Conflict models in graph form: Solution concepts and their interrelationships

86 European Journal of Operational Research 41 (1989) 86-100 North-Holland

Theory and Methodology

Conflict models in graph form: Solution concepts and their interrelationships

Lip ing F A N G and Ke i t h W. H I P E L Department of Systems Design Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

D. M a r c K I L G O U R Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5 and Department of Systems Design Engineering, University of Waterloo

Abstract: A comprehensive approach for modeling conflicts is the recently developed graph form of conflict models. Within the flexible structure of the graph model, solution concepts including Nash stability, the various metagame techniques, the sequential stability method of Fraser and Hipel, and Stackelberg equilibrium are defined and characterized. These stability principles are compared with the limited-move and non-myopic stabilities developed previously for the graph model framework and mathematical interrelationships in this very general context are derived. As well, some steps are taken toward the development of a taxonomy of solution concepts for the graph form.

Keywords: Games, graph model

1. Introduction

In a conflict, two or more decision-making groups are in dispute over some issue(s), and the resolution depends on the choices of all participants. The non-cooperative game is the mathematical model used to study behavior in conflicts, but its information can be represented in a variety of ways. It has been shown (see Kilgour, Hipel and Fraser, 1984, for references) that it is often most convenient to model real-world conflicts by describing the possible states of the conflict, the players' preferences over these states, which inter-state transitions are possible, and which player controls which transitions. In what

Received December 1986; revised June 1987

follows, this approach to modeling and analyzing conflicts will be developed and extended.

Two traditional methods can be adapted to change-of-state conflict modeling. The normal form representation of games was given by von Neu- mann and Morgenstern (1953). The option form, which is often more convenient to use than the normal form, was introduced by Howard (1971) and subsequently utilized by Fraser and Hipel (1979, 1984). Recently, the graph form of conflict model was developed by Kilgour, Hipel and Fang (1987) to provide simple, flexible, and comprehensive conflict models. The graph approach to change-of-state conflict analysis constitutes a genuinely new methodology because the graph form can conveniently represent all conflicts, in- eluding those which cannot be modeled by the normal or option form.

037%2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

L. Fang et al. / Conflict models in graph form 87

A solution concept describes participants' behavior patterns and uses this assumed behavior to predict the outcome of the conflict. But humans can react to conflict situations in many ways, and many different solution concepts have been pro- posed. Kilgour, Hipel and Fraser (1984) compared mathematically a wide range of solution concepts used in normal form (or, equivalently, option form) models. Subsequently, Fang, Hipel and Kilgour (1986) integrated various solution concepts into a decision support system for two-player conflicts.

For n-player normal form conflict models, common solution procedures include those of classical game theory (Nash, 1950, 1951; von Neu- mann and Morgenstern, 1953), metagame analysis (Howard, 1971), the approach of Fraser and Hipel (1979, 1984), limited-move stabilities (Kilgour, 1985; Kilgour, Hipel and Fang, 1987; Zagare, 1984), non-myopic stability (Brams and Wittman, 1981; Kilgour, 1984, 1985; Kilgour, Hipel and Fang, 1987) and Stackelberg equilibrium (von Stackelberg, 1934). In this paper, Nash stability, metagame analysis, sequential stability, and Stac- kelberg equilibrium are reformulated for the graph form. These solution concepts, along with the limited-move and non-myopic methods developed previously for the graph model framework, are compared mathematically for individual stabilities and group stability (equilibrium). The definitions developed for the graph form constitute natural extensions of the solution concepts in other forms, in the sense that if the graph representation hap- pens to be equivalent to normal form (or option form) then these definitions would be equivalent. However, it was demonstrated by a real-world application in Kilgour, Hipel and Fang (1987) that only the graph model can accurately and conveniently describe certain conflicts and, hence, solution concepts defined within the graph para- digm must sometimes be used for ascertaining the equilibria.

In Section 2, the graph model for conflicts is presented briefly in terms of both graphical and analytic representations. The idea of stability is introduced in Section 3 and the specific solution concepts referred to previously are defined for two-player conflicts in Section 4, where some interesting relationships among them are derived. In Section 5, these solution concepts are extended to n-player conflicts in graph form, and again mathematical comparisons are presented.

2. The graph model for conflicts

A graph model for a conflict consists of a set of directed graphs and a set of payoff functions. Let N = {1, 2 . . . . . n} denote the set of players and U = {1, 2 . . . . . u} the set of outcomes (states) of the conflict. A collection of finite directed graphs D i = ( U , A,), i ~ N, can be used to model the course of the conflict. The vertices of each graph are the possible outcomes (states) of the conflict, and hence the set of vertices is common to all graphs. In each directed graph, the arcs are defined as follows: if player i can (unilaterally) move (in one step) from outcome k to outcome q, there is an arc with orientation from k to q in Ap For convenience, it is assumed that there is no arc from outcome k to itself, i.e. there are no loops in any player's graph. For each player i ~ N, a payoff function Pi : U ---, R, where R is the set of real numbers, is defined on the set of outcomes. The payoff functions measure the worth of outcomes to the players. As described below, it is assumed that values of the payoff functions represent only the players' ordinal rankings.

A simple model of a superpower military confrontation is used as illustration. (The detailed justification of this model can be found in Richel- son, 1979.) The conflict is modeled by two players: USA (player 1) and USSR (player 2). Both players, 1 and 2, have three alternatives (strategies): a conventional attack (labeled C), a limited nuclear strike (labeled L), and a full nuclear attack (labeled F). The nine distinct outcomes that are possible are labeled (CC), (CL), (CF), (LC), (LL), (LF), (FC), (FL) and (FF). A graph model of this conflict is shown in Figure 1.

