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An Analysis of Innovation with Biform Games
Bachelor's Thesis
Written by: Attila Gyetvai
BA in Applied Economics
Supervisor:
Dr. Tamás Solymosi, associate professor
Department of Operations Research and Actuarial Sciences
Corvinus University of Budapest, Faculty of Economics
Corvinus University of Budapest
Faculty of Economics
2012
Contents
1 Introduction 1
2 Preliminaries 3
2.1 Notations and de�nitions regarding cooperative games . . . . . . . . . 3
2.2 Notations and de�nitions regarding non-cooperative games . . . . . . 6
2.3 The original biform game . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Market games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Innovation in monopolistic markets 10
3.1 Big boss games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 An alternative de�nition of the biform game 16
4.1 An example of the original biform game . . . . . . . . . . . . . . . . 16
4.2 The modi�ed example . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 The alternative biform game . . . . . . . . . . . . . . . . . . . . . . . 22
5 Innovation in oligopolistic markets 24
5.1 Transportation games . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Summary 30
A The characteristic functions of the game in Example 4.1.1 34
B The core of v(I, SQ)(S) in Example 4.2.1 35
C The characteristic functions of the game in Model 3.2 37
D The characteristic function of a 5-player transportation game 38
E The characteristic functions of the game in Example 5.2.1 39
II
List of Figures
3.1 The core of a 3-player big boss game. . . . . . . . . . . . . . . . . . . 11
3.2 The innovation game on a monopolistic market in normal form. . . . 14
4.1 The original game in normal form. . . . . . . . . . . . . . . . . . . . 18
4.2 Projection of the core of the game v(I, SQ)(S) in the modi�ed exam-
ple to the plane of pay-o�s of the decisive players. . . . . . . . . . . . 20
4.3 The modi�ed innovation game in normal form. . . . . . . . . . . . . . 21
4.4 Inconsistency of con�dence indices in Example 4.2.1. . . . . . . . . . 21
5.1 The innovation game on an oligopolistic market in normal form. . . . 28
5.2 The transportation game in assignment game form. . . . . . . . . . . 29
List of Tables
3.1 The second-stage cooperative games of Example 3.2.1. . . . . . . . . 14
3.2 The coalitional surpluses of the second-stage cooperative games of
Example 3.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 The original example of the biform game. . . . . . . . . . . . . . . . . 17
4.2 The modi�ed example of the biform game. . . . . . . . . . . . . . . . 19
5.1 An example of the model of oligopolistic markets. . . . . . . . . . . . 27
III
Chapter 1
Introduction
During its century-long unbroken span, game theory has been subject to overwhelm-
ing enthusiasm, especially among mathematicians and economists. While mathe-
maticians deal with the abstract structure of strategic decision making, economists
seek for its applications in economic systems.
Game theory is often divided into two branches, based on the type of strategic
interaction between the players. Non-cooperative game theory assumes that the
players make their strategic moves on their own in order to maximize their utilities,
anticipating the moves of the others. Its main focus is to determine strategies which
are chosen with certainty. Contrarily, cooperative game theory proceeds from the
assumption that the players' decision is to cooperate or not. The point of cooperation
is that it generates bene�ts. The aim of such games is to design fair allocation
schemes, with respect to di�erent conceptions of fairness.
However, there is no point in assuming that the strategic environment in which
the game is played is static. Moreover, it is quite a natural assumption that the
strategic decisions of the players may result in multiple di�erent strategic envi-
ronments in which they can cooperate in a latter stage of the game. This idea is
presented by Brandenburger and Stuart (2007), who introduce the class of biform
games to model such a situation. In our paper, we display their model and modify
it in order to make it applicable to an extended set of players as well.
As mentioned above, one can approach game theory from the side of theory or
from the side of applications. Our personal motivation brings us to the second ap-
proach. We use the two abstract de�nitions of the biform game (and other related
conepts) to model innovation in monopolistic and oligopolistic markets. Our inten-
tion is to show the modelling capability of biform games in industrial organization.
