat IEWDS effectively assumes this: a strategy survives IEWDS if andonly if it is a best response to a belief where one strategy is infinitely lesslikely than another if the former is eliminated at an earlier round thanthe latter. Thus, IEWDS adds extraneous and hard-to-justify restric-tions on beliefs, and may not appear to correspond to the most naturalformalization of deductive reasoning under admissibility. So what does?

Reproducing joint work with Martin Dufwenberg, cf. Asheim andDufwenberg (2003a), this chapter presents the concept of ‘fully permis-sible sets’ as an answer. In G

1 this concept agrees with the prediction of IEWDS, as seems natural. The procedure leading to this predictionis quite different, though, as is its interpretation . In G8, however, fullpermissibility predicts that 1’s set of rational choices is either {U } or {U, M }, while 2’s set of rational choices is either {} or {, r}. This hasinteresting implications. If 2 is certain that 1’s set is {U }, then—absentextraneous restrictions on beliefs—one cannot conclude that 2 prefers to r or vice versa. On the other hand, if 2 considers it possible that 1’sset is {U, M }, then weakly dominates r on this set and justifies {} as2’s set of rational choices. Similarly, one can justify that U is preferred

to M if and only if 1 considers it impossible that 2’s set is {, r}. Thus,full permissibility tells a consistent story of deductive reasoning underadmissibility, without adding extraneous restrictions on beliefs.

This chapter is organized as follows. Section 11.1 illustrates the keyfeatures of the requirement—called ’full admissible consistency’—that isimposed on players to arrive at full permissibility. Section 11.2 formallydefines the concept of fully permissible sets through an algorithm that



Capturing forward induction through full permissibility 135

eliminates strategy sets under full admissible consistency. General ex-istence as well as other properties are shown. Section 11.3 establishesepistemic conditions for the concept of fully permissible sets, and checksthat these conditions are indeed needed and thereby relates full permis-sibility to other concepts. Section 11.4 investigates examples, showinghow forward induction is promoted and how multiple fully permissiblesets may arise. Section 11.5 compares our epistemic conditions to thoseprovided in related literature. As elsewhere in this book, the analysiswill be limited to two-player games. In this chapter (and the next), thisis for ease of presentation, as everything can essentially be generalizedto n-player games (with n > 2).

11.1 Illustrating the key features

Our modeling captures three key features:

1 Caution. A player should prefer one strategy to another if the for-mer weakly dominates the latter. Such admissibility of a player’spreferences on the set of all opponent strategies is defended, e.g., inChapter 13 of Luce and Raiffa (1957) and is implicit in proceduresthat start out by eliminating all weakly dominated strategies.

2 Robust belief of opponent rationality. A player should deem any op-ponent strategy that is a rational choice infinitely more likely than

any opponent strategy not having this property. This is equivalentto preferring one strategy to another if the former weakly dominatesthe latter on the set of rational choices for the opponent. Such ad-missibility of a player’s preferences on a particular subset of oppo-nent strategies is an ingredient of the analyses of weak dominance bySamuelson (1992) and Borgers and Samuelson (1992), and is essen-tially satisfied by ‘extensive form rationalizability’ (EFR; cf. Pearce,1984 and Battigalli, 1996a, 1997) and IEWDS.

3 No extraneous restrictions on beliefs. A player should prefer onestrategy to another only if the former weakly dominates the latter

on the set of all opponent strategies or on the set of rational choicesfor the opponent. Such equal treatment of opponent strategies thatare all rational—or all irrational—have in principle been argued bySamuelson (1992, p. 311), Gul (1997), and Mariotti (1997).

These features are combined as follows. A player’s preferences overhis own strategies leads to a choice set (i.e., a set of maximal purestrategies; cf. Section 6.1). A player’s preferences is said to be fully




r

U

M

D

1, 1 1, 1

1, 1 1, 0

1, 0 0, 1

Figure 11.2. G9, illustrating the key features of full admissible consistency.

admissibly consistent with the game and the preferences of his opponentif one strategy is preferred to another if and only if the former weaklydominates the latter

on the set of all opponent strategies, or

on the union of the choice sets that are deemed possible for the op-ponent.

A subset of strategies is a fully permissible set if and only if it can bea choice set when there is common certain belief of full admissible con-sistency. Hence, the analysis yields a solution concept that determinesa collection of choice sets for each player. This collection can be foundvia a simple algorithm, introduced in the next section.

We use G9 of Fig. 11.2 to illustrate the consequences of imposing

‘caution’ and ‘robust belief of opponent rationality’. Since ‘caution’means that each player takes all opponent strategies into account, itfollows that player 1’s preferences over his strategies will be U ∼ M D(where ∼ and denote indifference and preference, respectively). Player1 must prefer each of the strategies U and M to the strategy D, becausethe former strategies weakly dominate D. Hence, U and M are maximal,implying that 1’s choice set is {U, M }.

The requirement of ‘robust belief in opponent rationality’ comes intoeffect when considering the preferences of player 2. Suppose that 2certainly believes that 1 is cautious and therefore (as indicated above)certainly believes that {U, M } is 1’s choice set. Our assumption that 2has robust belief of 1’s rationality captures that 2 deems each elementof {U, M } infinitely more likely than D. Thus, 2’s preferences respectweak dominance on 1’s choice set {U, M }, regardless of what happens if 1 chooses D . Hence, 2’s preferences over her strategies will be r.

Summing up, we get to the following solution for G9:

1’s preferences: U ∼ M D2’s preferences: r




Hence, {U, M } and {} are the players’ fully permissible sets.The third feature of full admissible consistency—‘no extraneous re-

strictions on beliefs’—means in G9 that 2 does not assess the relativelikelihood of 1’s maximal strategies U and M . This does not have anybearing on the analysis of G9, but is essential for capturing forward in-duction in G

1 of Figure 2.6. In this case the issue is not whether a playerassesses the relative likelihood of different maximal strategies, but ratherwhether a player assesses the relative likelihood of different non-maximalstrategies. To see the significance in G

1, assume that 1 deems r infinitelymore likely than , while 2 deems Out infinitely more likely than InRand InR infinitely more likely than InL. Then the players rank their

strategies as follows:

1’s preferences: Out InR InL2’s preferences: r

Both ‘caution’ and ‘robust belief of opponent rationality’ are satisfiedand still the forward induction outcome (InL, ) is not promoted. How-ever, the requirement of ‘no extraneous restrictions on beliefs’ is notsatisfied since the preferences of 2 introduce extraneous restrictions onbeliefs by deeming one of 1’s non-maximal strategies, InR, infinitelymore likely than another non-maximal strategy, InL. When we returnto G

1 in Sections 11.4 and 11.5, we show how the additional impositionof ‘no extraneous restrictions on beliefs’ leads to (InL, ) in this game.

Several concepts with natural epistemic foundations fail to matchthese predictions in G

1 and G9. In the case of rationalizability—cf.Bernheim (1984) and Pearce (1984)—this is perhaps not so surprisingsince this concept in two-player games corresponds to IESDS. It can beunderstood as a consequence of common belief of rational choice with-out imposing caution, so there is no guarantee that a player prefers onestrategy to another if the former weakly dominates the latter. In G9,for example, all strategies are rationalizable.

It is more surprising that the concept of ‘permissibility’ does notmatch our solution of G9. Permissibility can be given rigorous epis-temic foundations in models with cautious players—cf. Borgers (1994)and Brandenburger (1992), who coined the term ‘permissible’; see alsoBen-Porath (1997) and Gul (1997) as well as Propositions 24 and 27of this book. In these models players take into account all opponentstrategies, while assigning more weight to a subset of those deemed to




be rational choices. As noted earlier, permissibility corresponds to theDekel-Fudenberg procedure where one round of elimination of all weaklydominated strategies is followed by iterated elimination of strongly dom-inated strategies. In G9, this means that 1 cannot choose his weaklydominated strategy D. However, while 2 prefers to r in our solution,permissibility allows that 2 chooses r. To exemplify using Branden-burger’s (1992) approach, this will be the case if 2 deems U to be in-finitely more likely than D which in turn is deemed infinitely more likelythan M . The problem is that ‘robust belief of opponent rationality’ isnot satisfied: Player 2 deems D more likely than M even though M is in1’s choice set, while D is not. In Section 11.3 we establish in Proposition

40 that the concept of fully permissible sets refines the Dekel-Fudenbergprocedure.

11.2 IECFA and fully permissible sets

We present in this section an algorithm—‘iterated elimination of choicesets under full admissible consistency’ (IECFA)—leading to the conceptof ‘fully permissible sets’. This concept will in turn be given an epistemiccharacterization in Section 11.3 by imposing common certain belief of full admissible consistency. We present the algorithm before the epis-temic characterization for different reasons:

IECFA is fairly accessible. By defining it early, we can apply it early,and offer early indications of the nature of the solution concept wewish to promote.

By defining IECFA, we point to a parallel to the concepts of ratio-nalizable strategies and permissible strategies. These concepts aremotivated by epistemic assumptions, but turn out to be identical in2-player games to the set of strategies surviving simple algorithms:respectively, IESDS and the Dekel-Fudenberg procedure.

Just like IESDS and the Dekel-Fudenberg procedure, IECFA is eas-ier to use than the corresponding epistemic characterizations. Thealgorithm should be handy for applied economists, independently of the foundational issues discussed in Section 11.3.

IESDS and the Dekel-Fudenberg procedure iteratively eliminate domi-nated strategies. In the corresponding epistemic models, these strategiesin turn cannot be rational choices, cannot be rational choices given thatother players do not use strategies that cannot be rational choices, etc.




IECFA is also an elimination procedure. However, the interpretation of the basic item thrown out is not that of a strategy that cannot be arational choice, but rather that of a set of strategies that cannot be achoice set for any preferences that are in a given sense consistent withthe preferences of the opponent. The specific kind of consistency in-volved in IECFA—which will be defined in Section 11.3 and referred toas ‘full admissible consistency’—requires that a player’s preferences arecharacterized by the properties of ‘caution’, ‘robust belief of opponentrationality’ and ‘no extraneous restrictions on beliefs’. Thus, IECFAdoes not start with each player’s strategy set and then iteratively elim-inates strategies. Rather, IECFA starts with each player’s collection

of non-empty subsets of his strategy set and then iteratively eliminatessubsets from this collection.

Definition. Consider a finite strategic two-player game G = (S 1, S 2,u1, u2), and recall the following notation from Chapter 6: For any (∅ =)Y j ⊆ S j,

Di(Y j ) := {si ∈ S i| ∃ pi ∈ ∆(S i) such that

pi weakly dominates si on Y j or S j} .

Interpret Y j as the set of strategies that player i deems to be the set of rational choices for his opponent. Let i’s choice set be equal to S i\Di(Y j),

entailing that i’s choice set consists of pure strategies that are not weaklydominated by any mixed strategy on Y j or S j. In Section 11.3 we showhow this corresponds to a set of maximal strategies given the player’spreferences over his own strategies.

Let Σ = Σ1 × Σ2, where Σi := 2S i\{∅} denotes the collection of non-empty subsets of S i. Write σi (∈ Σi) for a subset of pure strategies. Forany (∅ =) Ξ = Ξ1 × Ξ2 ⊆ Σ, write α(Ξ) := α1(Ξ2) × α2(Ξ1), where

αi(Ξ j) := {σi ∈ Σi|∃(∅ =) Ψ j ⊆ Ξ j s.t. σi = S i\Di(∪σj∈Ψjσ

j)} .

Hence, αi(Ξ j) is the collection of strategy subsets that can be choicesets for player i if he associates Y j—the set of rational choices for his

opponent—with the union of the strategy subsets in a non-empty sub-collection of Ξ j.We can now define the concept of a fully permissible set.

Definition 22 Let G = (S 1, S 2, u1, u2) be a finite strategic two-playergame. Consider the sequence defined by Ξ(0) = Σ and, ∀g ≥ 1, Ξ(g) =α(Ξ(g − 1)). A non-empty strategy set σi is said to be fully permissible if




σi ∈

∞g=0

Ξi(g) .

Let Π = Π1 × Π2 denote the collection of profiles of fully permissiblesets. Since ∅ = αi(Ξ

j) ⊆ αi(Ξ j ) ⊆ αi(Σ j ) whenever ∅ = Ξ

j ⊆ Ξ j ⊆ Σ j

and since the game is finite, Ξ(g) is a monotone sequence that convergesto Π in a finite number of iterations. IECFA is the procedure that inround g eliminates sets in Ξ(g − 1)\Ξ(g) as possible choice sets. Asdefined in Definition 22, IECFA eliminates maximally in each round inthe sense that, ∀g ≥ 1, Ξ(g) = α(Ξ(g − 1)). However, it follows fromthe monotonicity of αi that any non-maximal procedure, where ∃g ≥ 1

such that Ξ(g − 1) ⊃ Ξ(g) ⊃ α(Ξ(g − 1)), will also converge to Π.A strategy subset survives elimination round g if it can be a choice setwhen the set of rational choices for his opponent is associated with theunion of some (or all) of opponent sets that have survived the procedureup till round g − 1. A fully permissible set is a set that survives inthis way for any g. The analysis of Section 11.3 justifies that strategysubsets that this algorithm has not eliminated by round g be interpretedas choice sets compatible with g − 1 order of mutual certain belief of fulladmissible consistency.

Applications. We illustrate IECFA by applying it. Consider G9 of Figure 11.2. We get:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{U, M }} × Σ2

Π = Ξ(2) = {{U, M } } × { {}} .

Independently of Y 2, S 1\D1(Y 2) = {U, M }, so for 1 only {U, M } survivesthe first elimination round, while S 2\D2({U, M }) = {}, S 2\D2({D}) ={r} and S 2\D2({U }) = {, r}, so that no elimination is possible forplayer 2. However, in the second round only {} survives since weaklydominates r on {U, M }, implying that S 2\D2({U, M }) = {}.

Next, consider G1 of Figure 2.6. Applying IECFA we get:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{Out}, {InL}, {Out, InL}} × Σ2

Ξ(2) = {{Out}, {InL}, {Out, InL} } × { {}, {, r}}Ξ(3) = {{InL}, {Out, InL} } × { {}, {, r}}Ξ(4) = {{InL}, {Out, InL} } × { {}}

Π = Ξ(5) = {{InL} } × { {}} .




Again the algorithm yields a unique fully permissible set for each player.Finally, apply IECFA to G8 of Figure 11.1:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{U }, {M }, {U, M }} × Σ2

Ξ(2) = {{U }, {M }, {U, M } } × { {}, {, r}}Π = Ξ(3) = {{U }, {U, M } } × { {}, {, r}} .

Here we are left with two fully permissible sets for each player. There isno further elimination, as {U } = S 1\D1({}), {U, M } = S 1\D1({, r}),{} = S 2\D2({U, M }), and {, r} = S 2\D2({U }).

The elimination process for G

1

and G8 is explained and interpretedin Section 11.4.

Results. The following proposition characterizes the strategy subsetsthat survive IECFA and thus are fully permissible, and is a straightfor-ward implication of Definition 22 (keeping in mind that Σ is finite and,for each i, αi is monotone).

Proposition 39 (i) For each i, Πi = ∅. (ii) Π = α(Π). (iii) For each i, σi ∈ Πi if and only if there exists Ξ = Ξ1 × Ξ2 with σi ∈ Ξi such that Ξ ⊆ α(Ξ).

Proposition 39(i) shows existence, but not uniqueness, of each player’s

fully permissible set(s). In addition to G2, games with multiple strictNash equilibria illustrate the possibility of such multiplicity; by Propo-sition 39(iii) any strict Nash equilibrium corresponds to a profile of fullypermissible sets. Proposition 39(ii) means that Π is a fixed point in termsof a collection of profiles of strategy sets as illustrated by G2 above. ByProposition 39(iii) it is the largest such fixed point.

We close this section by recording some connections between IECFAon the one hand, and IESDS, the Dekel-Fudenberg procedure (i.e., per-missibility), and IEWDS on the other. First, we note through the follow-ing Proposition 40 that IECFA has more bite than the Dekel-Fudenbergprocedure. Both G1 and G3 illustrate that this refinement may be strict.

Proposition 40 A pure strategy si is permissible if there exists a fully permissible set σi such that si ∈ σi.

Proof. Using Proposition 39(ii), the definitions of α(·) (given above)and a(·) (given in Chapter 6) imply, for each i,

P 0i := ∪σi∈Πiσi = ∪σi∈αi(Πj)σi ⊆ ai(P 0 j ) .




c r

UU

UD

DU

DD

1, 1 1, 1 0, 0

1, 1 0, 1 1, 0

0, 1 0, 0 2, 0

0, 0 0, 1 0, 2

Figure 11.3. G10, illustrating the relation between IECFA and IEWDS.

Since P 0 ⊆ a(P 0) implies P 0 ⊆ P , by Lemma 10(iii), it follows that, foreach i, ∪σi∈Πi

σi ⊆ P i.

It is a corollary that IECFA has also more cutting power than IESDS.However, neither of IECFA and IEWDS has more bite than the other,

as demonstrated by the game G10 of Fig. 11.3. It is straightforward toverify that UU and UD for player 1 and for player 2 survive IEWDS,while {UU } for 1 and {, c} for 2 survive IECFA and are thus the fullypermissible sets, as shown below:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{UU }, {DU }, {UU , UD }, {UU , DU }, {UD , DU },{UU , UD , DU } } × { {}, {r}, {, c}, {, r}, {c, r}, {,c,r}}

Ξ(2) = {{UU }, {DU }, {UU , UD }, {UU , DU }, {UD , DU },{UU , UD , DU } } × { {}, {, c}}

Ξ(3) = {{UU }, {UU , UD } } × { {}, {, c}}Ξ(4) = {{UU }, {UU , UD } } × { {, c}}

Π = Ξ(5) = {{UU } } × { {, c}} .

Strategy UD survives IEWDS but does not appear in any fully permissi-ble set. Strategy c appears in a fully permissible set but does not surviveIEWDS.

11.3 Full admissible consistency

When justifying rationalizable and permissible strategies through epis-

temic conditions, players are usually modeled as decision makers un-der uncertainty. Tan and Werlang (1988) characterize rationalizablestrategies by common belief (with probability one) of the event thateach player chooses a maximal strategy given preferences that are rep-resented by a subjective probability distribution. Hence, preferencesare both complete and continuous (cf. Proposition 1). Brandenburger(1992) characterizes permissible strategies by common belief (with pri-




mary probability one) of the event that each player chooses a maxi-mal strategy given preferences that are represented by an LPS with fullsupport on the set of opponent strategies (cf. Proposition 2). Hence,preferences are still complete, but not continuous due to the full sup-port requirement. Since preferences are complete and representable bya probability distribution or an LPS, these epistemic justifications differsignificantly from the corresponding algorithms, IESDS and the Dekel-Fudenberg procedure, neither of which makes reference to subjectiveprobabilities.1

When doing analogously for fully permissible sets, not only must con-tinuity of preferences be relaxed to allow for ‘caution’ and ‘robust belief

of opponent rationality’, as discussed in Section 11.1. One must alsorelax completeness of preferences to accommodate ‘no extraneous re-strictions on beliefs’, which is a requirement of minimal completenessand implies that preferences are expressed solely in terms of admissibil-ity on nested sets. Hence, preferences are not in general representable bysubjective probabilities (except through treating incomplete preferencesas a set of complete preferences; cf. Aumann, 1962; Bewley, 1986). Thismeans that epistemic operators must be derived directly from the un-derlying preferences—as observed by Morris (1997) and explored furtherin Chapter 4 of this book—since there is no probability distribution orLPS that represents the preferences. It also entails that the resulting

characterization, given in Proposition 41, must be closely related to thealgorithm used in the definition of fully permissible sets.