A player's strategic possibilities are represented graphically in the figure. An analytical representation is given, for player i, by i 's reachability matrix, R~, which displays exactly which unilateral moves are available to player i from any position. For i ~ N, R, is the u × u matrix defined by

R( q){i if player i can move

(in one step) from outcome k

to outcome q, otherwise,

(2.1a)

where k 4: q, and by convention

g , ( k , k ) - 0 . (2.1b)

88 L. Fang et al. / Conflict models in graph form

Assume that the outcomes of the superpower military confrontation conflict, (CC), (CL), (CF), (LC), (LL), (LF), (FC), (FL) and (FF), are num- bered 1, 2, 3, 4, 5, 6, 7, 8 and 9, respectively. By placing Ri(k l , k2) in the cell in row k 1 and column k2, the teachability matrices for players 1 and 2 can be written:

R 1

'0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

,0 0 0 0 0 0 0 0 0 (2.2a)

R 2 =

r0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

~0 0 0 0 0 0 0 0

0 ~

0 0 0 0 0 1 1 0,

(2.2b)

The teachability matrix R i provides one way to represent player i ' s graph analytically. In order to avoid recording the zero entries in a reachability matrix, one can employ player i ' s reachable list, which is an equivalent expression of player i 's decision possibilities. For i ~ N, player i 's reachable list for an outcome k is the set Si(k ) of all

Table 1 Reachable lists for the superpower military confrontation

k Sl(k) S2(k) 1 4 ,7 2,3 2 5,8 3 3 6,9 4 7 5,6 5 8 6 6 9 7 8,9 8 9 9

outcomes to which player i can move (in one step) from outcome k. Note that

S i (k ) - ( q : g , ( k , q) = 1}. (2,3)

The conflict is thus specified by nu reachable lists, one for each player and outcome (state). (Recall that n is the number of players, and u is the number of vertices or outcomes.)

The reachable lists of the conflict in Figure 1 are shown in Table 1. Note that there is no essential difference in the information about the conflict presented in Figure 1, Equations (2.2a, b) or Table 1.

The payoff function for player i, Pi, measures how preferred an outcome is for i. Thus, if k, q ~ U, then Pi(k)>~ P,(q) iff i prefers k to q, or is indifferent between k and q. When this in- equality is strict for all pairs of distinct outcomes for every player, the conflict is called strict ordinal; in other words, different outcomes have different payoffs for every player in a strict ordinal conflict. (It is assumed that, in any conflict, if two

f (a)

(b)

Figure 1, Simplified superpower military confrontation: (a) graph D 1 for player 1; (b) graph D 2 for player 2

L Fang et al. / Conflict models in graph form 89

outcomes u 1 and u 2 satisfy Pi(uO = Pi(u2) for every i ~ N, then u I = u2; in other words, different outcomes must have different payoffs for at least one player.) Beyond ordinal information about preference or indifference, nothing will be inferred from the values of Pr In summary,

P~ ( k ) = ordinal payoff to player i for outcome k.

(2.4)

For the superpower military confrontation the payoff function values are (Richelson, 1979)

P, = (5 ,4 , 1, 9, 6, 2, 8, 7, 3),

P2 = (8 , 9 , 7 , 4, 6, 5, 1, 2, 3). (2.5)

A directed graph is called transitive if there is an arc vlv 3 whenever arcs vav 2 and 0203 are in the graph, for any distinct vertices v~, v z, v 3. Each player's graph in a conflict model will be transitive since decision possibilities are transitive.

The graph form is more flexible than other forms for the change-of-state modeling of conflicts. The option and the normal forms are symmetric in the sense that if player i can move (in one step) from outcome x to outcome y, then player i can also move (in one step) from outcome y to outcome x. Therefore, irreversible moves cannot be properly represented using the option or normal form. However, irreversible moves can be readily modeled using the graph form. Further- more, the graph form can also model the possibil- ity of common moves, in which more than one player can move from one outcome to another. The extra flexibility to represent irreversible moves and common moves constitutes an important ad- vantage of the graph form for modeling conflicts.

a particular outcome for a specific player is a preferred outcome (for that player) to which he can unilaterally move. Note that the player must strictly prefer the resulting outcome to the initial outcome. To represent unilateral improvements, each player's teachability matrix, R,, can be replaced by R + , defined by

{10 i f R , ( k ' q ) = l a n d R+(k' q) =- Pi(q) > P , ( k ) , (3.1)

otherwise.

Similarly, player i ' s reachable list, S,(k), can be replaced by Si+(k), defined by

Si+(k) =- {q: R?(k, q ) = 1}. (3.2)

Thus, ST(k ) denotes the set of player i ' s unilateral improvements from outcome k and is called the unilateral improvement list of player i from outcome k.

For the superpower military confrontation described in last section, the unilateral improvement lists are shown in Table 2.

4. Stability and equilibrium concepts for two-player conflicts

In this section, the definitions of Nash stability, general and symmetric metarationality, F rase r - Hipel sequential stability, and Stackelberg equilibrium are extended to two-player conflicts modeled in graph form. The limited-move stability and non-myopic stabilities developed in Kilgour, Hipel and Fang (1987) are also presented here for com- parison. Some of the interrelationships of these stability definitions are also derived.