However, we remark that the theory of biform games is not mature. By date, the
article of Brandenburger and Stuart (2007) is cited 108 times, but practically all
1
consequent works apply the original model only. Thus, we provide a solid theoretic
background beside the applications.
Our paper is based on the lessons of research seminar "Cooperation and Allo-
cation", led by dr. Tamás Solymosi.1 Therefore, we have approached biform games
from the side of cooperative game theory. This is the reason why the concepts of
cooperative game theory are slightly overrepresented. However, focusing on coopera-
tive games does not mean the irrelevance of non-cooperative game theory. Moreover,
we highlight the importance of non-cooperative games in the biform context.
In Chapter 2, we provide a solid theoretic background of both cooperative and
non-cooperative game theory, their synthesis and the class of market games. In Chap-
ter 3, we show an economic application of biform games, the model of innovation
in monopolistic markets. In Chapter 4, we remark that the original de�nition of the
biform game, as introduced by Brandenburger and Stuart (2007), is not applicable
to feature multiple decisive players in the �rst non-cooperative stage, and we rede-
�ne the biform game. In Chapter 5, we apply our de�nition to model innovation in
oligopolistic markets. Finally, we conclude our �ndings in Chapter 6 and trace paths
of future research.
1We have submitted a paper (Gyetvai and Török, 2012) for the Conference of Scienti�c Students'
Associations with co-author and friend Tamás Török. That paper is a shorter version of Chapters
2 and 4 of this current work. Certainly, we take the exclusive responsibility for the followings.
2
Chapter 2
Preliminaries
2.1 Notations and de�nitions regarding cooperative
games
We start by declaring the basic notations we use in our paper. All notations and
de�nitions are based on the work of Solymosi (2007).1 We suppose there are n players
of a game. The set of players is N = {1, . . . , n}. We assume these players may form
coalitions without any hurdles, i.e. any coalition S ⊆ N may be formed. Speci�cally,
we call coalition S = ∅ the empty coalition, S = N the grand coalition and any
coalition S = {i} the singleton coalition of player i. We denote the set of all non-
empty coalitions by N so that N = N \ {∅}. Note that |N | = 2n − 1. We also use
the notation N+ to denote the set of all real sub-coalitions, i.e. N+ = N \{N}. Notethat |N | = 2n − 2. The pay-o� vector of players is denoted by x ∈ Rn i.e. xi is the
value player i captures when the game �nishes. x(S) is the pay-o� of a coalition S,
i.e. x(S) =∑i∈S
xi.
Now we give the de�nitions which we refer to in latter sections of our paper.
De�nition 2.1.1. Transferable utility (TU) game
An n-player transferable utility (TU) game is a function v ∈ GN , where v : 2N → Rand GN denotes the class of N-player games. By convention, v(∅) = 0.
We call function v ∈ GN the characteristic function of the game.
In the literature, another de�nition is accepted, i.e. a collection (N, v). We leave
the set of players of the de�nition of the cooperative game while we constrain v to
be in the class of N -player games.
1This teaching supplement is originally the work of Forgó et al. (2006). We refer to a latter
version of this work that has been largely extended by Solymosi (2007).
3
For convenience, when writing the value created by a coalition S, we leave the
curly braces and the commas from the notation of a set. E.g. we denote the value
created by the coalition of player 1 alone by v(1), players 1 and 2 by v(12), players
1, 2 and 3 by v(123), and so on.
We present two axioms regarding TU games which play an important role in the
followings.
Axiom 2.1.2. Monotonicity
A game v ∈ GN is called monotonic if for all coalitions S, T ⊆ N , S ⊆ T ⇒ v(S) ≤v(T ).
Axiom 2.1.3. Superadditivity
A game v ∈ GN is called superadditive if for all coalitions S, T ⊆ N , v(S) + v(T ) ≤v(S ∪ T ), whenever S ∩ T = ∅.
Monotonic games model situations where, when a coalition merges an additional
player, its contribution to the value created by the coalition does not decrease. In
superadditive games, two coalitions cannot come o� uniting badly whenever they
do not have any common players. Also, we state the following claim.