There is another fundamental difference. When characterizing ratio-nalizable and permissible strategies within the ‘rational choice’ approach,the event that is made subject to interactive epistemology is defined byrequiring that each player’s strategy choice is an element of his choice set(i.e. his set of maximal strategies) given his belief about the opponent’sstrategy choice.2 In contrast, in the characterization of Proposition 40,the event that is made subject to interactive epistemology is defined byimposing requirements on how each player’s choice set is related to hisbelief about the opponent’s choice set. Since a player’s choice set equals

the set of maximal strategies given the ranking that the player has overhis strategies, the imposed requirements relate a player’s ranking over

1However, as shown by Propositions 26 and 27 of this book, epistemic characterization of rationalizability and permissibility can be provided without using subjective probabilities.2As illustrated in Chapters 5 and 6 of this book, it is also possible to characterize rational-izable and permissible strategies within the ‘consistent preferences’ approach.




his strategies to the opponent’s ranking. Hence, fully permissible setsare characterized within the ‘consistent preferences’ approach.

The epistemic modeling is identical to the one given in Section 6.1;hence, this will not be recapitulated here. Recall, however, that κti

(⊆ {ti}×S j ×T j ) denotes the set of states that player i deems subjectivelypossible at ti, that β ti (⊆ κti) denotes the smallest set of states on whichplayer i’s preferences at ti, ti , are admissible, and that Assumption 2is imposed so that preferences are conditionally represented by a vNMutility function (cf. Proposition 4).

Characterizing full permissibility. To characterize the concept of

fully permissible sets, consider for each i,

B0i [rat j ] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

β ti = (projT i×S j×T j[rat j ]) ∩ κti , and

p ti q only if pE j weakly dominates qE j

for E j = projS j×T jβ ti or E j = projS j×T j

κti} ,

Define as follows the event that player i’s preferences over his strate-gies are fully admissibly consistent with the game G = (S 1, S 2, u1, u2)and the preferences of his opponent :

A0i := [ui] ∩ B0

i [rat j] ∩ [caui] .

Write A0 := A01 ∩ A0

2 for the event of full admissible consistency .

Proposition 41 A strategy set σi for i is fully permissible in a finite strategic two-player game G if and only if there exists an epistemic model with σi = S tii for some (t1, t2) ∈ projT 1×T 2CK A0.

Proof. Part 1: If σi is fully permissible, then there exists an epistemic model with σi = S tii for some (t1, t2) ∈ projT 1×T 2CK A0. It is sufficient toconstruct a belief system with S 1 × T 1 × S 2 × T 2 ⊆ CK A0 such that, foreach σi ∈ Πi of any player i, there exists ti ∈ T i with σi = S tii . Constructa belief system with, for each i, a bijection σi : T i → Πi from the set

of types to the the collection of fully permissible sets. By Proposition39(ii) we have that, for each ti ∈ T i of any player i, there exists Ψ jti ⊆ Π j

such that σi(ti) = S i\Di(Y jti), where Y j

ti := {s j ∈ S j | ∃σ j ∈ Ψ jti s.t.

s j ∈ σ j}. Determine the set of opponent types that ti deems subjectivelypossible as follows: T j

ti = {t j ∈ T j| σ j (t j) ∈ Ψ jti}. Let, for each ti ∈ T i

of any player i, ti satisfy

1. υtii ◦ z = ui (so that S 1 × T 1 × S 2 × T 2 ⊆ [u]), and




2. p ti q iff pE j weakly dominates qE j for E j = E jti := {(s j , t j )|s j ∈σ j(t j) and t j ∈ T ti j } or E j = S j × T j

ti, which implies that β ti ={ti}×E j

ti and κti = {ti}×S j ×T jti (so that S 1 ×T 1 ×S 2 ×T 2 ⊆ [cau]).

By the construction of E jti, this means that S tii = S i\Di(Y j

ti) = σi(ti)since, for any acts p and q on S j × T j satisfying that there exist mixedstrategies pi, q i ∈ ∆(S i) such that, ∀(s j, t j) ∈ S j ×T j, p(s j , t j ) = z( pi, s j)and q(s j, t j) = z (q i, s j), p ti q iff pE j weakly dominates qE j for E j =Y j

ti × T j or E j = S j × T j . This in turn implies, for each ti ∈ T i anyplayer i,

3. β ti = (projT i×S j×T j[rat j ]) ∩ κti (so that, in combination with 2., S 1 ×

T 1 × S 2 × T 2 ⊆ B0i [rat j] ∩ B0 j [rati]).

Furthermore, S 1 × T 1 × S 2 × T 2 ⊆ CK A0 since T jti ⊆ T j for each ti ∈ T i

of any player i. Since, for each player i, σi is onto Πi, it follows that,for each σi ∈ Πi of any player i, there exists ti ∈ T i with σi = S i

ti .Part 2: If there exists an epistemic model with σ∗

i = S it∗i for some

(t∗1, t∗

2) ∈ projT 1×T 2CK A0, then σ∗

i is fully permissible.Assume that there exists an epistemic model with σ ∗

i = S it∗i for some

(t∗1, t∗

2) ∈ projT 1×T 2CK A0. In particular, CK A0 = ∅. Let, for each i,

T i := projT iCK A0 and Ξi := {S tii | ti ∈ T i }. It is sufficient to show that,

for each i, Ξi ⊆ Πi. By Proposition 25(ii), for each ti ∈ T i of any playeri, β ti ⊆ κti ⊆ {ti} × S j × T j since CK A0 = KCK A0 ⊆ KiCK A0. By the

definition of A0, it follows that, for each ti ∈ T i of any player i,

1. ti is conditionally represented by υtii satisfying that υti

i ◦ z is apositive affine transformation of ui, and

2. p ti q iff pE j weakly dominates qE j for E j = E jti := projS j×T j

β ti

or E j = S j × T jti , where β ti = (projT i×S j×T j

[rat j]) ∩ κti .

Write Ψ jti := {S j

tj | t j ∈ T jti} and Y j

ti := {s j ∈ S j| ∃σ j ∈ Ψ jti s.t.

s j ∈ σ j}, and note that κti ⊆ {ti} × S j × T j implies Ψ jti ⊆ Ξ j. It

follows that, for any acts p and q on S j × T j satisfying that there existmixed strategies pi, q i ∈ ∆(S i) such that, ∀(s j, t j) ∈ S j × T j, p(s j, t j) =z( pi, s j) and q(s j , t j ) = z(q i, s j), p ti q iff pE

j

weakly dominates qE jfor E j = Y j

ti × T j or E j = S j × T j . Hence, S tii = S i\Di(Y jti). Since this

holds for each ti ∈ T i of any player i, we have that Ξ ⊆ α(Ξ). Hence,Proposition 39(iii) entails that, for each i, Ξi ⊆ Πi.

Interpretation. We now show how the event used to characterizefully permissible sets—full admissible consistency—can be interpreted interms of the requirements of ‘caution’, ‘robust belief of opponent ratio-




nality’, and ‘no extraneous restrictions on beliefs’. Following a commonprocedure of the axiomatic method, this will in turn be used to ver-ify that these requirements are indeed needed for the characterizationin Proposition 41 by investigating the consequences of relaxing one re-quirement at a time. These exercises contribute to the understandingof fully permissible sets by showing that the concept is related to prop-erly rationalizable, permissible, and rationalizable pure strategies in thefollowing manner:

When allowing extraneous restrictions on beliefs, we open for anyproperly rationalizable pure strategy, implying that forward induc-

tion is no longer promoted in G1 of Figure 2.6.3

When weakening ‘robust belief of opponent rationality’ to ‘belief of opponent rationality’, we characterize the concept of permissible purestrategies independently of whether a requirement of ‘no extraneousrestrictions on beliefs’ is retained.

When removing ‘caution’, we characterize the concept of rationaliz-able pure strategies independently of whether extraneous restrictionson beliefs are allowed and robust belief of opponent rationality isweakened.

Since it is clear that [cau] = [cau1] ∩ [cau2] corresponds to caution(cf. Section 6.3), it remains to split B0

1 [rat2]∩ B02 [rat1] into ‘robust belief

of opponent rationality’ and ‘no extraneous restrictions on beliefs’.To state the condition of ‘robust belief of opponent rationality’ we

need to recall the robust belief operator as defined and characterizedin Chapter 4. Since Assumption 2 is compatible with the frameworkof Chapter 4, we can in line with Section 4.2 define robust belief asfollows. If E does not concern player i’s strategy choice (i.e., E =S i × projT i×S j×S j

E ), say that player i robustly believes the event E at ti

if ti ∈ projT iB0

i E , where

B0i E := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

∃ ∈ {1, . . . , L} s.t. ρti = projT 1×S 2×T 2E ∩ κti} ,

3To relax ‘no extraneous restrictions on beliefs’ we need an epistemic model—as the oneintroduced in Section 6.1—that is versatile enough to allow for preferences that are morecomplete than being determined by admissibility on two nested sets.




and where (ρti1 , . . . , ρti

L ) is the profile of nested sets on which ti is ad-missible, and which satisfies:

∅ = β ti = ρti1 ⊂ · · · ⊂ ρti

⊂ · · · ⊂ ρtiL = κti ⊆ {ti} × S j × T j

(where ⊂ denotes ⊆ and =).If ti ∈ projT i

B0i [rat j], then i robustly believes at ti that j is rational.

By Proposition 6 this means that any (s j , t j ) that is deemed subjectivelypossible and where s j is a rational choice by j at t j is considered infinitelymore likely than any (s

j, t j) where s

j is not a rational choice by j at t j.

As ti ∈ projT iB0

i [rat j] entails that β ti = (projT i×S j×T j[rat j]) ∩ κti , it

follows that B0i [rat j] ⊆ B0i [rat j ]. Hence, relative to B01[rat2] ∩ B02[rat1],B0

1[rat2]∩B02[rat1] is obtained by imposing minimal completeness, which

in this context yields the requirement of ‘no extraneous restrictions onbeliefs’.

As established in Section 4.3, robust belief B0i is a non-monotone op-

erator which is bounded by the two KD45 operators, namely belief Bi

and certain belief Ki. Furthermore, as shown in Chapter 4, the robustbelief operator coincides with the notions of ‘absolutely robust belief’,as introduced by Stalnaker (1998), and ‘assumption’, as proposed byBrandenburger and Keisler (2002), and is closely related to the conceptof ‘strong belief’, as used by Battigalli and Siniscalchi (2002). However,

in contrast to the use of non-monotonic operators in these contributions,our non-monotonic operator B0

i is used only to interpret ‘full admissi-ble consistency’, while the KD45 operator K i is used for the interactive epistemology. The importance of this will be discussed in Section 11.5.There we also comment on how the present requirement of ‘no extra-neous restrictions on beliefs’ is related to Brandenburger and Keisler’sand Battigalli and Siniscalchi’s use of a ‘preference-complete’ epistemicmodel.

Allowing extraneous restrictions on beliefs. In view of the pre-vious discussion, we allow extraneous restrictions on beliefs by replacing,for each i, B0

i [rat j] by B0i [rat j]. Hence, let for each i,

A0i := [ui] ∩ B0

i [rat j] ∩ [caui] .

The following result is proven in Appendix C and shows that any prop-erly rationalizable pure strategy is consistent with common certain belief of A0 := A0

1 ∩ A02.




Proposition 42 Consider a finite strategic two-player game G. If a pure strategy si for i is properly rationalizable, then there exists an epis-temic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CKA0.

Note that both Out and r are properly rationalizable pure strategies(and, indeed (Out, r) is a proper equilibrium) in G

1, the ‘battle-of-the-sexes-with-an-outside-option’ game of Figure 2.6, while neither Out norr is consistent with common certain belief of full admissible consistency.This demonstrates that ‘no extraneous restrictions on beliefs’ is neededfor the characterization in Proposition 41 of the concept of fully per-missible sets, which in G

1 promotes only the forward induction outcome

(InL, ) (cf. the analysis of G

1 in Sections 11.2 and 11.4).Weakening robust belief of opponent rationality. By applying

the belief operator Bi, as defined in Section 6.1, we can weaken B 01[rat2]∩

B02[rat1] (i.e., robust belief of opponent rationality) to B1[rat2]∩B2[rat1]

(i.e., belief of opponent rationality). Moreover, we can weaken B01[rat2]

∩ B02[rat1] to B1[rat2] ∩ B2[rat1], where for each i,

Bi[rat j ] := {(s1, t1, s2, t2) ∈ S 1 × T 1 × S 2 × T 2|

β ti ⊆ (projT i×S j×T j[rat j ]), and

p ti q only if pE j weakly dominates qE j


κti} ,

Relative to B1[rat2] ∩ B2[rat1], B1[rat2] ∩ B2[rat1] is obtained by im-posing minimal completeness, which in the context of belief of opponentrationality yields the requirement of ‘no extraneous restrictions on be-liefs’.

To impose ‘caution’ and ‘belief of opponent rationality’, recall fromSection 6.3 that A = A1 ∩ A2 is the event of admissible consistencywhere, for each i,

Ai = [ui] ∩ Bi[rat j] ∩ [caui] ,

To add ‘no extraneous restrictions on beliefs’, consider for each i,

¯Ai := [ui] ∩

¯Bi[rat j] ∩ [caui] ,

and write A := A1 ∩ A2. Since A ⊆ A, the following proposition impliesthat permissibility (i.e., the Dekel-Fudenberg procedure; see Definition13) is characterized if ‘robust belief of opponent rationality’ is weakenedto ‘belief of opponent rationality’, independently of whether a require-ment of ‘no extraneous restrictions on beliefs’ is retained. This result,which is a strengthening of Proposition 27 and is proven in Appendix




C, shows that ‘robust belief of opponent rationality’ is needed for thecharacterization in Proposition 41 of the concept of fully permissiblesets.

Proposition 43 Consider a finite strategic two-player game G. If a pure strategy si for i is permissible, then there exists an epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2

CK A. A pure strategy si for iis permissible if there exists an epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CKA.

Removing caution. Recall from Section 6.2 that C = C 1 ∩ C 2 isthe event of consistency where, for each i,

C i = [ui] ∩ Bi[rat j ] .

To add ‘no extraneous restrictions on beliefs’ and ‘robust belief of oppo-nent rationality’, consider for each i,

C 0i := [ui] ∩ B0i [rat j ] ,

and write C 0 := C 01 ∩ C 02 . Since C 0 ⊆ C , the following strengthening of Proposition 25 means that the removal of ‘caution’ leads to a characteri-zation of rationalizability (i.e., IESDS; see Definition 11), independentlyof whether extraneous restrictions on beliefs are allowed and robust be-

lief of opponent rationality is weakened. Thus, ‘caution’ is necessary forthe characterization in Proposition 41.

Proposition 44 Consider a finite strategic two-player game G. If a pure strategy si for i is rationalizable, then there exists an epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CK C 0. A pure strategy si for i is rationalizable if there exists an epistemic model with si ∈ S tii

for some (t1, t2) ∈ projT 1×T 2CKC .

Also the proof of this result is contained in Appendix C.

11.4 Investigating examples

The present section illustrates the concept of fully permissible setsby returning to the previously discussed games G

1 and G8. Here, G1

will serve to show how our concept captures aspects of forward induc-tion, while G8 will be used to interpret the occurrence of multiple fullypermissible sets.

The two examples will be used to shed light on the differences between,on the one hand, the approach suggested here and, on the other hand,




IEWDS as characterized by Stahl (1995): A strategy survives IEWDS if and only if it is a best response to a belief where one strategy is infinitelyless likely than another if the former is eliminated at an earlier roundthan the latter.4

Forward induction. Reconsider G1 Figure 2.6, and apply our algo-

rithm IECFA to this “battle-of-the-sexes with an outside option” game.Since InR is a dominated strategy, InR cannot be an element of 1’schoice set. This does not imply, as in the procedure of IEWDS (givenStahl’s, 1995, characterization), that 2 deems InL infinitely more likelythan InR. However, 2 certainly believes that only {Out}, {InL} and

{Out, InL} are candidates for 1’s choice set. This excludes {r} as 2’schoice set, since {r} is 2’s choice set only if 2 deems {InR} or {Out, InR}possible. This in turn means that 1 certainly believes that only {} and{, r} are candidates for 2’s choice set, implying that {Out} cannot be1’s choice set. Certainly believing that only {InL} and {Out, InL} arecandidates for 1’s choice set does imply that 2 deems InL infinitely morelikely than InR. Hence, 2’s choice set is {} and, therefore, 1’s choiceset {InL}. Thus, the forward induction outcome (InL, ) is promoted.

To show how common certain belief of the event A0 is consistent withthe fully permissible sets {InL} and {}—and thus illustrate Proposition41—consider an epistemic model with only one type of each player; i.e.,

T 1 × T 2 = {t1} × {t2}. Let, for each i,

ti

satisfy that υ

ti

i ◦ z = ui. Also,letβ t1 = {t1} × {} × {t2} κt1 = {t1} × S 2 × {t2}β t2 = {t2} × {InL} × {t1} κt2 = {t2} × S 1 × {t1} .

Finally, let for each i, p ti q if and only if pE j weakly dominates qE j


κti . Then

S t11 = {InL} S t22 = {} .

Inspection will verify that C K A0 = A0 = S 1 × T 1 × S 2 × T 2.

Multiple fully permissible sets. Let us also return to G8 of Figure11.1, where IEWDS eliminates D in the first round, r in the second

round, and M in the third round, so that U and survive. Stahl’s(1995) characterization of IEWDS entails that 2 deems each of U andM infinitely more likely than D. Hence, the procedure forces 2 to deem

4Cf. Brandenburger and Keisler (2002, Theorem 1) as well as Battigalli (1996a) and Rajan(1998). See also Bicchieri and Schulte (1997), who give conceptually related interpretationsof IEWDS.




M infinitely more likely than D for the sole reason that D is eliminatedbefore M , even though both M and D are eventually eliminated by theprocedure.

Applying our algorithm IECFA yields the following result. Since Dis a weakly dominated strategy, D cannot be an element of 1’s choiceset. Hence, 2 certainly believes that only {U }, {M } and {U, M } arecandidates for 1’s choice set. This excludes {r} as 2’s choice set, since{r} is 2’s choice set only if 2 deems {D} or {U, D} possible. This in turnmeans that 1 certainly believes that only {} and {, r} are candidatesfor 2’s choice set, implying that {M } cannot be 1’s choice set. There isno further elimination. This means that 1’s collection of fully permis-

sible sets is {{U }, {U, M }} and 2’s collection of fully permissible sets is{}, {, r}}. Thus, common certain belief of full admissible consistencyimplies that 2 deems U infinitely more likely than D since U (respec-tively, D) is an element of any (respectively, no) fully permissible set for1. However, whether 2 deems M infinitely more likely than D dependson the type of player 2.