3. Stability analysis

The stability analysis of a conflict is carried out by determining the stability of each outcome (state) for every player. An outcome is stable for a player iff that player has no incentive to deviate from it unilaterally, under a particular stability definition. An outcome is an equilibrium iff all players in the conflict find it stable; an equilibrium constitutes a possible resolution for the conflict.

To represent various stability definitions in the graph form, the concept of unilateral improvement is invaluable. A unilateral improvement from

Table 2 Unilateral improvement lists for the superpower military confrontation

k S((k) &(k) 1 4 ,7 2 2 5 ,8 3 6 ,9 4 5 ,6 5 8 6 9 7 8 ,9 8 9 9

90 L Fang et al. / Conflict models in graph form

[]

I I

Figure 2. Player i's decision problem at initial outcome k in a two-player conflict where j is i's opponent, i.e. j = 2 if i = 1 and j =1 if i = 2; k, kl, kx are outcomes; and s means stay

In a two-player conflict, player i ' s decision problem at initial ou tcome k is illustrated in Fig- ure 2. In this figure, a special convent ion is introduced for convenience in two-player conflicts: if a player i E N has been identified, then i ' s co-player (opponent) is automatical ly denoted by j . I f player i seizes the initiative and moves to some outcome k I ~ Si(k), then player j , i ' s opponent , may per- haps move f rom k 1. Depending on what he expects player j might do f rom each possible k x S~(k), player i may prefer to stay at outcome k; if so, ou tcome k is stable for i. If ou tcome k is stable for bo th players, it is an equ i l i b r ium- -a state of a conflict which can be expected to persist if it arises. Fol lowing are several alternative definitions of stability for an ou tcome (state) of a two-player conflict.

4.1. Nash stability

The most basic axiom of game theory is the postulate of individual rationality. As characterized by Luce and Raiffa (1957), it is assumed that ' o f two alternatives which give rise to outcomes, a player will choose the one 'which yields the more preferred outcome' . Nash (1950, 1951) formalized this concept into an individual stability criterion.

Definition 4.1. Let i ~ N. An outcome k ~ U is Nash stable (or individually rational) (R ) for player i iff S~+(k)=~.

Under Nash stability, player i expects that player j will stay at any outcome i moves to, and

consequently that any state that i moves to will be the final state. The initial ou tcome k is therefore stable iff i cannot move f rom k to any ou tcome i prefers.

4.2. General metarationality

Definition 4.2. For i ~ N, an ou tcome k ~ U is general metarational (GMR) for player i iff for every k 1 ~ Si+(k) there exists at least one k 2 Sj(k 0 with P , (k2) ~< P~(k).

Thus, for general metarationali ty, player i expects that player j , i ' s opponent , will respond by hurt ing i if it is possible for j to do so, so that k is stable iff j can hurt i if i takes any unilateral improvement . Note that i anticipates that the conflict will end after j ' s move, and that j ' s move will be chosen without regard to j ' s payoffs.

4.3. Symmetric metarationality

Definition 4.3.. Let i E N. An outcome k ~ U is symmetric metarational ( S M R ) for player i iff for every k 1 ~ Si+(k), there exists k 2 ~ Sj(kl) , such that Pi(k2) <~ Pi(k) and Pi(k3) <~ Pi(k) for all k3 E Si( ka).

Thus, player i expects that he will have a chance to counter respond (k3) to his opponen t

j ' s response (k2) to i ' s original move (kl) . No te that i anticipates that the conflict will end after his counterresponse, and that j ' s response will be chosen both to hurt i and to make any counterresponse by i profitless, but again without regard to j ' s payoffs.

4. 4. Sequential stability

A n outcome is sequentially stable for a player iff he is deterred f rom taking any unilateral improvement f rom this ou tcome because a credible act ion (unilateral improvement) by the opponen t could result in an outcome less preferred (for the original player) than the initial outcome.

Definition 4.4. For i ~ N, an ou tcome k G U is sequentially stable (FHQ) for player i iff for every k~ ~ Si+(k) there exists k 2 ~ Sj+(kl) with Pi(k2) <~ P,(k).


This stabihty condition is similar to general metarationality, but admits only those 'sanctions ' ( k 2 ) which are 'credible' (k 2 ~ S S ( k l ) is required, rather than merely k 2 E S j ( k l ) ).

4.5. Limited-move stability

In the context of strict ordinal 2 × 2 games, Zagare (1984) suggested the concept of limited- move stability to show how the behavior of a player depends on the horizon of foresight, by making horizon distance a parameter supplied by the modeler. The player 's actions are determined under the assumption that subsequent moves, which alternate between the players, are always made in the best interest of the mover, who takes into account all possible courses for further moves (up to the original horizon). Let h be a positive integer. A player whose horizon is h moves distant is a player who foresees a game of (maximum possible) length h; limited-move stability is then referred to as L h stability. Kilgour (1985) gener- alized limited-move stability concepts to any finite two-player non-cooperative game in normal form. The graph form representation of limited-move stability for two-player and n-player conflicts was given by Kilgour, Hipel and Fang (1987) and is summarized here, along with an efficient solution technique.

Player i ' s decision problem at initial outcome k is illustrated in Figure 3 for L h stability. It is postulated that a player contemplating departure from a status quo outcome considers first his own move, then his opponent 's response, then his own counterresponse, etc., in strictly alternating sequence. Accordingly, player i, in deciding whether to move from k, views himself as the first player in an extensive game in which each player, in sequence, must decide whether to stay at the current outcome (state), or to move from it by some available (one step) move, in which case a new outcome is attained, and the opponent must now choose whether to stay at this new outcome (now the current outcome), or to move from it under the same conditions. For L h stability, the game ends as soon as either player chooses to stay, or after 'move ' has been chosen h times consecu- tively.