Claim 2.1.4. All non-negative (v(S) ≥ 0 for all S ⊆ N) superadditive games
v ∈ GN are monotonic.
We premise that market situations can typically be modelled by monotonic and
superadditive (thus non-negative) games as market actors do not act against their
self-interest.
Our main motivation in modelling such situations with games is to solve them,
hence capturing the essence of the modelled situation. To do so, we have to de�ne
what we mean by a solution.
De�nition 2.1.5. Solution
A solution of a game v ∈ GN on a class A ⊆ GN of games is a function ψ : A⇒ RN ,
where A⇒ B denotes a set-valued map from set A to set B.
In our paper, we are speci�cally interested in solutions which result in one single
value per player.
De�nition 2.1.6. Single-valued solution
A single-valued solution of a game v ∈ GN on a class A ⊆ GN of games is a function
ψ : A→ RN , i.e. |ψ(v)| = 1.
4
The following three de�nitions regard to a single-valued solution, the nucleolus.
This concept was introduced by Schmeidler (1969) and is widely accepted in the
literature. We de�ne two regarding concepts before the nucleolus itself.
De�nition 2.1.7. Imputation set
The imputation set of a game v ∈ GN is the set
I(v) = {x ∈ RN : x(N) = v(N), xi ≥ v(i), i ∈ N}.
De�nition 2.1.8. Excess function
The excess function of a coalition S ⊆ N at an imputation x ∈ I(v) in a game v ∈GN is the function e : GN ×RN → R2N so that for any v ∈ GN , S ⊆ N and x ∈ RN ,
e(v, x)S = v(S)− x(S).
We de�ne a vector θ by its components, i.e. these components are the excesses
of all coalitions in N+ in non-increasing order. Thus, we get a vector of excesses
θ ∈ R2n−2. We do not include e(v, x)∅ nor e(v, x)N , since e(v, x)∅ = e(v, x)N = 0 for
any imputation x ∈ I(v).
De�nition 2.1.9. Nucleolus (Schmeidler, 1969)
The nucleolus of a game v ∈ GN is a single-valued solution which lexicographi-
cally minimizes2 the vector of the non-increasing ordered excesses. Formally, the
nucleolus-map is set-valued, i.e.
N(v) = {x ∈ I(v) : θ(x) ≤lex θ(y)},
but it consists of a single imputation (Schmeidler, 1969).
We present an other solution concept as well, the core. Gillies (1959) introduced
this solution and it has been the most widely accepted set-valued solution ever since.
De�nition 2.1.10. Core (Gillies, 1959)
The core of a game v ∈ GN is the set
C(v) = {x ∈ RN : v(N) = x(N), e(v, x)S ≤ 0, S ⊆ N}.
The core consists such imputations that are acceptable for all coalitions. In other
words, no coalition is dissatis�ed when its members receive their core pay-o�s.
Finally, we de�ne the principle based on which players can capture value during
the bargaining modelled by a game.
2We say a vector a is lexicographically less than or equal to vector b, i.e. a ≤lex b ⇔ a = b or
ai < bi for the �rst component i in which a and b di�er (a, b ∈ Rn).
5
De�nition 2.1.11. Marginal contribution
The marginal contribution of player i in a game v ∈ GN is Mi(v) = v(N)−v(N \{i}).
In other words, player i's withdrawal from the grand coalition results in the
decrease of the value created. Therefore player i can threat the grand coalition more
when its marginal contribution is higher.
2.2 Notations and de�nitions regarding non-coope-
rative games
As we intend to synthesize elements of cooperative and non-cooperative game theory,
we give basic de�nitions of non-cooperative games we use later. In non-cooperative
game theory, players are assumed to have strategic moves and to decide which move
they choose independently from the others, but assuming their moves. Like in the
case of cooperative games, we suppose there are n players of a non-cooperative game;
we denote the set of players by N also. We denote a strategy of player i by si, and
its strategy set by Si, i.e. si ∈ Si. A strategy pro�le is a collection of n strategies,
one per player; we denote it by (s1, . . . , sn). We denote the set of strategies by
(S1 × · · · × Sn).