To show how common certain belief of the event A0 is consistent withthe collections of fully permissible sets {{U }, {U, M }} and {{}, {, r}}—and thus illustrate Proposition 41 also in the case of G8—consider anepistemic model with two types of each player; i.e., T 1 × T 2 = {t

1, t1} ×

{t2, t

2}. Let, for each type ti of any player i, ti satisfy that υtii ◦ z = ui.

Moreover, let

β t1 = {t

1} × {} × {t2} κt1 = {t

1} × S 2 × {t2}

β t1 = {t

1} × {(, t2), (, t

2), (r, t2)} κt1 = {t

1} × S 2 × T 2

β t2 = {t

2} × {(U, t1), (U, t

1), (M, t1)} κt2 = {t

2} × S 1 × T 1β t

2 = {t

2} × {U } × {t1} κt2 = {t

2} × S 1 × {t1} .

Finally, let for each type ti of any player i, p ti q if and only if pE j

weakly dominates qE j for E j = projS j×T jβ ti or E j = projS j×T j

κti . Then

S t11 = {U } S

t11 = {U, M } S

t22 = {} S

t22 = {, r} .

Inspection will verify that C K A0 = A0 = S 1 × T 1 × S 2 × T 2Our analysis of G8 allows a player to deem an opponent choice set

to be subjectively impossible even when it is the true choice set of theopponent. E.g., at (t

1, t2), player 1 deems it subjectively impossible that

player 2’s choice set is {, r} even though this is the true choice set of player 2. Likewise, at (t

1, t2), player 2 deems it subjectively impossible

that player 1’s choice set is {U, M } even though this is the true choice set




of player 1. This is an unavoidable feature of this game as there existsno pair of non-empty strategy subsets (Y 1, Y 2) such that Y 1 = S 1\D1(Y 2)and Y 2 = S 2\D2(Y 1). It implies that under full admissible consistencywe cannot have in G8 that each player is certain of the true choice setof the opponent.

Multiplicity of fully permissible sets arises also in the strategic form of certain extensive games in which the application of backward inductionis controversial, e.g. the ‘centipede’ game Γ

3 illustrated in Figure 2.4.For more on this, see Chapter 12 where the concept of fully permissiblesets is used to analyze extensive games.

11.5 Related literatureIt is instructive to explain how our analysis differs from the epistemic

foundations of IEWDS and EFR provided by Brandenburger and Keisler(2002) (BK) and Battigalli and Siniscalchi (2002) (BS), respectively. Itis of minor importance for the comparison that EFR makes use of theextensive form, while the present analysis is performed in the strategicform. The reason is that, by ‘caution’, a rational choice in the wholegame implies a rational choice at all information sets that are not pre-cluded from being reached by the player’s own strategy (cf. Lemma 11).

To capture forward induction players must essentially deem any op-ponent strategy that is a rational choice infinitely more likely than any

opponent strategy not having this property. An analysis incorporatingthis feature must involve a non-monotonic epistemic operator, which iscalled robust belief in the present analysis (cf. Section 11.3), while thecorresponding operators are called ‘assumption’ and ‘strong belief’ byBK and BS, respectively (see Chapter 4 for an analysis of the relation-ship between these non-monotonic operators).

We use robust belief only to define the event that the preferences of each player is ‘fully admissibly consistent’ with the preferences of hisopponent, while the monotonic certain belief operator is used for theinteractive epistemology:

each player certainly believes (in the sense of deeming the complement

subjectively impossible) that the preferences of his opponent are fullyadmissibly consistent,

each player certainly believes that his opponent certainly believesthat he himself has preferences that are fully admissibly consistent,

and so on. As the examples of Section 11.4 illustrate, it is here a centralquestion what opponent types (choice sets) a player deems subjectively




possible. Consequently, the certain belief operator is appropriate for theinteractive epistemology.

In contrast, BK and BS use their non-monotonic operators for theinteractive epistemology. In the process of defining higher order beliefsboth BK and BS impose that lower order beliefs are maintained. Thisis precisely how BK obtain Stahl’s (1995) characterization which—e.g.,in G8 of Figure 11.1—seems to correspond to extraneous and hard-to- justify restrictions on beliefs.

Stahl’s characterization provides an interpretation of IEWDS wherestrategies eliminated in the first round are completely irrational, while

strategies eliminated in later rounds are at intermediate degrees of ra-tionality. Likewise, Battigalli (1996a) has shown how EFR correspondsto the ‘best rationalization principle’, entailing that some opponentstrategies are neither completely rational nor completely irrational. Thepresent analysis, in contrast, differentiates only between whether a strat-egy is maximal (i.e., a rational choice) or not. As the examples of Section11.4 illustrate, although a strategy that is weakly dominated on the setof all opponent strategies is a “stupid” choice, it need not be “morestupid” than any remaining admissible strategy, as this depends on theinteractive analysis of the game.

The fact that a non-monotonic epistemic operator is involved when

capturing forward induction also means that the analysis must ensurethat all rational choices for the opponent are included in the epistemicmodel. BK and BS ensure this by employing ‘preference-complete’ epis-temic models, where all possible epistemic types of each player are rep-resented. Instead, the present analysis achieves this by requiring ‘noextraneous restrictions on beliefs’, meaning that the preferences are min-imally complete (cf. Section 11.3). Since an ordinary monotonic oper-ator is used for the interactive epistemology, there is no more need fora ‘preference-complete’ epistemic model here than in usual epistemicanalyses of rationalizability and permissibility.

Our paper has a predecessor in Samuelson (1992), who also presents

an epistemic analysis of admissibility that leads to a collection of setsfor each player, called a ‘generalized consistent pair’. Samuelson requiresthat a player’s choice set equals the set of strategies that are not weaklydominated on the union of choice sets that are deemed possible for theopponent; this implies our requirements of ‘robust belief of opponentrationality’ and ‘no extraneous restrictions on beliefs’ (cf. Samuelson,1992, p. 311). However, he does not require that each player deems no




opponent strategy impossible, as implied by our requirement of ‘caution’.Hence, his analysis does not yield {{U, M }} × {{}} in G9 of Figure11.2. Furthermore, he defines possibility relative to a knowledge operatorthat satisfies the truth axiom, while our analysis—as illustrated by thediscussion of G8 in Section 11.4—allows a player to deem an opponentchoice set to be subjectively impossible even when it is the true choiceset of the opponent. This explains why we in contrast to Samuelsonobtain general existence (cf. Proposition 39(i)).

If each player is certain of the true choice set of the opponent, oneobtains a ‘consistent pair’ as defined by Borgers and Samuelson (1992),a concept that need not exist even when a generalized consistent pair ex-

ists. Ewerhart (1998) modifies the concept of a consistent pair by adding‘caution’. However, since he allows extraneous restrictions on beliefs toensure general existence, his concept of a ‘modified consistent pair’ doesnot promote forward induction in G

1. A ‘self-admissible set’ in the ter-minology of Brandenburger and Friedenberg (2003) is a Cartesian prod-uct of strategy subsets, where each player’s subset consists of strategiesthat weakly dominated neither on the subset of opponent strategies noron the set of all opponent strategies. Also Brandenburger and Frieden-berg allow extraneous restrictions on beliefs. Hence, ‘modified consistentpairs’ and ‘self-admissible sets’ need not correspond to profiles of fullypermissible sets. However, if there is a unique fully permissible set for

each player, then the pair constitutes both a ‘modified consistent pair’and a ‘self-admissible set’. Basu and Weibull’s (1991) ‘tight curb* set’is another variant of a consistent pair that ensures existence withoutyielding forward induction in G

1, as they impose ‘caution’ but weaken‘robust belief of opponent rationality’ to ‘belief of opponent rationality’.In particular, the set of permissible strategy profiles is ‘tight curb*’.

‘Caution’ and ‘robust belief of opponent rationality’ are admissibilityrequirements on the preferences of players, thus positioning the analysisof the present chapter in the ‘consistent preferences’ approach. More-over, by imposing ‘no extraneous restrictions on beliefs’ as a requirementof minimal completeness, preferences are not in general representable by

subjective probabilities, thus showing the usefulness of an analysis thatrelax completeness. 5

5By not employing subjective probabilities, the analysis is related to the filter model of beliefspresented by Brandenburger (1997, 1998).



Chapter 12

APPLYING FULL PERMISSIBILITY

TO EXTENSIVE GAMES

In many economic contexts decision makers interact and take actionsthat extend through time. A bargaining party makes an offer, whichis observed by the adversary, and accepted, rejected or followed by acounter-offer. Firms competing in markets choose prices, levels of adver-tisement, or investments with the intent of thereby influencing the futurebehavior of competitors. One could add many examples. The standardeconomic model for analyzing such situations is that of an extensive

game . Reproducing joint work with Martin Dufwenberg, cf. Asheim andDufwenberg (2003b), this chapter revisits a question that was alreadyposed in Chapters 7–10: What happens in an extensive game if playersreason deductively by trying to figure out one another’s moves? We havein Asheim and Dufwenberg (2003a), incorporated in Chapter 11 of thisbook, proposed a model for deductive reasoning leading to the conceptof ‘fully permissible sets’, which can be applied to many strategic situa-tions. In the present chapter we argue that the model is appropriate foranalyzing extensive games and we apply it to several such games.

12.1 Motivation

There is already a literature exploring the implications of deductivereasoning in extensive games, but the answers provided differ and theissue is controversial. Much of the excitement concerns whether or notdeductive reasoning implies backward induction in games where thatprinciple is applicable. We next discuss this issue, since it provides auseful backdrop against which to motivate our own approach.




1 2 1 0F f F 3

D d D

1 0 30 2 0

d f

D

FD

FF

1, 0 1, 0

0, 2 3, 0

0, 2 0, 3

Figure 12.1. Γ11 and its pure strategy reduced strategic form.

Consider the 3-stage “take-it-or-leave-it”, introduced by Reny (1993)(a version of Rosenthal’s, 1981, “centipede” game, see Γ

3 of Figure 2.4),

and shown in Figure 12.1 together with its pure strategy reduced strate-gic form.1 What would 2 do in Γ11 if called upon to play? Backwardinduction implies that 2 would choose d, which is consistent with the fol-lowing idea: 2 chooses d because she “figures out” that 1 would chooseD at the last node. Many models of deductive reasoning support thisstory, starting with Bernheim’s concept of ‘subgame rationalizability’and Pearce’s concept of ‘extensive form rationalizability’ (EFR). Morerecently, Battigalli and Siniscalchi (2002) provide a rigorous epistemicfoundation for EFR, while Chapters 7–10 of this book epistemicallymodel rationalizability concepts that resemble ‘subgame rationalizabil-ity’.

However, showing that backward induction can be given some kindof underpinning does not imply that the underpinning is convincing.Indeed, skepticism concerning backward induction can be expressed bymeans of Γ11. Suppose that each player believes the opponent will playin accordance with backward induction; i.e., 1 believes that 2 chooses dif asked to play, and 2 believes that 1 plays D at his initial note. Thenplayer 1 prefers playing D to any of his two other strategies FD andFF . Moreover, if 2 is certain that 1 believes that 2 chooses d if she wereasked to play, then 2 realizes that 1 has not chosen in accordance with hispreferences if she after all is asked to play. Why then should 2 believethat 1 will make any particular choice between his two less preferredstrategies, FD and FF , at his last node? So why then should 2 prefer d

to f ?This kind of perspective on the “take-it-or-leave-it” game is much in-

spired by the approach proposed by Ben-Porath (1997), where similar

1We need not consider what players plan to do at decision nodes that their own strategyprecludes them from reaching (cf. Section 12.2).



Applying full permissibility to extensive games 157

objections against backward inductive reasoning are raised. We shalldiscuss his contribution in some detail, since the key features of our ap-proach can be appreciated via a comparison to his model. Applied toΓ11, Ben-Porath’s model captures the following intuition: Each playerhas an initial belief about the opponent’s behavior. If this belief iscontradicted by the play (a “surprise” occurs) he may subsequently en-tertain any belief consistent with the path of play. The only restrictionimposed on updated beliefs is Bayes’ rule. In Γ11, Ben-Porath’s modelallows player 2 to make any choice. In particular, 2 may choose f if sheinitially believes with probability one that player 1 will choose D, andconditionally on D not being chosen assigns sufficient probability on FF .

This entails that if 2 initially believes that 1 will comply with backwardinduction, then 2 need not follow backward induction herself.

In Γ11, our analysis captures much the same intuition as Ben-Porath’sapproach, and it has equal cutting power in this game. However, it yieldsa more structured solution as it is concerned with what strategy subsetsthat are deemed to be the set of rational choices for each player. Whileagreeing with Ben-Porath that deductive reasoning may lead to each of D and FD being rational for 1 and each of d and f being rational for 2,our concept of full permissibility predicts that 1’s set of rational choicesis either {D} or {D, F }, and 2’s set of rational choices is either {d} or{d, f }. This has appealing features. If 2 is certain that 1’s set is {D},

then—unless 2 has an assessment of the relative likelihood of 1’s lesspreferred strategies FD and FF —one cannot conclude that 2 prefers dto f or vice versa; this justifies {d, f } as 2’s set of rational choices. Onthe other hand, if 2 considers it possible that 1’s set is {D, FD }, thend weakly dominates f on this set and justifies {d} as 2’s set of rationalchoices. Similarly, one can justify that D is preferred to FD if and onlyif 1 considers it impossible that 2’s set is {d, f }.

This additional structure is important for the analysis of Γ6, illus-

trated in Figure 8.2. This game is due to Reny (1992, Figure 1) and hasappeared in many contributions. Suppose in this game that each playerbelieves the opponent will play in accordance with backward induction

by choosing FF and f respectively. Then both players will prefer FF and f to any alternative strategy. Moreover, as will be shown in Sec-tion 12.3, our analysis implies that {FF } and {f } are the unique sets of rational choices.

Ben-Porath’s approach, by contrast, does not have such cutting powerin Γ

6, as it entails that deductive reasoning may lead to each of thestrategies D and FF being rational for 1 and each of the strategies d and




f being rational for 2. The intuition for why the strategies D and d areadmitted is as follows: D is 1’s unique best strategy if he believes withprobability one that 2 plays d. Player 1 is justified in this belief in thesense that d is 2’s best strategy if she initially believes with probabilityone that 1 will choose D , and if called upon to play 2 revises this belief so as to believe with sufficiently high probability (e.g., probability one)that 1 is using FD . This belief revision is consistent with Bayes’ rule,and so is acceptable.

Ben-Porath’s approach is a very important contribution to the liter-ature, since it is a natural next step if one accepts the above critiqueof backward induction. Yet we shall argue below that it is too permis-

sive, using Γ6 as an illustration. Assume that 1 deems d infinitely morelikely than f , while 2 deems D infinitely more likely than FD and FD infinitely more likely than FF . Then the players rank their strategies asfollows:

1’s preferences: D FF FD 2’s preferences: d f

This is in fact precisely the justification of the strategies D and d givenabove when applying Ben-Porath’s approach to Γ

6. Here, ‘caution’ issatisfied since all opponent strategies are taken into account; in partic-ular, FF is preferred to FD as the former strategy weakly dominatesthe latter. Moreover, ‘robust belief of opponent rationality’ is satisfiedsince each player deems the opponent’s maximal strategy infinitely morelikely that any non-maximal strategy. However, the requirement of ‘noextraneous restrictions on beliefs’, as described in Chapter 11, is notsatisfied since the preferences of 2 introduce extraneous restrictions onbeliefs by deeming one of 1’s non-maximal strategies, FD , infinitely morelikely than another non-maximal strategy, FF . When we return to G

6

in Section 12.3, we show how the additional imposition of ‘no extrane-ous restrictions on beliefs’ means that deductive reasoning leads to theconclusion that {FF } and {f } are the players’ choice sets in this game.

As established in Chapter 11, our concept of fully permissible sets ischaracterized by ‘caution’, ‘robust belief of opponent rationality’, and

‘no extraneous restrictions on beliefs’. In Section 12.2 we prove resultsthat justify the claim that interesting implications of deductive reasoningin a given extensive game can be derived by applying this concept to thestrategic form of that game.

Sections 12.3 and 12.4 are concerned with such applications, with theaim of showing how our solution concept gives new and economically rel-evant insights into the implications of deductive reasoning in extensive






other if and only if the one weakly dominates the other on Y j — theset of strategies that player i deems to be the set of rational choices forhis opponent — or S j — the set of all opponent strategies. Hence, thestrategy is maximal at the outset of a corresponding extensive game.Corollary 2 makes the observation that this strategy is still maximalwhen the preferences have been updated upon reaching any informationset that the choice of this strategy does not preclude.

Assume that player i’s preferences over his own strategies satisfy that pi is preferred to q i if and only if pi weakly dominates q i on Y j or S j.Let, for any h ∈ H i, Y j(h) := Y j ∩ S j(h) denote the set of strategiesin Y j that are consistent with the information set h being reached. If

pi, q i (∈ ∆(S i(h))), then i’s preferences conditional on the informationset h ∈ H i being reached satisfy that pi is preferred to q i if and onlyif pi weakly dominates q i on Y j (h) or S j(h) (where it follows from thedefinition that weak dominance on Y j (h) is not possible if Y j(h) = ∅).Furthermore, i’s choice set conditional on h ∈ H i, S i

Y j (h), is given by

S iY j(h) := S i(h) \ {si ∈ S i(h)| ∃xi ∈ ∆(S i(h)) s.t.

xi weakly dominates si on Y j(h) or S j(h)} .

Write S iY j := S i

Y j (∅) (= S i\Di(Y j) in earlier notation). By the resultbelow, if si is maximal at the outset of an extensive game, then it is also

maximal at later information sets for i that si does not preclude.Corollary 2 Let (∅ =) Y j ⊆ S j . If si ∈ S i

Y j , then si ∈ S iY j (h) for

any h ∈ H i with S i(h) si.

Proof. This follow from Lemma 11 by letting i’s preferences (at ti)on i’s set of mixed strategies satisfy that pi is preferred to q i if and onlyif pi weakly dominates q i on Y j or S j.

By the assumption of ‘caution’, each player i takes into account thepossibility of reaching any information set for i that the player’s ownstrategy does not preclude from being reached. Hence, ‘rationality’ im-plies ‘weak sequential rationality’; i.e., that a player chooses rationallyat all information sets that are not precluded from being reached by theplayer’s own strategy.

Reduced strategic form. It follows from Proposition 45 below thatit is in fact sufficient to consider the pure strategy reduced strategicform when deriving the fully permissible sets of the game. The followingdefinition is needed.




Definition 23 Let G = (S 1, S 2, u1, u2) be a finite strategic two-playergame. The pure strategies si and s

i (∈ S i) are equivalent if for eachplayer k , uk(s

i, s j) = uk(si, s j ) for all s j ∈ S j. The pure strategy reduced strategic form (PRSF) of G is obtained by letting, for each player i,each class of equivalent pure strategies be represented by exactly onepure strategy.