Player i ' s decision can be interpreted as the first choice in a finite extensive game of perfect information, which can be solved completely by

/ I

I

\ I I

Figure 3. L h s tab i l i ty for p layer in a 2-player conf l ic t ( p is the player who can move las t so tha t p = i if h is odd and

p = j if h is even; k, k l , k 2 . . . . , k h are outcomes; and s

means stay)

' backward induction', that is, by induction pro- ceeding backwards in time. A simple specific illustration can be found in Section 4.9.

A method for assessing L h stability, adapted so that solutions at different lengths h can be obtained efficiently, is now outlined. (This adapta- tion is similar in spirit to the usual solution proce- dure used in dynamic programming.) Let G~(i, k ) denote i ' s anticipation vector for the limited-move family with horizon h, interpreted as follows:

Gh(i, k ) - f i n a l outcome of a game of length h beginning at outcome k with the initial move made by player i, (4.1a)

where, for convenience,

Go(i , k ) - k . (4.1b)

Following Figure 3, the anticipation vector of player i for L h stability can be constructed as


follows. Recall that Si(k ) = (q: Ri(k, q) = 1) represents the set of outcomes to which player i can move (in one step) from outcome k, or i ' s reachable list f rom k. If Si(k ) =JJ, the empty set, then player i is unable to move from outcome k, and outcome k is necessarily stable for i. The stability of an outcome from which i can move (if he so chooses) is now considered. If S i ( k ) ~ J , player i anticipates that the most he can achieve by moving from k (to Mh(i, k)) is Ah(i, k), defined as follows:

M h (i , k ) is the outcome ~ ~ Si ( k ) which satisfies

3)) =max{P,(Gh_l( j , q) ) : q ~ S , ( k ) ) . (4.2)

Then

Zh(i, k )=-e i (ah_ l ( j , Mh(i, k))). (4.3)

If the game is strict ordinal, ~ is uniquely defined by (4.2). General definitions for the non- strict ordinal case are given elsewhere.

The anticipation vector can be defined inductively. Go(i, k) is defined by assumption in (4.1b). Gh(i, k), h >1 1, can be found from Gh_l(j, k) by

Gh(i , k) = k if Si(k ) = ~ , otherwise:

( k if P~(k) >~Ah(i, k),

Gh(i, k ) - - ~ G h _ l ( j , Mh(i, k)) (4.4) I

if Pi(k) <Ah(i, k).

Note that since Go(i, k) = k, (4.2), (4.3) and (4.4) are well-defined for h = 1.

Definition 4.5. Let i ~ N. An outcome k ~ U is fimited-move, horizon h o r L h stable ( L h ) for player i iff Gh(i, k ) = k.

It is obvious (see Figure 3) that L 1 stability is identical to Nash stability.

4.6. Non-myopic stability

Brams and Wit tman's (1981) concept of non- myopic stability in 2 × 2 games assumes that players can look ahead and anticipate where a process might end up if they are allowed to make several sequential moves and countermoves start-

ing from any status quo position. Kilgour (1984) broadened this idea by endowing the players with sufficient foresight to envision the outcomes of arbitrarily long move-countermove sequences in any 2 × 2 game. If all sufficiently long sequences have the same outcome, then the result of a departure by a 'foresightful ' player is determined. The status quo is stable when the foresightful player chooses not to depart unilaterally from it. Non- myopic stability is thus the limiting case of limited-move stability as the horizon h increases without bound.

Definition 4.6. For i E N, an outcome k ~ U is non-myopically stable (NM) for player i iff there is a positive integer s such that Gt(i, k) = k for all t >~ s.

4. 7 Stackelberg equilibrium

All of the stability concepts discussed above are individual stability concepts in that they model a specific player 's decision to stay at or depart u- nilaterally from a status quo outcome k. An outcome is an equilibrium iff every player finds it stable. Most of the definitions of equilibrium ap- pearing in the literature are symmetric in the sense that the roles of the players are symmetric, that is to say, each player's stability definition is the same. It may well be appropriate to use an asym- metric equilibrium model if, say, one of the players has the ability to force his decision on another. In the formulation of von Stackelberg (1934), the player who holds the powerful position is called the leader, and the other player, who reacts to the leader's decision, is called the follower.

Definition 4.7. Let i E N. An outcome k ~ U is an equilibrium in the sense of yon Stackelberg with i as leader (ST(i)) iff k is L 2 stable for i and Nash (or L1) stable for j .

The original definition of von Stackelberg (1934) is in the context of strict ordinal two-player conflicts in normal form. The definition of L 2

stability given above thus permits Stackelberg's equilibrium concept to apply to any finite two- player conflict, in any form of model. If an outcome is a Stackelberg equilibrium with either player as leader, it is referred to as a dual Stackel- berg equilibrium (ST).

L. Fang et aL / Conflict models in graph form 93

4.8. Properties of the solution concepts

Because solution concepts defined in graph form are more general than in normal or opt ion form, mathemat ical relationships among the solution concepts may be different in graph form. Conse- quently, relationships among the solution concepts in graph model f ramework are now derived.

First, it is easy to prove that if k is Nash stable for i, then k is symmetr ic metarat ional for i, and if k is symmetr ic metarat ional for i, then k is general metarat ional for i. It is also true that if k is Nash stable for i, then k is sequentially stable for i, and if k is sequentially stable for i, then k is general metarat ional for i. The proofs of these properties are simply specializations of the proofs presented in the next section for corresponding relationships in n-player conflict models in graph form; therefore, they will not be given here.