Since the maximality of one of two equivalent strategies implies thatthe other is maximal as well, the following observation holds: If si ands

i are equivalent and σi is a fully permissible set for i, then si ∈ σi if and only if s

i ∈ σi. To see this formally, note that if si ∈ σi for some

fully permissible set σi, then, by Proposition 39(ii), there exists (∅ =)Ψ j ⊆ Π j such that si ∈ σi = S i

Y j for Y j = ∪σj∈Ψjσ

j . Since si and si are

equivalent, si ∈ S i

Y j = σi. This observation explains why the followingresult can be established.

Proposition 45 Let G = (S 1, S 2, u1, u2) be a finite strategic two-player game where si and s

i are two equivalent strategies for i. Consider G =(S 1, S 2, u1, u2) where S i = S i\{s

i} and S j = S j for j = i, and where, for each player k, uk is the restriction of uk to S = S 1 × S 2. Let, for each player k, Πk (Πk) denote the collection of fully permissible sets for k in G ( G). Then Πi is obtained from Πi by removing s

i from any σi ∈ Πi

with si ∈ σi, while, for j = i, Π j = ˜Π j.

Proof. By Proposition 39(iii) it suffices to show that

1 If Ξ ⊆ α(Ξ) for G, then Ξ ⊆ α(Ξ) for G, where Ξi is obtained fromΞi by removing s

i from any σi ∈ Ξi with si ∈ σi, while, for j = i,Ξ j = Ξ j .

2 If Ξ ⊆ α(Ξ) for G, then Ξ ⊆ α(Ξ) for G, where Ξi is obtained fromΞi by adding s

i to any σi ∈ Ξi with si ∈ σi, while, for j = i, Ξ j = Ξ j.

Part 1. Assume Ξ ⊆ α(Ξ). By the observation preceding Proposition45, if σi ∈ Πi, then s

i ∈ σi if and only if si ∈ σi. Pick any player k

and any σk ∈ Πk. Let denote the other player. By the definition of αk(·), there exists (∅ =) Ψ ⊆ Π such that σk = S k

Y for Y = ∪σ∈Ψσ

.Construct Ψi by removing s

i from any σi ∈ Ψi with si ∈ σi and replaceS i by S i, while, for j = i, Ψ j = Ψ j and S j = S j. Let Y = ∪σ∈Ψ

σ.

Then it follows from the definition of S kY that S k

Y = σk\{sk} if k = i

and S kY = σk if k = i. Since, for each player k , (∅ =) Ψk ⊆ Ξk, we have

that Ξ ⊆ α(Ξ). Part 2 is shown similarly.




Proposition 45 means that the PRSF is sufficient for analyzing com-mon certain belief of full admissible consistency, which is the epistemicfoundation for the concept of fully permissible sets. Consequently, in thestrategic form of an extensive game, it is unnecessary to specify actionsat information sets that a strategy precludes from being reached. Hence,instead of fully specified strategies, it is sufficient to consider what Ru-binstein (1991) calls plans of action . For a generic extensive game, theset of plans of action is identical to the strategy set in the PRSF.

In the following two sections we apply the concept of fully permissiblesets to extensive games. We organize the discussion around two themes:

backward and forward induction. Motivated by Corollary 2 and Proposi-tion 45, we analyze each extensive game via its PRSF (cf. Definition 23),given in conjunction to the extensive form. In each example, each planof action that appears in the underlying extensive game corresponds toa distinct strategy in the PRSF.

12.3 Backward induction

Does deductive reasoning in extensive games imply backward induc-tion? In this section we show that the answer provided by the conceptof fully permissible sets is “sometimes, but not always”.

‘Sometimes.’ There are many games where Ben-Porath’s approachdoes not capture backward induction while our approach does (and theconverse is not true). Ben-Porath (1997) assumes ‘initial common cer-tainty of rationality’ in extensive games of perfect information. As dis-cussed in Chapter 7 he proves that in generic games (with no payoff ties at terminal nodes for any player) the outcomes consistent with thatassumption coincide with those that survive the Dekel-Fudenberg proce-dure (where one round of elimination of all weakly dominated strategiesis followed by iterated elimination of strongly dominated strategies).

It is a general result that the concept of fully permissible sets refines

the Dekel-Fudenberg procedure (cf. Proposition 40). Game Γ6 of Figure8.2 shows that the refinement may be strict even for generic extensivegames with perfect information, and indeed that fully permissible setsmay respect backward induction where Ben-Porath’s solution does not.The strategies surviving the Dekel-Fudenberg procedure, and thus con-sistent with ‘initial common certainty of rationality’, are D and FF forplayer 1 and d and f for player 2. In Section 12.2 we gave an intuition for




why the strategies D and d are possible. This is, however, at odds withthe implications of common certain belief of full admissible consistency.

Applying IECFA to the PRSF of Γ6 of Figure 8.2 yields:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{D}, {FF }, {D, FF }} × Σ2

Ξ(2) = {{D}, {FF }, {D, FF } } × { {f }, {d, f }}

Ξ(3) = {{FF }, {D, FF } } × { {f }, {d, f }}

Ξ(4) = {{FF }, {D, FF } } × { {f }}

Π = Ξ(5) = {{FF } } × { {f }}

Interpretation: Ξ(1): ‘Caution’ implies that FD cannot be a maxi-mal strategy (i.e., an element of a choice set) for 1 since it is weaklydominated (in fact, even strongly dominated). Ξ(2): Player 2 certainlybelieves that only {D}, {FF } and {D, FF } are candidates for 1’s choiceset. By ‘robust belief of opponent rationality’ and ‘no extraneous re-strictions on beliefs’ this excludes {d} as 2’s choice set, since d weaklydominates f only on {FD } or {D, FD }. Ξ(3): 1 certainly believes thatonly {f } and {f, d} are candidates for 2’s choice set. By ‘robust belief of opponent rationality’ and ‘no extraneous restrictions on beliefs’ this ex-cludes {D} as 1’s choice set, since D weakly dominates FD and FF only

on {d}. Ξ(4): Player 2 certainly believes that only {FF } and {D, FF }are candidates for 1’s choice set. By ‘robust belief of opponent rational-ity’ this implies that 2’s choice set is {f } since f weakly dominates don both {FF } and {D, FF }. Ξ(5): 1 certainly believes that 2’s choiceset is {f }. By ‘robust belief of opponent rationality’ this implies that{FF } is 1’s choice set since FF weakly dominates D on {f }. No furtherelimination of choice sets is possible, so {FF } and {f } are the respectiveplayers’ unique fully permissible sets.

‘Not always.’ While fully permissible sets capture backward in-duction in Γ

6 and other games, the concept does not capture backwardinduction in certain games where the procedure has been considered con-

troversial.2 The background for the controversy is the following para-doxical aspect: Why should a player believe that an opponent’s futureplay will satisfy backward induction if the opponent’s previous play isincompatible with backward induction? A prototypical game for cast-

2See discussion and references in Chapter 7 of the present text.




ing doubt on backward induction is the “take-it-or-leave-it” game Γ11 of Figure 12.1, which we next analyze in detail.

Applying IECFA to the PRSF of Γ11 of Figure 12.1 yields:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{D}, {FD }, {D, FD }} × Σ2

Ξ(2) = {{D}, {FD }, {D, FD } } × { {d}, {d, f }}

Π = Ξ(3) = {{D}, {D, FD } } × { {d}, {d, f }}

Interpretation: Ξ(1): FF cannot be a maximal strategy for 1 since

it is strongly dominated. Ξ(2): Player 2 certainly believes that only{D}, {FD } and {D, FD } are candidates for 1’s choice set. This excludes{f } as 2’s choice set since {f } is 2’s choice set only if 2 deems {FF } or{FD , FF } subjectively possible. Ξ(3): 1 certainly believes that only {d}and {d, f } are candidates for 2’s choice set, implying that {FD } cannotbe 1’s choice set. No further elimination of choice sets is possible andthe collection of profiles of fully permissible sets is as specified.

Note that backward induction is not implied. To illustrate why, wefocus on player 2 and explain why {d, f } may be a choice set for her.Player 2 certainly believes that 1’s choice set is {D} or {D, FD }. Thisleaves room for two basic cases. First, suppose 2 deems {D, FD } subjec-tively possible. Then {d} must be her choice set, since she must considerit infinitely more likely that 1 uses FD than that he uses FF . Second,and more interestingly, suppose 2 does not deem {D, FD } subjectivelypossible. Then conditional on 2’s node being reached 2 certainly believesthat 1 is not choosing a maximal strategy. As player 2 does not assessthe relative likelihood of strategies that are not maximal (cf. the require-ment of ‘no extraneous restrictions on beliefs’), {d, f } is her choice setin this case. Even in the case where 2 deems {D} to be the only subjec-tively possible choice set for 1, she still considers it subjectively possiblethat 1 may choose one of his non-maximal strategies FD and FF (cf.the requirement of ‘caution’), although each of these strategies is in thiscase deemed infinitely less likely than the unique maximal strategy D .

Applied to (the PRSF of) Γ11, our concept permits two fully permis-sible sets for each player. How can this multiplicity of fully permissiblesets be interpreted? The following interpretation corresponds to the un-derlying formalism: The concept of fully permissible sets, when appliedto Γ11, allows for two different types of each player. Consider player 2.Either she may consider that {D, FD } is a subjectively possible choiceset for 1, in which case her choice set will be {d} so that she complies




with backward induction. Or she may consider {D} to be the only subjectively possible choice set for 1, in which case 2’s choice set is {d, f }.Intuitively, if 2 is certain that 1 is a backward inducter, then 2 need notbe a backward inducter herself! In this game, our model captures anintuition that is very similar to that of Ben-Porath’s model.

Reny (1993) defines a class of “belief consistent” games, and argues onepistemic grounds that backward induction is problematic only for gamesthat are not in this class. It is interesting to note that the game whereour concept of fully permissible sets differs from Ben-Porath’s analysisby promoting backward induction, Γ

6, is belief-consistent. In contrast,the game where the present concept coincides with his by not yieldingbackward induction, Γ11, is not belief-consistent. There are examples of games that are not belief consistent, where full permissibility still impliesbackward induction, meaning that belief consistency is not necessary forthis conclusion. It is, however, an as-of-yet unproven conjecture thatbelief consistency is sufficient for the concept of fully permissible sets topromote backward induction.

We now compare our results to the very different findings of Aumann(1995), cf. also Section 5 of Stalnaker (1998) as well as Chapter 7 of thisbook. In Aumann’s model, where it is crucial to specify full strategies(rather than plans of actions), common knowledge of rational choice im-plies in Γ11 that all strategies for 1 but DD (where he takes a payoff of 1at his first node and a payoff of 3 at his last node) are impossible. Hence,it is impossible for 1 to play FD or FF and thereby ask 2 to play. How-ever, in the counterfactual event that 2 is asked to play, she optimizes asif player 1 at his last node follows his only possible strategy DD , imply-ing that it is impossible for 2 to choose f (cf. Aumann’s Sections 4b, 5b,and 5c). Thus, in Aumann’s analysis, if there is common knowledge of rational choice, then each player chooses the backward induction strat-egy. By contrast, in our analysis player 2 being asked to play is seen tobe incompatible with 1 playing DD or DF . For the determination of 2’spreference over her strategies it is the relative likelihood of FD versusFF that is important to her. As seen above, this assessment depends on

whether she deems {D, FD } as a possible candidate for 1’s choice set.

Prisoners’ dilemma. We close this section by considering a finitelyrepeated “prisoners’ dilemma” game. Such a game does not have perfectinformation, but it can still be solved by backward induction to findthe unique subgame perfect equilibrium (no one cooperates in the lastperiod, given this no one cooperates in the penultimate period, etc.).




sN T 1

sN V 1

sN E 1

sRT 1

sRV 1

sRE 1

sN T 2 sNV

2 sN E 2 sRT

2 sRV 2 sRE

2

7, 7 4, 8 4, 8 5, 5 2, 6 2, 6

8, 4 5, 5 5, 5 4, 8 1, 9 1, 9

8, 4 5, 5 5, 5 5, 5 2, 6 2, 6

5, 5 8, 4 5, 5 3, 3 6, 2 3, 3

6, 2 9, 1 6, 2 2, 6 5, 5 2, 6

6, 2 9, 1 6, 2 3, 3 6, 2 3, 3

Figure 12.2. Reduced form of Γ12 (a 3-period “prisoners’ dilemma” game).

This solution has been taken to be counterintuitive; cf, e.g. Pettit andSugden (1989). We consider the case of a 3-period “prisoners’ dilemma”game (Γ12) and show that, again, the concept of fully permissible setsdoes not capture backward induction. However, the fully permissiblesets nevertheless have considerable cutting power. Our solution refinesthe Dekel-Fudenberg procedure and generates some special “structure”on the choice sets that survive.

The payoffs of the stage game are given as follows, using Aumann’s(1987b, pp. 468–9) description: Each player decides whether he willreceive 1 (defect ) or the other will receive 3 (cooperate ). There is nodiscounting. Hence, the action defect is strongly dominant in the stagegame, but still, each player is willing to cooperate in one stage if this in-duces the other player to cooperate instead of defect in the next stage. Itfollows from Proposition 45 that we need only consider what Rubinstein(1991) calls plans of action.

There are six plans of actions for each player that survive the Dekel-Fudenberg procedure. In any of these, a player always defects in the 3rdstage, and does not always cooperate in the 2nd stage. The six plansof actions for each player i are denoted sN T

i ,sN V i , sN E

i , sRT i , sRV

i andsRE

i , where N denotes that i is nice in the sense of cooperating in the 1ststage, where R denotes that i is rude in the sense of defecting in the 1st

stage, where T denotes that i plays tit-for-tat in the sense of cooperating in the 2nd stage if and only j = i has cooperated in the 1st stage, whereV denotes that i plays inverse tit-for-tat in the sense of defecting in the2nd stage if and only if j = i has cooperated in the 1st stage, and whereE denotes that i is exploitive in the sense of defecting in the 2nd stageindependently of what j = i has played in the 1st stage. The strategicform after elimination of all other plans of actions is given in Figure




12.2. Note that none of these plans of actions are weakly dominated inthe full strategic form.

Proposition 40 shows that any fully permissible set is a subset of theset of strategies surviving the Dekel-Fudenberg procedure. Hence, onlysubsets of

{sN T i , sN V

i , sN E i , sRT

i , sRV i , sRE

i }

can be i’s choice set under common certain belief of full admissible con-sistency. Furthermore, under common certain belief of full admissibleconsistency, we have for each player i that

any choice set that contains sN T i must also contain sN E

i , since sN T i is

a maximal strategy only if sN E

i is a maximal strategy,any choice set that contains sN V

i must also contain sN E i , since sN V

i

is a maximal strategy only if sN E i is a maximal strategy,

any choice set that contains sRT i must also contain sRE

i , since sRT i is

a maximal strategy only if sRE i is a maximal strategy,

any choice set that contains sRV i must also contain sRE

i , since sRV i is

a maximal strategy only if sRE i is a maximal strategy,

Given that the choice set of the opponent satisfies these conditions, thisimplies that

if sN E i is included in i’s choice set, only the following sets are candi-

dates for i’s choice set: {sN T i , s

N E i , s

RT i , s

RE i }, {s

N V i , s

N E i , s

RV i , s

RE i },

or {sN E i , sRE

i }. The reason is that sN E i is a maximal strategy only

if i considers it subjectively possible that j ’s choice set contains sN T j

(and hence, sN E j ) or sRT

j (and hence, sRE j ).

if sRE i , but not sN E

i , is included in i’s choice set, only the follow-ing sets are candidates for i’s choice set: {sRT

i , sRE i }, {sRV

i , sRE i }, or

{sRE i }. The reason is that sRE

i is a maximal strategy only if i con-siders it subjectively possible that j’s choice set contains sNV

j , sN E j ,

sRV j , or sRE

j .

This in turn implies that

i’s choice set does not contain sN V i or s

RV i since any candidate for j ’s

choice set contains sRE j , implying that sN E

i is preferred to sN V i and

sRE i is preferred to sRV

i .

Hence, the only candidates for i’s choice set under common certain be-lief of full admissible consistency are {sN T

i , sNE i , sRT

i , sRE i }, {sN E

i , sRE i },

{sRT i , sRE

i }, and {sRE i }. Moreover, it follows from Proposition 39(iii)

that all these sets are indeed fully permissible since




{sNT i , sNE

i , sRT i , sRE

i } is i’s choice set if he deems {sRT j , sRE

j }, but not{sNE

j , sRE j } and {sN T

j , sNE j , sRT

j , sRE j }, as possible candidates for j’s

choice set,

{sNE i , sRE

i } is i’s choice set if he deems {sN T j , sN E

j , sRT j , sRE

j } as apossible candidate for j ’s choice set,

{sRT i , sRE

i } is i’s choice set if he deems {sRE j } as the only possible

candidate for j ’s choice set,

{sRE i } is i’s choice set if he deems {sN E

j , sRE j }, but not {sRT

j , sRE j }

and {sN T j , sN E

j , sRT j , sRE

j }, as possible candidates for j’s choice set.

While play in accordance with strategies surviving the Dekel-Fudenbergprocedure does not provide any prediction other than both players defecting in the 3rd stage, the concept of fully permissible sets has morebite. In particular, a player cooperates in the 2nd stage only if the oppo-nent has cooperated in the 1st stage. This implies that only the followingpaths can be realized if players choose strategies in fully permissible sets:

((cooperate , cooperate ), (cooperate , cooperate ), (defect , defect ))

((cooperate , cooperate ), (cooperate , defect ), (defect , defect )) and vice versa

((cooperate , defect ), (defect , cooperate ), (defect , defect )) and vice versa

((cooperate , cooperate ), (defect , defect ), (defect , defect ))((cooperate , defect ), (defect , defect ), (defect , defect )) and vice versa

((defect , defect ), (defect , defect ), (defect , defect )).

That the path ((cooperate , defect ), (cooperate , defect ), (defect , defect ))or vice versa cannot be realized if players choose strategies in fully per-missible sets can be interpreted as an indication that the present analysisseems to produce some element of reciprocity in the 3-period “prisoners’dilemma” game.

12.4 Forward induction

In Chapter 11 we have already seen how the concept of fully per-missible sets promotes the forward induction outcome, (InL, ), in thePRSF of the “battle-of-the-sexes with an outside option” game Γ

1, il-lustrated in Figure 2.6. In this section we first investigate whether thisconclusion carries over to two other variants of the “battle-of-the-sexes”game, before testing the concept of fully permissible sets in an economicapplication.




r r rr

NU

ND

BU

BD

3, 1 3, 1 0, 0 0, 0

0, 0 0, 0 1, 3 1, 3

2, 1 -1, 0 2, 1 -1, 0

-1, 0 0, 3 -1, 0 0, 3

Figure 12.3. G13 (the pure strategy reduced strategic form of “burning money”).

The “battle-of-the-sexes” game with variations. Consider firstthe “burning money” game due to van Damme (1989) and Ben-Porath

and Dekel (1992). Game G13 of Figure 12.3 is the PRSF of a “battle-of-the-sexes” game with the addition that 1 can publicly destroy 1 unitof payoff before the “battle-of-the-sexes” game starts. BU (NU ) is thestrategy where 1 burns (does not burn), and then plays U , etc., while ris the strategy where 2 responds with conditional on 1 not burning andr conditional on 1 burning, etc. The forward induction outcome (sup-ported e.g. by IEWDS) involves implementation of 1’s preferred “battle-of-the-sexes” outcome, with no payoff being burnt .

One might be skeptical to the use of IEWDS in the “burning money”game, because it effectively requires 2 to infer that BU is infinitely morelikely than BD based on the sole premise that BD is eliminated before

BU , even though all strategies involving burning (i.e. both BU andBD ) are eventually eliminated by the procedure. On the basis of thispremise such an inference seems at best to be questionable. As shownin Table 12.1, the application of our algorithm IECFA yields a sequenceof iterations where at no stage need 2 deem BU infinitely more likelythan BD , since {NU } is always included as a candidate for 1’s choice set.The procedure uniquely determines {NU } as 1’s fully permissible set and{, r} as 2’s fully permissible set. Even though the forward inductionoutcome is obtained, 2 does not have any assessment concerning therelative likelihood of opponent strategies conditional on burning; hence,she need not interpret burning as a signal that 1 will play according with

his preferred “battle-of-the-sexes” outcome.3

3Also Battigalli (1991), Asheim (1994), and Dufwenberg (1994), as well as Hurkens (1996) ina different context, argue that (NU , r) in addition to (NU , ) is viable in “burning money”.




Table 12.1. Applying IECFA to “burning money”.

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{NU }, {ND }, {BU }, {NU , ND }, {ND , BU }, {NU , BU }, {NU , ND , BU }}

× Σ2

Ξ(2) = {{NU }, {ND }, {BU }, {NU , ND }, {ND , BU }, {NU , BU }, {NU , ND , BU }}

× {{}, {r}, {,r}, {r,rr}, {,r}, {,r,r,rr}}

Ξ(3) = {{NU }, {BU }, {ND , BU }, {NU , BU }, {NU , ND , BU }}

× {{}, {r}, {,r}, {r,rr}, {,r}, {,r,r,rr}}

Ξ(4) = {{NU }, {BU }, {ND , BU }, {NU , BU }, {NU , ND , BU }}

× {{}, {r}, {,r}, {,r}}Ξ(5) = {{NU }, {BU }, {NU , BU } } × { {}, {r}, {,r}, {,r}}

Ξ(6) = {{NU }, {BU }, {NU , BU } } × { {}, {,r}, {,r}}

Ξ(7) = {{NU }, {NU , BU } } × { {}, {,r}, {,r}}

Ξ(8) = {{NU }, {NU , BU } } × { {}, {,r}}

Ξ(9) = {{NU } } × { {}, {,r}}

Π = Ξ(10) = {{NU } } × { {,r}}

We next turn now to a game introduced by Dekel and Fudenberg(1990) (cf. their Figure 7.1) and discussed by Hammond (1993), andwhich is reproduced here as Γ

1 of Figure 12.4. It is a modification of

Γ1 which introduces an “extra outside option” for player 2. In thisgame there may seem to be a tension between forward and backward

induction: For player 2 not to choose out may seem to suggest that 2“signals” that she seeks a payoff of at least 3/2, in contrast to the payoff of 1 that she gets when the subgame structured like Γ

1 is considered inisolation (as seen in the analysis of Γ

1). However, this intuition is notquite supported by the concept of fully permissible sets.

Applying our algorithm IECFA to the PRSF of Γ1 yields:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{Out}, {InL}, {Out, InL}}

× {{out}, {inr}, {out, in}, {out, inr}, {in, inr}, {out, in, inb}}Ξ(2) = {{Out}, {InL}, {Out, InL} } × { {out}, {out, in}, {in, inr}}

Ξ(3) = {{InL}, {Out, InL} } × { {out}, {out, in}, {in, inr}}

Π = Ξ(4) = {{InL}, {Out, InL} } × { {out}, {out, in}} .

The only way for Out to be a maximal strategy for player 1 is thathe deems {out} as the only subjectively possible candidate for 2’s choice








This situation is depicted by the extensive game Γ14. Again, we ana-lyze the PRSF (cf. Figure 12.5). Application of IECFA yields:

Ξ(0) = Σ1 × Σ2

Ξ(1) = {{Out}, {InH }, {Out, InH } } × { {out}, {inh}, {out, inh}}

Ξ(2) = {{InH }, {Out, InH } } × { {inh}, {out, inh}}

Π = Ξ(3) = {{InH } } × { {inh}} .

Interpretation: Ξ(1): Shirking cannot be a maximal strategy for eitherworker since it is weakly dominated. Ξ(2): This excludes the possibilitythat a worker’s choice set contains only the outside option. Ξ(3): Since

each worker certainly believes that hard work is , while shirking is not , anelement of the opponent’s choice set, it follows that each worker deems itinfinitely more likely that the opponent chooses hard work rather thanshirking. This means that, for each worker, only hard work is in hischoice set, a conclusion that supports Schotter’s argument.

12.5 Concluding remarks

In this final chapter of the book we have explored the implications of the concept of fully permissible sets in extensive games. In Chapter 11we have already seen—based on Asheim and Dufwenberg (2003a)—thatthis concept can be characterized as choice sets under common certain

belief of full admissible consistency. Full admissible consistency consistsof the requirements

‘caution’,

‘robust belief of opponent rationality’, and

‘no extraneous restrictions on beliefs’,

and entails that one strategy is preferred to another if and only if theformer weakly dominates the latter on the union of the choice sets thatare deemed possible for the opponent, or on the set of all opponentstrategies.

The requirement of ‘robust belief of opponent rationality’ is concerned

with strategy choices of the opponent only initially, in the whole game ,not with choices among the remaining available strategies at each and every information set . To illustrate this point, look back at Γ11 andconsider a type of player 2 who deems {D} as the only subjectivelypossible choice set for 1. Conditional on 2’s node being reached it isclear that 1 cannot be choosing a strategy that is maximal given hispreferences. Conditional on 2’s node being reached, the modeling of the




current chapter imposes no constraint on 2’s assessment of likelihoodconcerning which non-maximal strategy FF or FD that 1 has chosen.This crucially presumes that 2 assesses the likelihood of different strate-gies as chosen by player 1 initially, in the whole game .

It is possible to model players being concerned with opponent choicesat all information sets . In Γ11 this would amount to the following whenplayer 2 is of a type who deems {D} as the only possible choice setfor 1: Conditional on 2’s node being reached she realizes that 1 cannotbe choosing a strategy which is maximal given his preferences. Still,2 considers it infinitely more likely that 1 at his last node chooses astrategy that is maximal among his remaining available strategies given

his conditional preferences at that node. In Section 12.2 we arguedwith Ben-Porath (1997) that this is not necessarily reasonable, a viewwhich permeates the working hypotheses on which the current chapterin grounded.

Yet, research on the basis of this alternative approach is illuminatingand worthwhile. Indeed, Chapters 7–9 of this book have reproduced theepistemic models of Asheim (2002) and Asheim and Perea (2004) whereeach player believes that his opponent chooses rationally at all informa-tion sets. The former model yields an analysis that is related to Bern-heim’s (1984) subgame rationalizability, while the latter model demon-strates how it—in accordance with Bernheim’s conjecture—is possible to

define sequential rationalizability. Moreover, Chapter 10 has consideredthe closely related strategic form analyses of Schuhmacher (1999) andAsheim (2001) that define and characterize proper rationalizability as anon-equilibrium analog to Myerson’s (1978) proper equilibrium.

Analysis that goes in this alternative direction promotes concepts thatimply backward induction without yielding forward induction. Thus,they lead to implications that are significantly different from those of the current final chapter, where forward induction is promoted withoutinsisting on backward induction in all games.

The tension between these two approaches to extensive games can-not be resolved by formal epistemic analysis alone. It is worth noting,

though, that the analysis—independently of this issue—makes use of the‘consistent preferences’ approach to deductive reasoning in games.



Appendix A

Proofs of results in Chapter 4

Proof of Proposition 6. Only if. Assume that d is admissible on E . Let

e ∈ E and f ∈ ¬E . It now follows directly that e is not Savage-null at and that

p d{e} q implies p d

{e,f } q. If. Assume that e ∈ E and f ∈ ¬E imply e d f . Let

p and q satisfy that pE weakly dominates qE at d. Then there exists e0 ∈ E such

that υd(p(e0)) > υd(q(e0)). Write ¬A = {f 1, . . . , f n}. Let, for m ∈ {0, . . . , n},

pm(d) =

n+1−mn+1

p(d) + mn+1

q(d) if d = e0

p(d) if d ∈ E \e0

q(d) if d = f m and m ∈ {1, . . . , m}

p(d) if d = f m and m ∈ {m + 1, . . . , n}.

Then p = p0, pm−1 d pm for all m = {1, . . . , n} (since e ∈ E and f ∈ ¬E

imply that e d f ), and p(n) d q (since p(n) weakly dominates q at d with

υd(pn(e0)) > υd(q(e0))). By transitivity of e, it follows that p d q.

Proof of Proposition 7. ( Q serial.) If d is Savage-null at d, then there exists

e ∈ τ d such that e is not Savage-null at d since d is nontrivial. Clearly, d is not

infinitely more likely than e at d, and dQe. If d is not Savage-null at d, then dQd

since d is not infinitely more likely than itself at d.

( Q transitive.) We must show that dQe and eQf imply dQf . Clearly, dQe and

eQf imply d ≈ e ≈ f , and that f is not Savage-null at d. It remains to be shown

that d d f does not hold if dQe and eQf . Suppose to the contrary that d d f .

It suffices to show that dQe contradicts eQf . Since f is not Savage-null at d ≈ e,

e d f is needed to contradict eQf . This follows from Axiom 11 because dQe entailsthat d d e does not hold.

( Q satisfies forward linearity.) We must show that dQe and dQf imply eQf or

fQe. From dQe and dQf it follows that d ≈ e ≈ f and that both e and f are not

Savage-null at e ≈ f . Since e e f and f f e cannot both hold, we have that eQf

or f Qe.

( Q satisfies quasi-backward linearity.) We must show that dQf and eQf imply

dQe or eQd if ∃d ∈ F such that dQe. From dQf and eQf it follows that d ≈ e ≈ f ,




while dQe implies that e is not Savage-null at d ≈ d ≈ e. If d is Savage-null at d,

then d d e cannot hold, implying that dQe. If d is not Savage-null at d ≈ e, then

d d e and e e d cannot both hold, implying that dQe or eQd.

Proof of Proposition 8. ( R serial.) For all d ∈ F , ρd = ∅.

( R transitive.) We must show that dRe and eRf imply dRf . Since dRe

implies that d ≈ e, we have that ρd = ρe . Now, eRf (i.e., f ∈ ρe) implies dRf (i.e.,

f ∈ ρd ).

( R Euclidean.) We must show that dRe and dRf imply eRf . Since dRe

implies that d ≈ e, we have that ρd = ρe . Now, dRf (i.e., f ∈ ρd ) implies eRf (i.e.,

f ∈ ρe).

( dRe implies dR+1e.) This follows from the property that ρd ⊆ ρd+1.

( ∃f such that dR+1f and eR+1f ) implies ( ∃f such that dRf and eRf ). Since

dR+1f implies that d ≈ f and eR+1f implies that e ≈ f , we have that d ≈ e andρd = ρe . By the non-emptiness of this set, ∃f such that dRf and eRf .

Proof of Proposition 9. (i) ( dQd is equivalent to d being not Savage-null at

d.) If dQd, then it follows directly from Definition 2 that d is not Savage-null at d.

If d is not Savage-null at d, then by Definition 2 it follows that dQd since d ≈ d and

not d d d. ( dRLd is equivalent to d being not Savage-null at d.) By Definition 3,

dRLd iff d ∈ ρdL = κd, which directly establishes the result.

(ii) Only if. Assume that dQe and not eQd. From dQe it follows that d ≈ e and

e is not Savage-null at d, i.e. e ∈ κd(⊆ τ d). Consider E := {e ∈ F | eQe}. Clearly,

e ∈ E ⊆ κd(⊆ τ d) and d ∈ τ d\E = ∅. If e ∈ E and f ∈ τ d\E , then not eQf ,

since otherwise it would follow from eQf and the transitivity of Q that eQf , thereby

contradicting f /∈ E . If, on the one hand, f ∈ κd\E , then e d f since f is not

Savage-null at d ≈ e

and e

Qf does not hold. If, on the other hand, f /∈ κd

, thene d f since f is Savage-null at d and e is not. Hence, e ∈ E and f ∈ ¬E imply

e d f . By Proposition 6, d is admissible on E , entailing that ∃ ∈ {1, . . . , L} such

that ρd = E . By Definition 3, dRe and not eRd since e ∈ E and d ∈ τ d\E .

If. Assume that ∃ ∈ {1, . . . , L} such that dRe and not eRd. From dRe it

follows that d ≈ e and e ∈ ρd (⊆ κd); in particular, e is not Savage-null at d. Since

eRd does not hold, however, d /∈ ρe = ρd . By construction, d is admissible on ρd ,

and it now follows from Proposition 6 that e d d. Furthermore, e d d implies that

d d e does not hold. Hence, dQe since d ≈ e, e is not Savage-null at d and d d e

does not hold, while not eQd since e d d.

Proof of Lemma 3. Since κd = {e ∈ τ d|eQe}, it follows that ∃e1 ∈ τ d ∩ φ

such that e1Qe1 if κd ∩ φ = ∅. Either, ∀f ∈ τ d ∩ φ, fQe1 – in which case we are

through – or not. In the latter case, ∃e2 ∈ τ d ∩ φ such that e2Qe1 does not hold.

Since e1, e2 ∈ τ d, ∃e2 ∈ τ d such that e1Qe2 and e2Qe2. Since e1Qe1 and not e2Qe1 it

now follows from quasi-backward linearity that e1Qe2. Moreover, not e2Qe1 implies

e2 = e1. Either ∀f ∈ τ d ∩ φ, fQe2 – in which case we are through – or not. In the

latter case we can, by repeating the above argument and invoking transitivity, show

the existence of some e3 ∈ τ d ∩ φ such that e1Qe3, e2Qe3, and e3 = e1, e2. Since

τ d ∩ φ is finite, this algorithm converges to some e satisfying, ∀f ∈ τ d ∩ φ,fQe.

To prove Proposition 11 it suffices to show the following lemma.



Appendix A: Proofs of results in Chapter 4 177

Lemma 14 If φ ∈ Φ, then β d(φ) = ρd ∩ φ, where := min{k ∈ {1, . . . , L}| ρdk ∩ φ =∅}.

Proof. (β d(φ) ⊆ ρd ∩ φ) Assume that (τ d ∩ φ)\ρd = ∅. Let e ∈ (τ d ∩ φ)\ρd .

Since ρd ∩ φ = ∅, ∃f ∈ ρd ∩ φ. Then, by Definition 3 eRf and not fRe, which

by Proposition 9(ii) implies eQf and not fQe. Hence, e ∈ (τ d ∩ φ)\β d(φ), and

ρd ∩ φ = (τ d ∩ φ) ∩ ρd ⊇ (τ d ∩ φ) ∩ β d(φ) = β d(φ). Assume then that (τ d ∩ φ)\ρd = ∅.

In this case, ρd ∩ φ = (τ d ∩ φ) ∩ ρd = τ d ∩ φ ⊇ β d(φ).

(ρd ∩ φ ⊆ β d(φ)) Let e ∈ ρd ∩ φ. If f ∈ ρd ∩ φ, then fRLf since ρd ⊆ ρdL, and

fQf by Proposition 9(i). Since e, f ∈ τ d and fQf , it follows by quasi-backward

linearity of Q that f Qf or eQf . However, since by construction, ∀k ∈ {1, . . . − 1},

ρdk ∩ φ = ∅, there is no k ∈ {1, . . . − 1} such that f Rke and not eRkf or vice versa,

and Proposition 9(ii) implies that both f Qe and eQf must hold. In particular, f Qe.

If, on the other hand, f ∈ (τ d ∩ φ)\ρd , then by Definition 3 fRe and not eRf ,

implying by Proposition 9(ii) that fQe. Thus, ∀f ∈ τ d ∩ φ, fQe, and e ∈ β d(φ)

follows.

Proof of Proposition 12. Recall that B0E := ∩φ∈ΦEB(φ)E , where ΦE :=

∩d∈F ΦdE is non-empty and defined by, ∀d ∈ F , Φd

E := {φ ∈ Φd| E ∩ κd ∩ φ =∅ if E ∩ κd = ∅ }.

(If ∃ ∈ {1, . . . , L} such that ρd = E ∩ κd, then d ∈ B0E .) Let ρd = E ∩ κd

and consider any φ ∈ ΦE. We must show that d ∈ B(φ)E . By the definition of ΦE ,

E ∩ κd ∩ φ = ∅ since φ ∈ ΦE and E ∩ κd = ρd = ∅. Since ρd ∩ φ = E ∩ κd ∩ φ, it

follows that ∅ = ρd ∩ φ ⊆ E , so by Proposition 11, d ∈ B(φ)E .

(If d ∈ B0E , then ∃ ∈ {1, . . . , L} such that ρd = E ∩ κd.) Let d ∈ B0E ; i.e.,

∀φ ∈ ΦE, d ∈ B(φ)E . We first show that ρd1 ⊆ E . Consider some φ ∈ ΦE satisfying

τ d

∩ φ

= (E ∩ τ d

) ∪ ρd1. Since d ∈ B(φ

)E , ∃k ∈ {1, . . . , L} such that ∅ = ρdk ∩

φ = ρdk ∩ (E ∪ ρd1) ⊆ E . Since ρd1 ⊆ ρdk, ρd1 ⊆ E . Let = max{k|ρdk ⊆ E }. If

= L, then ρd = κd, and ρd ⊆ E implies ρd = E ∩ κd. If < L, then, since

ρd ⊂ ρdL = κd, ρd = ρd ∩ κd ⊆ E ∩ κd. To show that ρd = E ∩ κd also in this

case, suppose instead that (E ∩ κd)\ρd = ∅, and consider some φ ∈ ΦE satisfying

τ d∩φ = ((E ∩κd)∪ρd+1)\ρd . Since, ∀k ∈ {1, . . ,}, ρdk ⊆ ρd , it follows from ρd∩φ = ∅

that, ∀k ∈ {1, . . ,}, ρdk ∩ φ = ∅. Since by construction, ρd ⊆ E , while ρd+1 ⊆ E does

not hold, ρd+1 ∩ φ = ρd+1\ρd is not included in E . Since ρd1 ⊂ · · · ⊂ ρdL, there is

no k ∈ {0, . . . , L} such that ∅ = ρdk ∩ φ ⊆ E , contradicting by Proposition 11 that

d ∈ B(φ)E . Hence, ρd = E ∩ κd.