Some other properties of stability in two-player conflicts are presented now.

Theorem 4.8. For i ~ N and k ~ U, i f outcome k is L 2 stable for i, then k is sequentially stable for player i.

Proof. Similar to the proof of Theorem 5.13. []

A corollary of Theorem 4.8 is that an outcome which is L 2 stable for i is general metarat ional for i. The converse of Theorem 4.8 is false.

Theorem 4.9. For i ~ N and k ~ U, if outcome k is Lh stable for player i, h ~ (1, 2, 3 , . . . ), then k is general metarational for i; also if k is non-myopically stable for i, then k is general metarational for i.

Proof. Assume k is L h stable for i, for some h ~ {1, 2, 3 . . . . ). For h = 1, L~ stability is identical to Nash stability, and the theorem is true as noted above. The theorem is also true for h = 2, as noted following Theorem 4.8. N o w fix h > 2, i, and k ~ U. The proof is complete if it can be shown that k is G M R for i. This is certainly true if S~ (k ) = ~[. Therefore assume that S i ( k ) -~ ~. For k I ~ S~(k), it may happen that Sj(k~) =~J. Then P,(ka)<~Pi(k) , since k is Lh stable for i, and therefore k~ ~ Si÷ ( k ). Otherwise Sj( k~) ~ fJ. Then either k~ q~ S~+(k), or, if k~ ~ S~+(k), there exists a

sanction k 2 E ~ ( k l ) . Specifically, either k 2 is Lh_ 2 stable for i, in which c a s e Pi(k2)<~ Pi(k), since k is L h stable for i, or k 2 is n o t Lh_ 2 stable for i; in the latter c a s e , Gh_2(i, k 2 ) = k x for some k x ~ U, and Pi(kx) > Pi(k2). But since k is L h stable for i, P~(k) >~ Pi(kx) . Again, it follows that P~(kR) <~ P, (k ) .

In summary, if k is L h stable for i, then for every k I ~ S , + ( k ) there exists k 2 ~ S j ( k l ) with Pi(k2)<~Pi(k) . Therefore k is general metarational for i by Definit ion 4.2. By Defini t ion 4.6, the second statement of the theorem is now obvious. []

Theorem 4.10. For i ~ N and k ~ U, i f outcome k & L 3 stable for player i, then k is svmmetric metarational for i.

Proof. Assume k is L 3 stable for i. First, if Si (k ) = ~, the theorem is proved. Therefore assume that Si( k ) :/: fJ. For k 1 ~ Si( k ), it may happen that S j ( k l ) = ~ . Then Pi(kl)<~ Pi(k) , since k is t 3 stable for i, and therefore k I ~ S~+(k). Otherwise S j (k l )~J~ . Then either k 1 ~ti S~+(k) or if k 1 ~ S~+(k), there exists a sanct ion k 2 ~ Sj (k l ) . In particular, either k 2 is L 1 stable for i, so that

m a x ( P ~ ( k 3 ) : k 3 ~ Sg(k2) } ~< P , ( k 2 ) <<. P , ( k )

because k is L 3 stable for i, or k 2 is not L 1 stable for i; in the latter case, let k3 ~ S~(k2) satisfy

P~(~:3) = max( P~(k3): k 3 ~ S~(k2) } .

N o w Pi (k ) >~ Pi(k3) >~ Pi(k2) and P~(k) >! P~(~c3) >1 e l (k3) for all k 3 E Si (k2) , since k is L 3 stable for i.

It follows that if k is L 3 stable for i, then for every k I ~S~+(k) there exists k 2 ~ S j ( k l ) with P~(k2) <~ P i ( k ) and Pi(k3) <~ Pi(k) for all k 3 E

S~ (k 2). Therefore, k is symmetr ic metarat ional for i by Definit ion 4.3. []

It follows that, if k is L x o r L 3 stable, the k is symmetric metarational . However it can be shown by examples that an ou tcome can be L 5 stable but not symmetr ic metarational . (This shows that a proper ty found by Kilgour, 1985, does not generalize to the graph form.) Also, if k is a Stackelberg equilibrium, then k is general metarat ional for the leader and symmetr ic metarat ional for the follower.

94 L. Fang et aL / Conflict models in graph form

f A l l o u t c o m e s ""

fGMR -'~ fLh' h> 3/.~_~

FHO

1 ,J

J

I AII outcomes GMR

f L h , h > 3 ~'~ - - FHQ~" 2 - -

f ~T(;] SMR

. . J

Figure 4. Individual stability concepts in two-player conflicts Figure 5. Group stability concepts in two-player conflicts

[]

y ~G,O,5):8

P, B):7

(a)

G2 0,5) :5

( 9 ) : 3

,(2,8): 9

r,~ (9) :3

[]

s ~ ] ) G3(I,5) :5 ( ~ ~(9):3

P;~) ~ .,.~) G2(2,8) :9 = / I

P~G~ 0,9):9

pi(9) :3 Co)

(b) Figure 6. Player l 's decision problem at initial outcome 5: (a) length 1; (b) length 2; (c) length 3. (Preferred selections are indicated

by the arrows)

L. Fang et at. / Conflict models in graph form 95

Table 3 Stability analysis of the superpower mili tary confrontation

Outcome R G M R SMR FHQ L l L 2 t 3 L h, h >/4 ST(l) ST(2)

1 E E E 2 2 2 2 2 E E 2 2 2 2 2

3 2 2 2 2 2 2 2 2 4 1 E E E 1 E 1 1

5 2 E E E 2 E E E 6 2 2 2 2 2 2 2 2

7 1 1 1 1 1 1 1 1

8 1 1 1 1 1 1 1 1

9 E E E E E E E E

E

E E

The consequences of above theorems are illustrated in the Venn diagram in Figure 4. To interpret the figure, recall that R, GMR, SMR, and F H Q correspond to Nash stability, general metarationality, symmetric metarationality, and sequential stability, respectively.