Proof of Proposition 14. (KE ∩ KE = K(E ∩ E )) To prove KE ∩ KE ⊆

K(E ∩ E ), let d ∈ KE and d ∈ KE . Then, by Definition 4, κd ⊆ E and κd ⊆ E and

hence, κd ⊆ E ∩ E , implying that d ∈ K(E ∩ E ). To prove KE ∩ KE ⊇ K(E ∩ E ),

let d ∈ K(E ∩ E ). Then κd ⊆ E ∩ E and hence, κd ⊆ E and κd ⊆ E , implying thatd ∈ KE and d ∈ KE .

(B(φ)E ∩ B(φ)E = B(φ)(E ∩ E )) Using Defintion 5 the proof of conjunction for

B(φ) is identical to the one for K except that β d(φ) is substituted for κd.

(KF = F ) KF ⊆ F is obvious. That KF ⊇ F follows from Definition 4 since,

∀d ∈ F , κd ⊆ τ d ⊆ F .

(B(φ)∅ = ∅) This follows from Definition 5 since, ∀d ∈ F , β d(φ) = ∅, implying

that there exists no d ∈ F such that β d(φ) ⊆ ∅.




(KE ⊆ KKE ) Let d ∈ KE . By Definition 4, d ∈ KE is equivalent to κd ⊆ E .

Since ∀e ∈ τ d, κe = κd, it follows that τ d ⊆ KE . Hence, κd ⊆ τ d ⊆ KE , implying by

Definition 4 that d ∈ KKE .

(B(φ)E ⊆ KB(φ)E ) Let d ∈ B(φ)E . By Definition 5, d ∈ B(φ)E is equivalent

to β d(φ) ⊆ E . Since ∀e ∈ τ d, β e(φ) = β d(φ), it follows that τ d ⊆ B(φ)E . Hence,

κd ⊆ τ d ⊆ B(φ)E , implying by Definition 4 that d ∈ KB(φ)E .

(¬KE ⊆ K(¬KE )) Let d ∈ ¬KE . By Definition 4, d ∈ ¬KE is equivalent to

κd ⊆ E not holding. Since ∀e ∈ τ d, κe = κd, it follows that τ d ⊆ ¬KE . Hence,

κd ⊆ τ d ⊆ ¬KE , implying by Definition 4 that d ∈ K(¬KE ).

(¬B(φ)E ⊆ K(¬B(φ)E )) Let d ∈ ¬B(φ)E . By Definition 5, d ∈ ¬B(φ)E is

equivalent to β d(φ) ⊆ E not holding. Since ∀e ∈ τ d, β e(φ) = β d(φ), it follows

that τ d ⊆ ¬B(φ)E . Hence, κd ⊆ τ d ⊆ ¬B(φ)E , implying by Definition 4 that

d ∈ K(¬B(φ)E ).

Proof of Proposition 15. (1.) β d(φ) ⊆ φ follows by definition since, ∀e ∈

β d(φ), e ∈ φ.

(2.) By Definitions 2 and 3 and Proposition 9, β d = ρd1. Hence, β d ∩ φ = ∅ implies

ρd1 ∩ φ = ∅ and min{|ρd ∩ φ = ∅} = 1. By Lemma 14, β d(φ) = ρd1 ∩ φ = β d ∩ φ.

(3.) This follows directly from Lemma 3, since φ ∈ Φ implies that, ∀d ∈ F ,

κd ∩ φ = ∅.

(4.) Let β d(φ) ∩ φ = ∅. By Lemma 14, β d(φ) = ρd ∩ φ = ∅ where := min{k|ρdk ∩

φ = ∅}. Likewise, β d(φ ∩ φ) = ρd ∩ φ ∩ φ, where := min{k|ρdk ∩ φ ∩ φ = ∅}.

It suffices to show that = . Obviously, ≤ . However, ∅ = β d(φ) ∩ φ =

(ρd ∩ φ) ∩ φ = ρd ∩ φ ∩ φ implies that ≥ .

Proof of Proposition 16. That KE ⊆ B0E follows from Definition 4 and

Propositions 9 and 12 since κd ⊆ E implies that ρdL

= κd = κd ∩ E . That B0E ⊆

B(F )E follows from Definition 6 since F ∈ ΦE .

Proof of Proposition 17. (B0E ∩ B0E ⊆ B0(E ∩ E )) Let d ∈ B0E and

d ∈ B0E . Then, by Proposition 12, there exist k such that ρd = E ∩ κd and k such

that ρd = E ∩ κd. Since ρd1 ⊂ · · · ⊂ ρdL, either ρd ⊆ ρd or ρd ⊇ ρd , or equivalently,

E ∩ κd ⊆ E ∩ κd or E ∩ κd ⊇ E ∩ κd. Hence, either ρd = E ∩ κd = E ∩ E ∩ κd or

ρd = E ∩ κd = E ∩ E ∩ κd, implying by Proposition 12 that a ∈ B0(E ∩ E ).

(B0E ⊆ KB0E ) Let d ∈ B0E . By Proposition 12, d ∈ B0E is equivalent to

∃ ∈ {1, . . . , L} such that ρd = E ∩ κd. Since ∀e ∈ τ d, ρe = ρd and κe = κd, it follows

that τ d ⊆ B0E . Hence, κd ⊆ τ d ⊆ B0E , implying by Definition 4 that d ∈ KB0E .

(¬B0E ⊆ K(¬B0E )) Let d ∈ ¬B0E . By Proposition 12, d ∈ ¬B0E is equivalent to

there not existing k ∈ {1, . . . , L} such that ρd = E ∩ κd. Since ∀e ∈ τ d, ρe = ρd and

κe = κd, it follows that τ d ⊆ ¬B0E . Hence, κd ⊆ τ d ⊆ ¬B0E , implying by Definition

4 that d ∈ K(¬B0E ).

To prove Proposition 18 the following lemma is helpful.

Lemma 15 Assume that d satisfies Axioms 1 and 4 (in addition to the assump-

tions made in Section 4.1), and let , ∈ {1, . . . , Ld} satisfy < . Then pd

πd

q

implies pd

πd∪πd

q.

Proof. This follows from Proposition 3.



Appendix A: Proofs of results in Chapter 4 179

Proof of Proposition 18. (If E is assumed at d, then d ∈ B0E .) Let E be

assumed at d. Then it follows that dE is nontrivial; hence, E ∩ κd = ∅. Assume that

pE∩κd weakly dominates qE∩κd at d. Since E ∩ κd = ∅, we have that p dE q. Hence,

it follows from the premise (viz., that E is assumed at d) that p d q. This shows

that d is admissible on E ∩ κd, and, by Proposition 12, d ∈ B0E .

(If d ∈ B0E , then E is assumed at d.) Let d ∈ B0E , so by Proposition 12 d is

admissible on E ∩ κd (= ∅). Hence, by Proposition 6, e ∈ E ∩ κd and f ∈ ¬(E ∩ κd)

implies e d f . By Axiom 4 this in turn implies that ∃ such that

E ∩ κd =

k=1πdk ,

since the first property of Axiom 4 – the Archimedean property of d within each

partitional element – rules out that e and f are in the same element of the partition

{πd1 , . . . , πdLd} if e d f .

Assume that p dE q. Then p d

E∩κd q, and, by the above argument,

p d

k=1

πdk

q .

By completeness and the partitional Archimedean property, Lemma 15 entails that

∃ ∈ {1, . . . , } such that

pd

πd

q and, ∀k ∈ {1, . . . , − 1}, p∼d

πdk

q .

By Lemma 15, p d q since ∪Ld

k=1πdk = κd. Hence, p d

E q implies p d q. Moreover,

dE is nontrivial since E ∩ κd = ∅, and it follows from Definition 7 that E is assumed

at d.





Appendix B

Proofs of results in Chapters 8–10

For the proofs of Propositions 30, 34, 36, and 37 we need two results from Blume

et al. (1991b). To state these results, introduce the following notation. Let λ =

(µ1,...,µL) be an LPS on a finite set F and let r = (r1,...,rL−1) ∈ (0, 1)L−1. Then,

r λ denotes the probability distribution on F given by the nested convex combination

(1 − r1)µ1 + r1 [(1 − r2)µ2 + r2 [(1 − r3)µ3 + r3 [. . . ] . . . ]] .

The first is a restatement of Proposition 2 in Blume et al. (1991b).

Lemma 16 Let (x(n))n∈

be a sequence of probability distributions on a finite set

F . Then, there exists a subsequence x(m) of (x(n))n∈

, an LPS λ = (µ1

,...,µL), and

a sequence r(m) of vectors in (0, 1)L−1 converging to zero such that x(m) = r(m) λ

for all m.

The second is a variant of Proposition 1 in Blume et al. (1991b).

Lemma 17 Consider a type ti of player i whose preferences over acts on S j × T jare represented by υ ti

i — with υ tii ◦ z = ui — and λti = (µti

1 , . . . , µtiL ) ∈ L∆(S j × T j).

Then, for every sequence (r(n))n∈

in (0, 1)L−1 converging to zero there is an n such

that, ∀si, si ∈ S i, si ti si if and only if

sj

tj

(r(m) λti)(sj, tj)ui(si, sj) >

sj

tj

(r(m) λti)(sj , tj)ui(si, sj)

for all n ≥ n.

Proof. Suppose that si ti si. Then, there is some ∈ {1,...,L} such that

sj

tj

µtik (sj, tj)ui(si, sj) =

sj

tj

µtik (sj , tj)ui(si, sj) (B.1)

for all k < and

sj

tj

µti (sj , tj)ui(si, sj) >

sj

tj

µti (sj, tj)ui(si, sj). (B.2)




Let (r(n))n∈

be a sequence in (0, 1)L−1 converging to zero. By (B.1) and (B.2),

sj

tj

(r(n) λti)(sj , tj)ui(si, sj) >

sj

tj

(r(n) λti)(sj , tj)ui(si, sj)

if n is large enough. Since S i is finite, this is true if n is large enough for any si,

si ∈ S i satisfying si ti si. The other direction follows from the proof of Proposition

1 in Blume et al. (1991b).

For the proofs of Propositions 30 and 34 we need the following definitions. Let

the LPS λi = (µi1, . . . , µi

L) ∈ L∆(S j) have full support on S j . Say that the behavior

strategy σj is induced by λi if for all h ∈ H j and a ∈ A(h),

σj(h)(a) := µi(S j(h, a))

µi(S j(h))

,

where = min{k| supp λik ∩ S j(h) = ∅}. Moreover, say that player i’s beliefs over past

opponent actions β i are induced by λi if for all h ∈ H i and x ∈ h,

β i(h)(x) := µi

(S j(x))

µi(S j(h))

,

where = min{k| supp λik ∩ S j(h) = ∅}.

Proof of Proposition 30. (Only if.) Let (σ, β ) be a sequential equilibrium.

Then (σ, β ) is consistent and hence there is a sequence (σ(n))n∈

of completely mixed

behavior strategy profiles converging to σ such that the sequence (β (n))n∈

of in-

duced belief systems converges to β . For each i and all n, let pi(n) ∈ ∆(S i) be the

mixed representation of σi(n). By Lemma 16, the sequence ( pj(n))n∈

of probabil-

ity distributions on S j contains a subsequence pj(m) such that we can find an LPSλi = (µi1, . . . , µi

L) with full support on S j and a sequence of vectors r (m) ∈ (0, 1)L−1

converging to zero with

pj(m) = r(m) λi

for all m. W.l.o.g., we assume that pj(n) = r(n) λi for all n ∈

.

We first show that λi induces the behavior strategy σj. Let σj be the behavior

strategy induced by λi. By definition, ∀h ∈ H j , ∀a ∈ A(h),

σj(h)(a) = µi

(S j(h, a))

µi(S j(h))

= limn→∞

(r(n) λi)(S j(h, a))

(r(n) λi)(S j(h))

= limn→∞

pj(n)(S j(h, a))

pj(n)(S j(h)) = lim

n→∞σj(n)(h)(a) = σj(h)(a) ,

where = min{k| supp λ

i

k ∩ S j(h) = ∅}. For the fourth equation we used the fact that pj(n) is the mixed representation of σj(n). Hence, for each i, λi induces σj .

We then show that λi induces the beliefs β i. Let β i be player i’s beliefs over past

opponent actions induced by λi. By definition, ∀h ∈ H i, ∀x ∈ h,

β i(h)(x) = µi

(S j(x))

µi(S j(h))

= limn→∞

r(n) λi(S j(x))

r(n) λi(S j(h))

= limn→∞

pj(n)(S j(x))

pj(n)(S j(h)) = lim

n→∞β i(n)(h)(x) = β i(h)(x),



Appendix B: Proofs of results in Chapters 8–10 183

where = min{k| supp λik ∩ S j(h) = ∅}. For the fourth equality we used the facts that

pj(n) is the mixed representation of σj(n) and β i(n) is induced by σj(n). Hence, for

each i, λi induces β i.

We now define the following epistemic model. Let T 1 = {t1} and T 2 = {t2}.

Let, for each i, υtii satisfy υtii ◦ z = ui, and (λti , ti) be the SCLP with support

S j×{tj}, where (1) λti coincides with the LPS λi constructed above, and (2) ti(E j) =

min{| supp λti ∩ E j = ∅} for all (∅ =) E j ⊆ S j × {tj}. Then, it is clear that

(t1, t2) ∈ [u], there is mutual certain belief of {(t1, t2)} at (t1, t2), and for each i, σiis induced for ti by tj . It remains to show that (t1, t2) ∈ [isr].

For this, it is sufficient to show, for each i, that σi is sequentially rational for ti.

Suppose not. By the choice of ti , it then follows that there is some information set

h ∈ H i and some mixed strategy pi ∈ ∆(S i(h)) that is outcome-equivalent to σi|hsuch that there exist si ∈ S i(h) with pi(si) > 0 and si ∈ S i(h) having the property

that

ui(si, µti |Sj(h)) < ui(si, µti

|Sj(h)) ,

where = min{k| supp λtik ∩ (S j(h) × {tj}) = ∅} and µti |Sj(h) ∈ ∆(S j(h)) is the

conditional probability distribution on S j(h) induced by µti . Recall that µti

is the -th

level of the LPS λti . Since the beliefs β i and the behavior strategy σj are induced by λi,

it follows that ui(si, µti |Sj(h)) = ui(si, σj ; β i)|h and ui(si, µti

|Sj(h)) = ui(si, σj ; β i)|hand hence

ui(si, σj ; β i)|h < ui(si, σj; β i)|h,

which is a contradiction to the fact that (σ, β ) is sequentially rational.

(If) Suppose that there is an epistemic model with (t1, t2) ∈ [u] ∩ [isr] such that

there is mutual certain belief of {(t1, t2)} at (t1, t2), and for each i, σi is induced for

ti by tj . We show that σ = (σ1, σ2) can be extended to a sequential equilibrium.For each i, let λi = (µi

1, . . . , µiL) ∈ L∆(S j) be the LPS coinciding with λti , and let

β i be player i’s beliefs over past opponent choices induced by λi. Write β = (β 1, β 2).

We first show that (σ, β ) is consistent.

Choose sequences (r(n))n∈

in (0, 1)L−1 converging to zero and let the sequences

( pj(n))n∈

of mixed strategies be given by pj(n) = r(n) λi for all n. Since λi has

full support on S j for every n, pj(n) is completely mixed. For every n, let σj(n)

be a behavior representation of pj(n) and let β i(n) be the beliefs induced by σj(n).

We show that (σj(n))n∈

converges to σj and that (β i(n))n∈

converges to β i, which

imply consistency of (σ, β ).

Note that the inducement of σj by ti depends on λti through, for each h ∈ H j ,

µti , where = min{k| supp λtik ∩ (S j(h) × {tj}) = ∅}. This implies that σj is induced

by λi. Since σj(n) is a behavior representation of pj(n) and σj is induced by λi, we

have, ∀h ∈ H j , ∀a ∈ A(h),

limn→∞

σj(n)(h)(a) = limn→∞

pj(n)(S j(h, a))

pj(n)(S j(h)) = lim

n→∞

r(n) λi(S j(h, a))

r(n) λi(S j(h))

= µi

(S j(h, a))

µi(S j(h))

= σj(h)(a),

where = min{k| supp λik ∩ S j(h) = ∅}. Hence, (σj(n))n∈

converges to σj .




Since β i(n) is induced by σj(n) and σj(n) is a behavior representation of pj(n),

and furthermore, β i is induced by λi, we have, ∀h ∈ H i, ∀x ∈ h,

limn→∞

β i(n)(h)(x) = limn→∞

pj(n)(S j(x))

pj(n)(S j(h)) = lim

n→∞

r(n) λi(S j(x))

r(n) λi(S j(h))

= µi

(S j(x))

µi(S j(h))

= β i(h)(x),

where = min{k| suppλik ∩ S j(h) = ∅}. Hence, (β i(n))n∈

converges to β i.

This establishes that (σ, β ) is consistent.

It remains to show that for each i and ∀h ∈ H i,

ui(σi, σj ; β i)|h = maxσi

ui(σi, σj ; β i)|h .

Suppose not. Then, ui(σi, σj ; β i)|h < ui(σi, σj ; β i)|h for some h ∈ H i and some σi.

Let pi ∈ ∆(S i(h)) be outcome-equivalent to σi|h. Then, there is some si ∈ S i(h) with

pi(si) > 0 and some s i ∈ S i(h) such that

ui(si, σj ; β i)|h < ui(si, σj; β i)|h.

Since the beliefs β i and the behavior strategy σj are induced by λi, it follows (us-

ing the notation that has been introduced in the ‘only if’ part of this proof) that

ui(si, σj ; β i)|h = ui(si, µti |Sj(h)) and ui(si, σj; β i)|h = ui(si, µti

|Sj(h))|h and hence

ui(si, µti |Sj(h)) < ui(si, µti

|Sj(h)),

which contradicts the fact that σi is sequentially rational for ti. This completes the

proof of this proposition.

Proof of Proposition 34. (Only if.) Let (σ1, σ2) be a quasi-perfect equilibrium.

By definition, there is a sequence (σ(n))n∈

of completely mixed behavior strategy

profiles converging to σ such that for each i and every n ∈ and h ∈ H i,

ui(σi, σj(n))|h = maxσi

ui(σi, σj(n))|h .

For each j and every n, let pj(n) be the mixed representation of σj(n). By Lemma

16, the sequence ( pj(n))n∈

of probability distributions on S j contains a subsequence

pj(m) such that we can find an LPS λi = (µi1, . . . , µi

L) with full support on S j and a

sequence of vectors r (m) ∈ (0, 1)L−1 converging to zero with

pj(m) = r(m) λi

for all m. W.l.o.g., we assume that pj(n) = r(n) λi for all n ∈ .By the same argument as in the proof of Proposition 30, it follows that λi induces

the behavior strategy σj. Now, we define an epistemic model as follows. Let T 1 = {t1}

and T 2 = {t2}. Let, for each i, υtii satisfy υtii ◦ z = ui, and (λti , ti) be the SCLP

with support S j × {tj}, where (1) λti coincides with the LPS λi constructed above,

and (2) ti(S j × {tj}) = L. Then, it is clear that (t1, t2) ∈ [u], there is mutual certain

belief of {(t1, t2)} at (t1, t2), and for each i, σi is induced for ti by tj . It remains to

show that (t1, t2) ∈ [isr] ∩ [cau].




Since, obviously, (t1, t2) ∈ [cau], it suffices to show, for each i, that σi is sequentially

rational for ti. Fix a player i and let h ∈ H i be given. Let pi (∈ ∆(S i(h))) be outcome-

equivalent to σi|h and let pj(n) be the mixed representation of σj(n). Then, since

(σ1, σ2) is a quasi-perfect equilibrium, it follows that

ui( pi, pj(n)|h) = maxpi∈∆(Si(h))

ui( pi, pj(n)|h)

for all n. Hence, pi(si) > 0 implies that

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj) = maxsi∈Si(h)

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj) (B.3)

for all n. Let λtih be i’s preferences at ti conditional on h. Since ti ∈ projT i [caui]—so

that i’s system of conditional preferences at ti satisfies Axiom 6 (Conditionality)—and

pj(n) = r(n) projSjλti for all n, there exist vectors r(n)|h converging to zero suchthat pj(n)|h = r(n)|h

projSj λtih for all n. Together with equation (B.3) we obtain

that pi(si) > 0 implies

sj∈Sj(h)

(r(n)|h projSjλtih )(sj)ui(si, sj)

= maxsi∈Si(h)

sj∈Sj(h)

(r(n)|hprojSjλtih )(sj)ui(si, sj) .(B.4)

We show that pi(si) > 0 implies si ∈ S tii (h). Suppose that si ∈ S i(h)\S tii (h). Then,

there is some si ∈ S i(h) with si tih si. By applying Lemma 17 in the case of acts on

S j(h) × {tj}, it follows that r(n)|h has a subsequence r(m)|h for which

sj∈Sj(h)

(r(m)|h projSjλtih )(sj)ui(si, sj) >

sj∈Sj(h)

(r(m)|h projSj λtih )(sj)ui(si, sj)

for all m, which is a contradiction to (B.4). Hence, si ∈ S tii (h) whenever pi(si) > 0,

which implies that pi ∈ ∆(S tii (h)). Hence, σi|h is outcome equivalent to some pi ∈

∆(S tii (h)). This holds for every h ∈ H i, and hence σi is sequentially rational for ti.

(If) Suppose, there is an epistemic model with ( t1, t2) ∈ [u] ∩ [isr] ∩ [cau] such that

there is mutual certain belief of {(t1, t2)} at (t1, t2), and for both i, σi is induced for

ti by tj . We show that (σ1, σ2) is a quasi-perfect equilibrium.

For each i, let λi = (µi1, . . . , µi

L) ∈ L∆(S j) be the LPS coinciding with λti . Choose

sequences (r(n))n∈

in (0, 1)L−1 converging to zero and let the sequences ( pj(n))n∈

of mixed strategies be given by pj(n) = r(n) λi for all n. Since λi has full support

on S j for every n, pj(n) is completely mixed. For every n, let σj(n) be a behavior

representation of pj(n). Since λi induces σj , it follows that (σj(n))n∈

converges to

σj; this is shown explicitly under the ‘if’ part of Proposition 30. Hence, to establish

that (σ1, σ2) is a quasi-perfect equilibrium, we must show that, for each i and ∀n ∈ N

and ∀h ∈ H i,

ui(σi, σj(n))|h = maxσi

ui(σi, σj(n))|h . (B.5)

Fix a player i and an information set h ∈ H i. Let pi (∈ ∆(S i(h))) be outcome-

equivalent to σi|h. Then, equation (B.5) is equivalent to

ui( pi, pj(n)|h) = maxpi∈∆(Si(h))

ui( pi, pj(n)|h)




for all n. Hence, we must show that pi(si) > 0 implies that

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj) = maxsi∈Si(h)

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj) (B.6)

for all n. In fact, it suffices to show this equation for infinitely many n, since in this

case we can choose a subsequence for which the above equation holds, and this would

be sufficient to show that (σ1, σ2) is a quasi-perfect equilibrium.

Since, by assumption, σi is sequentially rational for ti, σi|h is outcome equivalent

to some mixed strategy in ∆(S tii (h)). Hence, pi ∈ ∆(S tii (h)). Let pi(si) > 0. By

construction, si ∈ S tii (h). Suppose that si would not satisfy (B.6) for infinitely many

n. Then, there exists some si ∈ S i(h) such that

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj) <

sj∈Sj(h)

pj(n)|h(sj)ui(si, sj)

for infinitely many n. Assume, w.l.o.g., that it is true for all n. Let λtih be i’s prefer-

ences at ti conditional on h. Since ti ∈ projT i [caui]—so that i’s system of conditional

preferences at ti satisfies Axiom 6 (Conditionality)—and pj(n) = r(n) projSjλti for

all n, there exist vectors r(n)|h converging to zero such that pj(n)|h = r(n)|h projSjλtihfor all n. This implies that

sj∈Sj(h)

(r(n)|h projSjλtih )(sj)ui(si, sj) <

sj∈Sj(h)

(r(n)|h projSjλtih )(sj)ui(si, sj)

for all n. By applying Lemma 17 in the case of acts on S j(h)×{tj}, it follows that i at

ti strictly prefers si to si conditional on h, which contradicts the fact that si ∈ S tii (h).

Hence, pi(si) > 0 implies (B.6) for infinitely many n, and as a consequence, (σ1, σ2)

is a quasi-perfect equilibrium.

Proof of Proposition 36. (Only if.) Let ( p1, p2) be a proper equilibrium. Then,

by Definition 7, there is a sequence ( p(n))n∈

of ε(n)-proper equilibria converging to

p, where ε(n) → 0 as n → ∞. By the necessity part of Proposition 5 of Blume et

al. (1991b), there exists an epistemic model with T 1 = {t1} and T 2 = {t2} where, for

each i,

υtii satisfies that υ tii ◦ z = ui,

the SCLP (λti , ti) has the properties that λti = (µti1 , . . . , µti

L ) with support

S j × {tj} satisfies that, ∀sj ∈ S j , µti1 (sj , tj) = pj(sj), and ti satisfies that

ti(S j × T j) = L,

such that (t1, t2) ∈ [resp]. This argument involves Lemma 16 (which yields, for each

i, the existence of λti with full support on ∆(S j × {tj}) by means of a subsequence

pj(m) of ( pj(n))n∈

) and Lemma 17 (which yields that, for m large enough, i havingthe conjecture pj(m) leads to the same preferences over i’s strategies as ti). The

only-if part follows since it is clear that (t1, t2) ∈ [u] ∩ [cau], that there is mutual

certain belief of {(t1, t2)} at (t1, t2), and that, for each i, pi is induced for ti by tj .

(If.) Suppose that there exists an epistemic model with (t1, t2) ∈ [u]∩ [resp] ∩ [cau]

such that there is mutual certain belief of {(t1, t2)} at (t1, t2), and, for each i, piis induced for ti by tj . Then, by the sufficiency part of Proposition 5 in Blume

et al. (1991b), there exists, for each i, a sequence of completely mixed strategies




( pi(n))n∈ℵ converging to pi, where, for each n, ( p1(n), p2(n)) is an ε(n)-proper equi-

librium and (n) → 0 as n → ∞. This argument involves Lemma 17 (which yields,

for each j , the existence of (xj(n))n∈

so that, for all n, i having the conjecture pj(n)

leads to the same preferences over i’s strategies as ti).

Proof of Proposition 37. Part 1: If p∗i is properly rationalizable, then there

exists an epistemic model with (t∗1, t∗2) ∈ CK([u] ∩ [resp] ∩ [cau]) such that p∗i is

induced for t∗i by t∗j . In the definition of proper rationalizability, g in Kg[- prop trem]

goes to infinity for each ε, and then ε converges to 0. The strategy for the proof

of the ‘only if’ part of Proposition 37 is to reverse the order of g and ε, by first

noting that ε-proper rationalizability implies ε-proper g-rationalizability for all g,

then showing that ε-proper g-rationalizability as ε converges to 0 corresponds to

the gth round of a finite algorithm, and finally proving that any mixed strategy

surviving all rounds of the algorithm is rational under common certain belief of [u] ∩[resp] ∩ [cau] in some epistemic model. The algorithm eliminates preference relations

on the players’ strategy sets. It is related to, but differs from, Hammond’s (2001)

‘rationalizable dominance relations’, which are recursively constructed by gradually

extending a single incomplete binary relation on each player’s strategy set.

Say that a mixed strategy pi for i is ε-properly g-rationalizable if there exists an

∗-epistemic model with piti = pi for some ti ∈ projT iKg([u] ∩ [ind] ∩ [ε- prop trem]).

Since, for all g ,

CK[ε- prop trem]) ⊆ Kg[ε- prop trem]) ,

it follows from Definition 21 that if p∗i is an ε-properly rationalizable strategy, then,

for all g, there exists an ∗-epistemic model with piti = p∗i for some ti ∈ projT iKg([u] ∩

[ind] ∩ [ε- prop trem]). Consequently, if a mixed strategy p∗i for i is ∗-properly ra-

tionalizable, then, for all g, there exists a sequence ( pi(n))n∈

of ε(n)-properly g-

rationalizable strategies converging to p∗i , where (n) → 0 as n → ∞. This means

that it is sufficient to show that if p∗i satisfies that, for all g , there exists a sequence

( pi(n))n∈

of ε(n)-properly g -rationalizable strategies converging to p∗i and (n) → 0

as n → ∞, then p∗i is rational under common certain belief of [u] ∩ [resp] ∩ [cau] in

some epistemic model. This will in turn be shown in two steps:

1 If a sequence of (n)-properly g-rationalizable strategies converges to p∗i , then p∗isurvives the g th round of a finite algorithm.

2 Any mixed strategy surviving all rounds of the algorithm is rational under common

certain belief of [u] ∩ [resp] ∩ [cau] in some epistemic model.

To construct the algorithm, note that any complete and transitive binary relation

on S i can be represented by a vector of sets (S i(1), . . . , S i(L)) (with L ≥ 1) that

constitute a partition of S i. The interpretation is that si is preferred or indifferent to

si if and only if si ∈ S i(), si ∈ S i() and ≤ . Let, for each i, Σi := 2Si\{∅} be

the collection of non-empty subsets of S i and

Πi := {πi = (S i(1), . . . , S i(Lπi)) ∈ ΣiLπi | {S i(1), . . . , S i(Lπi)} is a partition of S i}

denote the collection of vectors of sets that constitute a partition of S i. Define the

algorithm by, for each i, setting Π−1i = Πi and determining, ∀g ≥ 0, Πg

i as follows:

πi = (S i(1), . . . , S i(Lπi)) ∈ Πgi if and only if πi ∈ Πi and there exists an LPS λπi ∈

L∆(S j × Πj) with suppλπi = S j × Ππij for some Πj

πi ⊆ Πg−1j , satisfying that

(sj, πj) (sj , πj) according to πi




if πj = (S j(1), . . . , S j(Lπj )) ∈ Πjπi , sj ∈ S j(), s j ∈ S j() and < , and

si πi si

if and only if si ∈ S i(), si ∈ S i() and < , where πi is represented by υπiisatisfying υπi

i ◦ z = ui and λπi .

Write Π := Π1 × Π2 and, ∀g ≥ 0, Πg = Πg1 × Πg

2. Since Π0 ⊆ Π, it follows by

induction that, ∀g ≥ 0, Πg ⊆ Πg−1. Moreover, since the finiteness of S = S 1 × S 2implies that Π is finite, it follows that Πg converges to Π∞ in a finite number of

rounds. Say that pi survives the gth round of the algorithm if there exists πi =

(S i(1), . . . , S i(Lπi)) ∈ Πgi with ∆(S i(1)) pi.

Step 1. We first show that p∗i survives the gth round of the algorithm if there

exists a sequence ( pi(n))n∈

of ε(n)-properly g-rationalizable strategies converging to

p∗i , where ε(n) → 0 as n → ∞. Say that the probability distribution µ ∈ ∆(S j × T j) is

an ε-properly g -rationalizable belief for i if there is an ∗-epistemic model with µti = µ

for some ti ∈ projT iKg([u] ∩ [ind] ∩ [ε- prop trem]). It is sufficient to establish the

following result:

If πi = (S i(1), . . . , S i(Lπi)) ∈ Πi satisfies that there exists a sequence (µπi(n))n∈ℵ of

ε(n)-properly g-rationalizable beliefs for i, where ε(n) → 0 as n → ∞, and where, for

all n,

sj

tj

µπi(n)(sj , tj)ui(si, sj) >

sj

tj

µπi(n)(sj , tj)ui(si, sj) (B.7)

if and only if si ∈ S i(), si ∈ S i() and < , then πi ∈ Πgi .

This result is established by induction.

If (µπi(n))n∈

is a sequence of ε(n)-properly g-rationalizable beliefs for i, then, for

each n, there exists an ∗-epistemic model with T 1(n)×T 2(n) as the set of type vectors,

such that µπi(n) ∈ ∆(S j × T j(n)). For the inductive proof we can w.l.o.g. partitionT j(n) into Πj , where πj = (S j(1), . . . , S j(Lπj )) ∈ Πj corresponds to the subset of

j-types in T j(n) satisfying that

si

ti

µtj (n)(si, ti)uj(sj, si) >

si

ti

µtj (n)(si, ti)uj(sj, si)

if and only if sj ∈ S j(), sj ∈ S j() and < , since i’s certain belief of j’s ε(n)-

proper trembling only matters through j-types’ preferences over j ’s pure strategies .

Hence, we can w.l.o.g. assume that µπi(n) ∈ ∆(S j × Πj).

(g = 0) Let (µπi(n))n∈

be a sequence of ε(n)-properly 0-rationalizable beliefs for

i, where (n) → 0 as n → ∞, and where, for all n, (B.7) is satisfied. By Lemma 16,

the sequence (µπi(n))n∈

contains a subsequence µπi(m) such that one can find an

LPS λπi ∈ L∆(S j × Πj) and a sequence of vectors rπi(m) ∈ (0, 1)L−1 (for some L)

converging to 0 withµπi(m) = rπi(m)

λπi

for all m. By Definition 20, suppλπi = S j × Πjπi for some Πj

πi ⊆ Πj . Let πi

be represented by υπii satisfying υπii ◦ z = ui and λπi . Since Definition 20 is the

only requirement on (µπi(n))n∈

for g = 0, we may, for each πj ∈ Πjπi , associate πj

with (S j(1), . . . , S i(Lπj )) ∈ Π−1j satisfying that (sj , πj) (sj , πj) according to πi

if sj ∈ S j(), sj ∈ S j() and < . By Lemma 2, πi yields the same preferences

on S i as µπi(n) (for any n). Hence, πi ∈ Π0i .




(g > 0) Suppose the result holds for g = 0, . . . , g − 1. Let (µπi(n))n∈

be a

sequence of ε(n)-properly g-rationalizable beliefs for i, where ε(n) → 0 as n → ∞,

and where, for all n, (B.7) is satisfied. As for g = 0, use Lemma 16 to construct an

LPS λπi ∈ L∆(S j × Πj), where suppλπi = S j × Πjπi for some Πj

πi ⊆ Πj, and where

πi is represented by υπii satisfying υπi

i ◦ z = ui and λπi . Since

Kg[- prop trem] ⊆ Ki

[- prop tremj ] ∩ Kg−1[- prop trem]

,

the induction hypothesis implies that Πjπi ⊆ Πg−1

j and (sj , πj) (sj, πj) according

to πi if πj = (S j(1), . . . , S i(Lπj )) ∈ Πjπi , sj ∈ S j(), sj ∈ S j() and < . By

Lemma 17, πi yields the same preferences on S i as µπi(n) (for any n). Hence,

πi ∈ Πgi . This concludes the induction and thereby Step 1.

Step 2. We then show that if a mixed strategy p∗i survives all rounds of the

algorithm, then there exists an epistemic model with p∗i

∈ ∆(S it∗i ) for some t∗

i

∈

projT iCK([u] ∩ [resp] ∩ [cau]). It is sufficient to show that one can construct an

epistemic model with T 1 × T 2 ⊆ CK([u] ∩ [resp] ∩ [cau]) such that, for each i, ∀πi =

(S i(1), . . . , S i(Lπi)) ∈ Π∞i , there exists ti ∈ T i satisfying that si

ti si if and only if

si ∈ S i(), si ∈ S i() and < . Construct an epistemic model with, for each i, a

bijection i : T i → Π∞i from the set of types to the collection of vectors in Π∞

i . Since

∃g such that Πg = Π∞ for g ≥ g, it follows from the definition of the algorithm

(Πg)g≥0 that, for each i, Π∞i is characterized as follows: πi = (S i(1), . . . , S i(Lπi)) ∈

Π∞i if and only if there exists ti ∈ T i such that

i(ti) = πi, and an LPS λti =

(µti1 , . . . µti

L ) ∈ L∆(S j × T j) with suppλti = S j × T jti for some T j

ti ⊆ T j , satisfying

for each tj ∈ T jti that

(sj, tj) (sj , tj) according to ti

if

j(tj) = (S j(1), . . . , S j(L

j(tj)

), sj ∈ S j(), s

j ∈ S j(

) and <

, andsi

ti si

if and only if si ∈ S i(), si ∈ S i() and < , where υtii satisfies υtii ◦ z = ui and

the SCLP (λti , ti) has the property that ti satisfies ti(S j × T j) = L (so that ti

is represented by υtii and λti). Consider any πi = (S i(1), . . . , S i(Lπi)) ∈ Π∞i . By

the construction of the type sets, there exists ti ∈ T i such that i(ti) = πi, and

si ti si if and only if si ∈ S i(), si ∈ S i() and < ; in particular, S i(1) = S tii . It

remains to be shown that, for each i, T 1 × T 2 ⊆ [ui] ∩ [respi] ∩ [caui], implying that

T 1 × T 2 ⊆ CK([u] ∩ [resp] ∩ [cau]) since T jti ⊆ T j for each ti ∈ T i of any player i.

It is clear that T 1 × T 2 ⊆ [ui] ∩ [caui]. That T 1 × T 2 ⊆ [respi] follows from the

property that, for any ti ∈ T i, (sj , tj) (sj , tj) according to ti whenever tj ∈ T jti if

sj ∈ S j(), sj ∈ S j() and < , while sj tj sj if and only if sj ∈ S j(), sj ∈ S j()

and <

(where

j(tj) = (S j(1), . . . , S j(L

j(tj)

)). This concludes Step 2.In the construction in Step 2, let t∗i ∈ T i satisfy that p∗i ∈ ∆(S it∗i ). To conclude

Part 1 of the proof of Proposition 37, add type t∗j to T j having the property that p∗i is

induced for t∗i by t∗j . Assume that υjt∗j satisfies υj

t∗j ◦z = uj and the SCLP (λti , ti) on

S i × T j with support S i × {t∗i } has the property that λt∗

j = (µ1t∗j , . . . , µL

t∗j ) satisfies,

∀si ∈ S i, µ1t∗j (si, t∗i ) = p∗i (si) and ti satisfies ti(S j × T j) = L (so that t∗j is

represented by υjt∗j and λt∗j ). Furthermore, assume that

(si, t∗i ) (si, t∗i ) according to t∗j




if i(t∗i ) = (S i(1), . . . , S i(L i(t∗i ))), si ∈ S i(), si ∈ S i() and < . Then t∗j ∈

projT j ([uj ] ∩ [respj ] ∩ [cauj ]), and since T it∗j ⊆ T i, T i × (T j ∪ {t∗j}) ⊆ CK([u] ∩ [resp] ∩

[cau]). Hence, (t∗1, t∗2) ∈ CK([u] ∩ [resp] ∩ [cau]) and p∗i is induced for t∗i for t∗j .