If an outcome is individually stable for a player according to some stability criterion, then that particular player prefers to remain there rather than to move unilaterally to some other outcome. Recall that an outcome is an equilibrium of a particular type iff it has individual stability of that type for every player. Theorems 4.8, 4.9, and 4.10 can be restated for equilibria. In particular, the following theorem is established for equilibria.

Theorem 4.11. Let k ~ U. I f k is a Stackelberg equilibrium, then k is a sequential equilibrium, and if k is a sequential equilibrium, then k is a general metarational equilibrium.

Proof. Immediate from Definition 4.7 and Theo- rem 4.8. []

It also follows that if k is a dual Stackelberg equilibrium, then k is a Nash (or L1) equilibrium, and is also an L 2 equilibrium.

Figure 5 illustrates the relationships of group stability (equilibrium) concepts. In the figure, ST(i) means Stackelberg equilibrium with player i as leader and ST means dual Stackelberg equilibrium.

4. 9. A n example

To demonstrate how the developments presented in Section 4 can be used in practice, they are now applied to the military conflict described

in Section 2. For example, the calculation of L h stability for h = 1, 2, and 3, is shown for outcome 5 in Figure 6. First, Gl(i, k ) is calculated for each player i and outcome k, as illustrated in Figure 6(a). Note that outcome 5 is not L 1 stable for 1, since I prefers to move to outcome 8 in this length 1 g a m e - - i n other words, GI(1, 5) = 8. Similarly, G1(2, 8) = 9, and this fact is used in Figure 6(b) to show that G2(1 , 5 ) = 5, i.e. that outcome 5 is L 2

stable for 1. More specifically, $1(5)= {8}, so M2(1, 5) = 8 in (4.2) and A2(1, 5) = PI(G~(2, 8)) = P1(9)= 3 in (4.3). By (4.4), G2(1, 5 ) = 5, since P1(5) = 6 > A2(1, 5) = 3. The demonstrat ion that outcome 5 is also L 3 stable for 1 is similar and is shown in Figure 6(c). (A similar computat ion for a much larger example is given in Kilgour, Hipel and Fang, 1987.)

The individual stability and equilibria of the military conflict are shown in Table 3. In this table, '1' or '2 ' indicates that the outcome is stable for the indicated player but not the opponent, and 'E ' indicates that the outcome is an equilibrium.

In Table 3, the effects of Theorems 4.8, 4.9, 4.10, and 4.11 can be seen. For instance, player 1 finds outcomes 4, 5, 7, 8, and 9 to be L 2 stable, and, as predicted by Theorem 4.8, they are all sequentially stable as well. However, outcome 1 is sequentially stable for player 1 but n o t L 2 stable, illustrating that the converse of Theorem 4.8 is false.

5. Stability and equilibrium concepts for n-player conflicts

In the previous section, Nash stability, the two metagame stabilities, sequential stability, and Stackelberg equilibrium were defined for two-

96 L. Fang et aL / Conflict models m graph form

[]

I I

I I I I I ii

I-q, ,,, ! I I I I I I

jEN-i

pEN-i

Figure 7. Player i ' s decision problem at initial outcome k in an n-player conflict

player conflicts in graph form. The definitions given here generalize these stability concepts to the n-player case.

In an n-player conflict, player i ' s decision problem at initial outcome k is illustrated in Fig- ure 7. If player i seizes the initiative and moves, say to outcome kl ~ Si(k), then some other player j , j ~ N - i , may move from k~, say to k 2 Sj(kl) . Depending on j ' s move, yet another player p, p ~ N - i , may move from k2, say to k 3 Sp(k2) , and so on. Depending on what player i expects the other players N - i to do from each k 1 ~ Si(k), player i may prefer to stay at outcome k. Note that in this sanction sequence the same player may move more than once; however, the (original) player i does not take part in the sequence of moves.

Let H _c N be any subset of the players, and let Sn(k ) denote the set of all outcomes that can result from any sequence of unilateral moves, by some or all of the players in H, starting at outcome k. Formally:

Definit ion 5.1. Let k ~ U and H _ N, H 4= ~[. A unilateral move by H is a member of Sr i (k) _ U, defined inductively by

i f j ~ H and k l ~ S j ( k ), then k I ~ Sri(k), (5.1a)

and

i f j ~ H , ka ~ S H ( k ) , and k2~Sj (ka) ,

then k 2 E S n ( k ). (5.1b)

Analogously, let S~(k) denote the set of outcomes that can result from any sequence of unilateral improvements by some or all of the players in the set H from outcome k.

Definit ion 5.2. Let k ~ U and H c N, H v~ £[. A unilateral improvement by H is a member of S~ (k ) _ U, defined inductively by

i f j ~ H and k l ~ S f - ( k ) , then k l ~ S ~ ( k ) ,

(5.2a)

and

if j ~ H, ka ~ S~ ( k ) , and k= ~ S~-(kl),

then k z ~ S~ (k ). (5.2b)

Sn(k ) and S~(k) can be thought of as H 's reachable list and unilateral improvement list, re-


spectively. In particular, the sets SN_, (k) and +

SN_,(k ) represent the possible outcomes of ' response sequences' of i ' s opponents against a move by i to k.