Part 2: If there exists an epistemic model with (t∗1, t∗2) ∈ CK([u] ∩ [resp] ∩ [cau])

such that p∗1 is induced for t∗1 by t∗2, then p∗1 is properly rationalizable. Schuhmacher

(1999) considers a set of type profiles T = T 1 × T 2, where each type ti of either player

i plays a completely mixed strategy piti and has a subjective probability distribution

on S j × T j, for which the conditional distribution on S j × {tj} coincides with pjtj

whenever the conditional distribution is defined. His formulation implies that all

types of a player agrees not only on the preferences but also on the relative likelihood

of the strategies for any given opponent type. In contrast, the characterization given

in Proposition 37 requires the types of a player only to agree on the preferences of

any given opponent type. This difference implies that expanded type sets must be

constructed for the ‘if’ part of the proof of Proposition 37.Assume that there exists an epistemic model with (t∗1, t∗2) ∈ CK([u] ∩ [resp] ∩ [cau])

such that p∗1 is induced for t∗1 by t∗2. In particular, CK([u] ∩ [resp] ∩ [cau]) = ∅,

and p∗1 ∈ ∆(S 1t∗1 ) since CK([u] ∩ [resp] ∩ [cau]) ⊆ [resp2]. Let, for each i, T i :=

projT iCK([u] ∩ [resp] ∩ [cau]). Note that, for each ti ∈ T i of any player i, ti deems

(sj, tj) subjectively impossible if tj ∈ T j\T j since CK([u] ∩ [ir] ∩ [cau]) = KCK([u] ∩

[ir] ∩ [cau]) ⊆ KiCK([u] ∩ [ir] ∩ [cau]), implying T jti ⊆ T j .

We first construct a sequence, indexed by n, of ∗-epistemic models. By Definition

20 and Assumption 3 this involves, for each n and for each player i, a finite set of

types—which we below denote by T i and which will not vary with n—and, for each n,

for each i, and for each type τ i ∈ T i , a mixed strategy and a probability distribution

( pτ ii (n), µτ i(n)) ∈ ∆(S i) × ∆(S j × T j ) that will vary with n.

For either player i and each type ti ∈ T i of the original epistemic model, make as

many “clones” of ti as there are members of T j : For each i, T i := {τ i(ti, tj)| ti ∈ T iand tj ∈ T j}, where τ i(ti, tj) is the “clone” of ti associated with tj . The term “clone”

in the above statement reflects that, ∀tj ∈ T j , τ i(ti, tj) is assumed to “share” the

preferences of ti in the sense that

1 the set of opponent types that τ i(ti, tj) does not deem subjectively possible,

T jτ i(ti,tj), is equal to {τ j(tj , ti)|tj ∈ T j

ti} (⊆ T j since T jti ⊆ T j), and

2 the likelihood of (sj, τ j(tj , ti)) according to τ i(ti,tj) is equal to the likelihood of

(sj , tj) according to ti .

Since T jτ i(ti,tj) = {τ j(tj , ti)| tj ∈ T j

ti} is independent of tj , but corresponds to disjoint

subsets of T j for different ti’s, we obtain the following conclusion for any pair of type

vectors (t1, t2), (t1, t2) ∈ T 1 × T 2:

T jτ i(ti,tj) = T j

τ i(t

i,t

j) if ti = ti

,

T jτ i(ti,tj) ∩ T j

τ i(t

i,t

j) = ∅ if ti = ti .

This ends the construction of type sets in the sequence of ∗-epistemic models.

Fix a player i and consider any τ i ∈ T i . Since CK([u] ∩ [resp] ∩ [cau]) ⊆ [ui], τ i

can be represented by a vNM utility function υτ ii satisfying υτ i

i ◦ z = ui and an LPS

λτ i on S j × T jτ i . Since CK([u] ∩ [resp] ∩ [cau]) ⊆ [caui], this LPS yields, for each

τ j ∈ T jτ i , a partition {E τ ij (1), . . . , E j

τ i(Lτ i)} of S j × T jτ i , where (sj , τ j) (sj, τ j)

according to τ i if and only (sj , τ j) ∈ E jτ i(), (sj , τ j) ∈ E j

τ i() and < . Since




CK([u] ∩ [resp] ∩ [cau]) ⊆ [respi], it follows that sj is a most preferred strategy for τ jin {sj ∈ S j |(sj , τ j) ∈ E j

τ i() ∪ · · · ∪ E jτ i(Lτ i)} if (sj , τ j) ∈ E j

τ i().

Consider any i and τ i ∈ T i . Construct the sequence (µτ i(n))n∈

as follows. Choose

∀τ i ∈ {τ i(ti, tj)|tj ∈ T j} one common sequence (rτ i(n))n∈

in (0, 1)Lτ i−1 converging

to 0 and let the sequence of probability distributions (µτ i(n))n∈

be given by µτ i(n) =

rτ i(n) λτ i . For all n, suppµτ i(n) = S j × T jτ i . By Lemma 17 (rτ i(n))n∈ℵ can be

chosen such that, for all n,

sj

τ j

µτ i(n)(sj , τ j)ui(si, sj) >

sj

τ j

µτ i(n)(sj, τ j)ui(si, sj)

if and only if si τ i si. Hence, for all n, the belief µτ i(n) leads to the same preferences

over i’s strategies as τ i . This ends the construction of the sequences ( µτ i(n))n∈

in

the sequence of ∗-epistemic models.

Consider now the construction of the sequence ( pτ ii (n))n∈

for any i and τ i ∈ T i .

There are two cases. Case 1: If there is τ j ∈ T j such that τ i ∈ T iτ j , implying that

S i × {τ i} ⊆ suppµτ j (n), then let pτ ii (n) be determined by

pτ ii (n)(si) = µτ j (n)(si, τ i)

µτ j (n)(S i, τ i) .

Moreover, for each n, there exists (n) such that, for each player i, the (n)-proper

trembling condition is satisfied at all such types in T i : Since

pτ ii (n)(si)

pτ ii (n)(si) =

µτ j (n)(si, τ i)

µτ j (n)(si, τ i) → 0 as n → ∞

if (si, τ i) ∈ E τ ji (), (si, τ i) ∈ E

τ ji () and < , and since si is a most preferred

strategy for τ i in {si ∈ S i|(s

i, τ i) ∈ E

τ ji () ∪ · · · ∪ E

τ ji (L

τ j )} if (si, τ i) ∈ E

τ ji (), it

follows that there exists a sequence (τ i(n))n∈

converging to 0 such that, for all n,

τ i(n) pτ ii (n)(si) ≥ pτ ii (n)(si) whenever

sj

τ j

µτ i(n)(sj , τ j)ui(si, sj) >

sj

τ j

µτ i(n)(sj , τ j)ui(si, sj) .

Let, for each n,

(n) := max {τ 1(n)|∃τ 2 ∈ T 2 s.t. τ 1 ∈ T τ 21 } ∪ {τ 2(n)|∃τ 1 ∈ T 1 s.t. τ 2 ∈ T τ 12 } .

Since the type sets are finite, (n) → 0 as n → ∞. Case 2: If there is no τ j ∈ T jsuch that τ i ∈ T i

τ j , then let pτ ii (n) be any mixed strategy having the property that

τ i satisfies the (n)-proper trembling condition given the belief µτ i(n). This ends the

construction of the sequences ( pτ ii (n))n∈ in the sequence of ∗-epistemic models.

We then turn to the construction of a sequence ( p1τ ∗1 (n))n∈

converging to p∗1. Add

type τ ∗1 to T 1 having the property that µτ ∗1 (n) = µτ 1(t

∗

1,t2)(n) for some t2 ∈ T 2, but

where p1τ ∗1 (n) = (1 − 1

n) p∗1 + 1

n p1

τ 1(t∗

1,t2)(n). For all n, we have that the belief µτ ∗

1 (n)

leads to the same preferences over 1’s strategies as t∗i . This in turn implies that i

satisfies the (n)-trembling condition at τ ∗1 since p∗1 ∈ ∆(C 1t∗1 ).

Consider the sequence, indexed by n, of ∗-epistemic models,

with T 1 ∪ {τ ∗1 } as the type set for 1 and T 2 as the type set for 2,




with, for each type τ i of any player i, ( pτ ii (n), µτ i(n)) as the sequence of a mixed

strategy and a probability distribution,

as constructed above. Furthermore, it follows that, for all n, the (n)-proper trembling

condition is satisfied at all types in T 1 ∪ {τ ∗1 } and at all types in T 2 , where (n) → 0

as n → ∞. Hence, for all n,

(T 1 ∪ {τ ∗1 }) × T 2 ⊆ CK[(n)- prop trem] ;

in particular, p1τ ∗1 (n) is (n)-properly rationalizable. Moreover, ( p1

τ ∗1 (n))n∈

con-

verges to p∗1. By Definition 21, p∗1 is a properly rationalizable strategy.



Appendix C

Proofs of results in Chapter 11

Proof of Proposition 42. Assume that the pure strategy si for i is properly

rationalizable in a finite strategic two-player game G. Then, there exists an epistemic

model satisfying Assumption 1 with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CK([u] ∩

[resp] ∩ [cau]) (this follows from Proposition 37 since CK([u] ∩ [resp] ∩ [cau]) =

KCK([u] ∩ [resp] ∩ [cau]) ⊆ KjCK([u] ∩ [resp] ∩ [cau])). In particular, CK([u] ∩

[resp] ∩ [cau]) = ∅.

By Proposition 20(ii), for each i, CK([u] ∩ [resp] ∩ [cau]) = KCK([u] ∩ [resp] ∩

[cau]) ⊆ KiCK([u] ∩ [resp] ∩ [cau]). Hence, we can construct a new epistemic model

(S 1, T 1, S 2, T 2) where, for each i, T i := projT iCK([u] ∩ [resp] ∩ [cau]), as for each

ti ∈ T i of any player i, κti = {ti} × S j × T jti ⊆ {ti} × S j × T j . Since T 1 × T 2 ⊆ [cau],according to the definition of caution given in Section 5.3, it follows that the new

epistemic model satisfies Axiom 6 for each ti ∈ T i of any player i. Therefore, the

new epistemic model satisfies Assumption 2 with S 1 × T 1 × S 2 × T 2 ⊆ [cau] according

to the definition of caution given in Section 6.3. Also, S 1 × T 1 × S 2 × T 2 ⊆ [u]. It

remains to be shown that, for each i, S 1 × T 1 × S 2 × T 2 ⊆ B0i [ratj ], since by the fact

that κti ⊆ {ti} × S j × T j for each ti ∈ T i of any player i, we then have an epistemic

model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CKA0.

Since T 1 × T 2 ⊆ [resp], we have that, for each ti ∈ T i of any player i, (sj , tj) ti

(sj , tj) whenever tj ∈ T iti and sj tj sj . In particular, for each ti ∈ T i of any player

i, (sj, tj) ti (sj , tj) whenever tj ∈ T iti , sj ∈ S j

tj and sj /∈ S jtj . By Proposition 6 this

means that, for each ti ∈ T i of any player i, ti is admissible on projT 1×S2×T 2 [ratj] ∩

κti , showing that S 1 × T 1 × S 2 × T 2 ⊆ (B01[rat2] ∩ B0

2[rat1]).

Proof of Proposition 43. Part 1: If si is permissible, then there exists an

epistemic model with si ∈ S tii for some (t1, t2) ∈ projT 1×T 2CK A. It is sufficient to

construct a belief system with S 1 × T 1 × S 2 × T 2 ⊆ CK A such that, for each si ∈ P iof any player i, there exists ti ∈ T i with si ∈ S tii . Construct a belief system with,

for each i, a bijection si : T i → P i from the set of types to the the set of permissible

pure strategies. By Lemma 10(i) we have that, for each ti ∈ T i of any player i, there

exists Y jti ⊆ P i such that si(ti) ∈ S i\Di(Y j

ti). Determine the set of opponent types





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Index

Accessibility relation, 39–44, 46

Act

Anscombe-Aumann act, 8, 22, 26, 32–33,39–40, 49, 54, 56, 70–71, 74–75,77–78, 83–84, 92, 102, 122, 126, 145,

181, 185–186, 194

Admissibility, 39, 41–42, 46, 49–50, 70, 72,85–86, 133–135, 143–144, 147,

153–154, 175–176, 179, 193

Backward induction, 2, 6–7, 10–11, 14–17,20–21, 23–24, 38, 69, 78–80, 82–83,87–88, 91–97, 99–100, 112–113, 115,118, 121, 123, 128, 132, 152, 155–159,

162–166, 170, 174Belief operators

absolutely robust belief, 38, 40, 45, 49–50,

147

assumption, 38, 40, 45, 48–50, 147, 152

certain belief, 19, 39, 44, 46–48, 57–61,

63–66, 72–73, 76, 78, 81–82, 87–88,90–96, 99, 103, 105–106, 108–109,113, 115, 117–118, 122–123, 125,127–128, 136, 138, 140, 147–148,150–153, 162–163, 167, 173, 183–188

conditional belief, 39–40, 44–45, 47, 50, 86

full belief, 38, 40, 45–46, 50

robust belief, 40, 44–46, 48–51, 135–139,

143, 145–149, 152–154, 158, 163, 173

strong belief, 38, 40, 45, 48, 50–51, 147,152

Caution, 10, 14, 23, 62–63, 75, 103, 115–116,123, 135–137, 139, 143, 145–146,148–149, 152, 154, 158, 160, 163–164,173, 193

Consistency of preferences

(ordinary) consistency, 5, 12, 53, 58–59,73, 149

admissible consistency, 53, 63, 75–76,

87–88, 97, 148

admissible subgame consistency, 90–96

full admissible consistency, 134–140,144–145, 147–148, 151–152, 162–163,167, 173

proper consistency, 122–123, 125, 128

quasi-perfect consistency, 117

sequential consistency, 104

weak sequential consistency, 108

Consistent preferences approach, 1–7, 11–12,

15–17, 21, 53, 81, 144, 154, 174

Epistemic independence, 90, 94, 96

Epistemic model, 3–5, 8–9, 15, 41–42, 48,50, 53–55, 58–62, 64–67, 69, 73–74,76–77, 83, 91–92, 94, 100, 102,

104–106, 109–111, 113–115, 117–119,121, 125–128, 130, 138, 144–145,147–151, 153, 174, 183–190, 193–194

Epistemic priority, 38–39, 42–46

Equilibrium

Nash equilibrium, 2–6, 11–13, 18, 53,58–60, 64–65, 115, 124, 130, 141,171–172

perfect equilibrium, 18, 53, 62–65, 115,

124, 130

proper equilibrium, 16, 18–19, 121–122,124–125, 127, 130, 148, 174, 186

quasi-perfect equilibrium, 18–19, 24,115–118, 127, 184–186

sequential equilibrium, 18–19, 24, 100,104–107, 115, 118, 182–183

subgame-perfect equilibrium, 87, 91, 94,113–114

weak sequential equilibrium, 18

Forward induction, 2, 6–7, 10–11, 17, 21, 24,38, 69, 97, 112, 133, 135, 137, 146,148–150, 152–154, 159, 162, 168–172,



202 REFERENCES

174

Gameextensive game, 6, 14–15, 17, 23, 50–51,

56, 80–85, 87, 89–91, 94, 99, 101–103,105–106, 108–109, 113, 117–119, 152,155, 158–160, 162, 173

of perfect information, 20, 79–82, 84–85,87–92, 94, 97, 100–101, 113, 118, 121,123, 128, 132, 162

strategic game, 2–4, 7–8, 53–54, 56–57,59–61, 63–66, 69, 71, 73, 76, 83–85,88, 90–91, 94, 101–103, 121, 125–130,139, 144, 148–149, 159, 161, 193

pure strategy reduced strategic form(PRSF), 133, 156, 159–164, 168–170,

173Inducement (of rationality)

of a rational mixed strategy, 5, 58

of a sequentially rational behaviorstrategy, 104

of a weak sequentially rational mixed

strategy, 107Iterated elimination

Dekel-Fudenberg procedure, 13–14, 23–24,65, 69, 78, 81, 83, 88, 112, 124, 138,141, 143, 148, 162, 166–168, 171

of choice sets under full admissibleconsistency (IECFA), 138–142,150–151, 163–164, 169–170, 173

of strongly dominated strategies (IESDS),

13, 23–24, 60, 69, 83, 133, 137–138,141–143, 149

of weakly dominated strategies (IEWDS),14, 16–17, 129–130, 133–135, 141–142,150, 152–153, 159, 169

No extraneous restrictions on beliefs, 135,137, 139, 143, 146–149, 153–154, 158,163–164, 173

Probability systemconditional probability system (CPS),

24–25, 34–36, 50, 109

lexicographic conditional probability

system (LCPS), 31, 35–36, 49–50

lexicographic probability system (LPS),24–25, 30–33, 36, 43, 49, 56, 60,62–63, 65, 67, 76, 88, 93–94, 96, 102,104, 106–107, 110, 116, 122, 131, 143,

181–185, 187–190

system of conditional lexicographicprobabilities (SCLP), 25, 32, 35–36,56–57, 59, 61–64, 66, 100, 102–104,

109–110, 114–116, 121–122, 183–184,186, 189

Rational choice approach, 1–3, 6, 11–12, 143

Rationalizability

(ordinary) rationalizability, 8, 13, 18, 53,

60, 69, 73, 124, 137–138, 142–143,146, 149, 153

extensive form rationalizability, 99,112–113, 135, 152–153, 156, 159

full permissibility, 17, 112, 134–146,148–152, 154–155, 157–173

permissibility, 13–15, 17–18, 53, 62, 65–67,69, 75–77, 81, 87, 112, 115, 119, 124,137–138, 141–143, 146, 148–149,153–154, 193–194

proper rationalizability, 1, 16, 18–19,

121–125, 127–131, 146–148, 174, 187,190, 192–193

quasi-perfect rationalizability, 1, 15–16,18, 24, 101, 115–116, 118–119, 128

sequential rationalizability, 1, 15–16, 18,99–101, 104, 106–107, 111–115,118–119, 128, 174

weak sequential rationalizability, 18, 20,107–112, 119

Strategic manipulation, 171

Strategically independent set, 84–85, 110,119

Subjective possibility, 38–39, 43



About the Author

Geir B. Asheim is Professor of Economics at the University of Oslo,Norway. In additional to investigating epistemic conditions for game-theoretic solution concepts, he is doing research on questions relating tointergenerational justice.



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Asheim-The Consistent Preferences Approach to Deductive Reasoning in Games (2)