5.1. Nash stability

Definition 5.3. Let i ~ N. An outcome k ~ U is Nash stable ( R ) for player i iff S~+ ( k ) = tJ.

Since Nash stability (individual rationality) does not take into account the other players' possible responses to a unilateral improvement, there is no difference between the two-player and n-player definitions (see Definition 4.1).

5.2. General metarationality

Definition 5.4. For i E N, an outcome k ~ U is general metarational (GMR) for player i iff for every k I ~ S~+(k) there is at least one outcome k x ~ SN_~(kl) with P~(kx) <<. P~(k).

them to do so, by a sequence of unilateral moves. As before, i anticipates that the conflict will end after the players of N - i have responded. As well, i ' s opponents are assumed to ignore their own payoffs in making their sanctioning move.

5.3. Symmetric metarationality

Definition 5.5. Let i ~ N. An outcome k ~ U is symmetric metarational ( S M R ) for player i iff for all k 1 ~ Si+(k), there exists k x ~ SN_i(kl) , such that Pi(kx) <<. P,(k) and Pi(k3) <~ Pi(k) for all

Thus, player i expects that he will have a chance to counterrespond (k 3) to the other players' response (kx) to i ' s original move (kl) . Note that i anticipates that the conflict will end after his counterresponse.

5.4. Sequential stability

Thus, player i expects that the other players (N - i) will respond to hurt i, if it is possible for

An outcome is sequentially stable for a given player iff he is deterred from making a unilateral

[ ]

$

i I

I I I

Figure 8. L h stability for player i in an n-player conflict

jEN-i

pEN-i

98 L. Fang et at / Conflict models in graph form

improvement because a sequence of individual unilateral improvements by the other players could result in an outcome less preferred (for the original player) than the initial outcome.

Definition 5.6. For i ~ N, an outcome k ~ U is sequentially stable (FHQ) for player i iff for every k 1 ~ S~÷(k) there is at least one outcome k X S~_~(ka) with P~(kx) <~ PAk).

Again, the main difference between the general metarationality and sequential stability is that the credibility of sanctions is important only in sequential stability.

5.5. Limited-move stability

Player i ' s decision problem at initial outcome k is illustrated in Figure 8. Because the player with the initiative cannot control other players' actions, all he can do is anticipate, conservatively, that he will achieve the least preferred of all those outcomes which could arise as a result of each player seizing the initiative. This anticipation is represented by the minimum operation in Figure 8.

From Figure 8, it can be seen that each sub- graph represents an ( n - 1)-player game with length h - l . Let N-i - Gh_ 1 (J, k) represent the vector Gh_I(j, k) for this ( n - 1)-player game. Then,

G N - i . • h - l ( J , k): j ~ N - i } for h >l l can be used to find Gh(i, k).

For h >~ 1, the minimum postulate implies that i anticipates that, if he can move from the status quo, the final outcome will be given by the vector G'~_a(i , k), defined by

" i Gh_, ( , k)=-GNh_-?(W, k), (5.3a)

where w ~ N - i satisfies

N--i Pi (Gh- I ( W, k ) )

=min{ei(Gff--li(j, k ) ) : j ~ N - i } . (5.3b)

Once G~'_l(i, k) is obtained, Gh(i , k) can be defined as follows: if Si(k ) @~J,

M h ( i, k ) is the outcome ~ ~ Sg ( k )

which satisfies

P,(Gh"-,(i, 3))

=max{Pg(G~'_~(i, q) ) : q~S i ( k ) } . (5.4)

Then

ah(i , k)=-Pi(a'~_l(i, Mh(i , k))), (5.5)

and Gh(i, k) can be found inductively from G~_l(i, k) by

G,(i, k ) = k if S , (k )=~, otherwise:

( k i fPi(k)>~Ah(i ,k ),

Gh(i , k ) - ~G7_1(i, Mh(i, k)) (5.6)

if Pi(k) <Ah(i, k).

For h = 1, the minimum operation in Equation (5.3) is superfluous because Gg-i(j, k)= k for every player j ~ N - i.

Definition 5.7. Let i ~ N. An outcome k ~ U is limited-move, horizon h or L h stable (Lh) for player i iff Gh(i, k )= k.

5.6. Non-myopic stability

Consider the sequence of anticipation vectors Gh(i, k) for h = 1, 2, 3 . . . . I f it should happen that Gs(i, k )= Gs+l(i, k) . . . . . k for some s >~ 1, the outcome k is non-myopically stable for i. Non-myopic stability is the limiting case of limited-move stability as the horizon h increases without bound.

Definition 5.8. For i ~ N, an outcome k ~ U is non-myopically stable (NM) for player i iff there exists a positive integer s such that Gt(i, k )= k for all t >/s.

5. 7. Stackelberg equilibrium

The generalization of Stackelberg equilibrium to n-player conflicts could be accomplished in a number of ways. Basar and Olsder (1982) discuss possible extensions in the case n = 3. For three- player conflicts, there are three possible modes of play among the players within the framework of non-cooperative decision making.

(1) There are two levels in the h ie ra rchy- -one leader and two followers.

(2) There are two levels in the h ie ra rchy- - two leaders and one follower.

(3) There are three levels in the h ie ra rchy- -denote the leader by P1 and the followers, in order, by P2 and P3.


For m o d e (1), a defini t ion of Stackelberg equi- l ibr ium is immedia te using the l imi ted-move stability definitions. For modes (2) and (3), defini- t ions of Stackelberg equi l ibr ium could be developed using the f r amework of l imi ted-move stability, bu t d ropp ing the m i n i m u m opera t ion when the order of response is predetermined.

Definition 5.9. For the 3-player conflict with one leader and two followers, an ou tcome k ~ U is a Stackelberg equilibrium iff k is L2 stable for the leader and N a s h (or L t) stable for the followers.

Fur ther extensions to n > 3 would then be con- ceptual ly s t raightforward.

5.8. Properties of the solution concepts

In Section 4.8, interrelat ionships a m o n g the solut ion concepts in graph fo rm for two-player conflicts were established. In this section, the corresponding results for n-player conflicts are established.

Theorem 5.10. Let i ~ N and k ~ U. I f k is Nash stable for i, then k is symmetric metarational for i; if k is symmetric metarational for i, then k is general metarational for i.

Proo | . If k is N a s h stable for i, then S i + ( k ) = ~ and Defini t ion 5.5 is satisfied. I f k is symmetr ic metara t iona l for i, Defini t ion 5.5 implies that Def ini t ion 5.4 is satisfied and therefore that k is general meta ra t iona l for i. []

Theorem 5.11. Let i ~ N and k ~ U. I f k is Nash stable for i, then k is sequentially stable for i; if k is sequentially stable for i, then k is general metarational for i.

Proof . If k is N a s h stable for i, then Def ini t ion 5.6 is satisfied trivially since S~+(k)=~[. Since S ~ _ i ( k l ) c S N _ i ( k 0 , an o u t c o m e which is sequential ly stable for i mus t also be general meta ra t iona l for i. []

Theorem 5.12. For i ~ N and k ~ U, if outcome k is L h stable for player i for h ~ (1, 2, 3 . . . . }, then k is general metarational for i; if k is non-myopically stable for i, then k is general metarational for i.

Proof. Assume k is L h stable for i for some h ~ (1, 2, 3 . . . . ). If h = 1, L 1 stabil i ty is identical to Nash stability, and the theorem is true as noted above. N o w fix h >~ 2. First, k is G M R for i if S i ( k ) = ~ . Therefore assume that S i ( k ) 4 ~ . For k I E S i ( k ) , if S N _ i ( k l ) = J ~ , then P i ( k l ) < ~ P i ( k ) , since k is L h stable for i, and it follows that k I f~ Si÷(k). N o w suppose that if S N _ _ ~ ( k l ) ~ . Therefore either k L ~ Si+ ( k ) or, if k 1 ~ Si+__( k ), then there exists k x ~ S N _ ~ ( k l ) , where k ~ = Gh~_l(i, k l ) and Pi(-k~)<<.P,(k), since k is L h stable for i.

Therefore, if h >/2, then for every k~ ~ S~ + ( k ) there exists k x = - k x ~ S N _ ~ ( k a ) with P,(k~)<~ P~(k), so that k is general meta ra t iona l for i by Defini t ion 5.4. By Def ini t ion 5.8, the second par t of the theorem is obvious. []

Theo rem 5.13. For i ~ N and k ~ U, if outcome k is L 2 stable for player i, then k is sequentially stable for i.

Proof. Assume k is L 2 stable for i. First, if Si (k ) = ~, the theorem is proved. There fore assume that Si( k ) 4~ ~J. For k I ~ Si( k ), if S/- ( k l ) = fJ for all j ~ N - i, then Pi (k t ) <~ Pi (k ) , since k is L 2 stable for i, and therefore k 1 ~ S,+(k). Alter- nately, ~ + ( k l ) 4:~J for some j ~ N - i. Therefore either k 1 ~ Si+(k) or, if k 1 E S,+(k), then there exists k x ~ S j ( k l ) , for some j ~ N - i, such that kx = G~'(i, k l ) . But then Pi(kx)~< P, ( k ) since k is L 2 stable for i.

('--/ii- o~ t~oo,,eS- . . . . . . . \ r f 6 M R . . . . i . . . . . ~ - _ .~

• - \

Figure 9. Stability concepts in n-player conflicts


Therefore, if k is L 2 stable for i, then for every k s ~ Si+(k) there exists k x = k x ~ S N - i ( k l ) with Pi(kx) <~ Pi(k) since, if j ~ N - i, Sj+(kl) _ S~_i(k l ) . Then k is sequentially stable for i by Definition 5.6. []

A corollary of Theorem 5.13 is that an outcome which is L 2 stable for i is general metarational for i. The consequences of above theorems are illustrated in Figure 9. Figure 9 also describes the interrelationships of the corresponding equilibrium definitions for the n-player case.

6. Conclusions

The graph model constitutes a flexible and comprehensive framework for conveniently deft- ning and developing, meaningfully characterizing, and rigorously comparing solution concepts. Here, the solution concepts of Nash stability, metagame analysis, sequential stability, and Stackelberg equilibrium are formulated for the first time within the graph model context. Additionally, theorems are derived for describing the interesting relationships among these solution concepts, as well as the limited-move and non-myopic concepts, for both two-player and n-player games.

The results of this paper should make it easier to apply the graph model for conflicts, and to interpret the results of those applications. This is an important contribution, because the graph form is the most flexible and comprehensive method for carrying out change-of-state conflict modeling.

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References

Basar, T., and Oisder, G.J. (1982), Dynamic Noncooperative Game Theory, Academic Press, London.

Documents

Conflict models in graph form: Solution concepts and their interrelationships