We note that in the literature of non-cooperative game theory, the following
notations of strategy pro�les are used: s = (s1, . . . , sn), S = (S1 × · · · × Sn). We
remark two di�erences between the standard notation system and ours. First, we
put the index of the players in subscript while superscript is standard. The reason
why we do so is that Brandenburger and Stuart (2007) use this notation, thus we
consider obligatory to follow their lead. Second, we avoid the contracted notations
of the strategy pro�les as we use the notation S for denoting a coalition of players of
a cooperative game. Consequently, we refer to the set of strategy pro�les of a non-
cooperative game as (S1 × · · · × Sn). Admitted, this longer form is rather elegant,
but this is the only way we can avoid any confusion. The elaboration of appropriate
notations of a biform game is subject to future research.
Now we de�ne the concepts we use in latter sections of our paper. We give the
de�nitions based on the works of Simonovits (2007) and Mas-Colell et al. (1995).
De�nition 2.2.1. Non-cooperative game in normal form
An n-player non-cooperative game is a collection ΓNnorm = (S1 , . . . , Sn; u1(s1, . . . , sn),
. . . , un(s1, . . . , sn)) where
a) Si denotes the set of player i's strategies (with si ∈ Si); and
6
b) ui : S1 × · · · × Sn → R denotes player i's pay-o� function.
The subscript norm refers to the normal form representation of the non-cooperative
game and the superscript N indicates the set of players N . Note that player i's pay-
o� is subject to the strategies of players N \ {i}.We do not require the players to choose their strategies with certainty. We can
assign probabilities to the strategies with which the players randomize their choices.
De�nition 2.2.2. Mixed strategy (Mas-Colell et al., 1995, p. 232)
Given player i's (�nite) pure strategy set Si, a mixed strategy for player i, σi : Si →[0, 1] assigns to each pure strategy si ∈ Si a probability σi(si) ≥ 0 that it will be
played, where∑si∈Si
σi(si) = 1.
We give the de�nition of the most widely known concept in game theory, the
Nash equilibrium. Named after Nash (1950), the concept is a standard solution of a
non-cooperative game.
De�nition 2.2.3. Nash equilibrium (Mas-Colell et al., 1995, p. 246)
A strategy pro�le (s1, . . . , sn) constitutes a Nash equilibrium of a game ΓNN = (S1, . . . , Sn ;
u1(s1, . . . , sn), . . . , un(s1, . . . , sn)) if for every i = 1, . . . , n
ui(si, sN\{i}) ≥ ui(s˜i, sN\{i})for all s˜i ∈ Si.
When playing a non-cooperative game, the players assume what strategies the
others will play and they choose theirs based on their assumption. Consequently,
each player knows the others' equilibrium strategies, and there is no player who
gains by its unilateral change of strategy. This solution can be applied to games
with pure and mixed strategies as well. In these cases, we call this concept pure
strategy and mixed strategy Nash equilibrium, respectively.
Now we synthesize elements of cooperative and non-cooperative game theory and
de�ne the biform game.
2.3 The original biform game
Brandenburger and Stuart (2007) introduce the class of biform games. These games
are a synthesis of non-cooperative and cooperative games. They are two-stage games,
i.e. in the �rst stage the players choose their strategies in a non-cooperative way,
then in the second stage they cooperate in the environment that has been formed by
7
the �rst-stage decisions. Hence there are multiple cooperative environments which
the players have to envisage when they make their strategic choices. We foreshadow
that we show an example to illustrate the fail of the original concept to model this
envision and we give an alternative de�nition of the biform game to eliminate this
misapprehension.
We display the original concept of Brandenburger and Stuart (2007).
De�nition 2.3.1. Biform game (Brandenburger and Stuart, 2007)
An n-player biform game is a collection (S1, . . . , Sn; v;α1, . . . , αn), where
a) Si is a �nite set of strategies, for all players i = 1, . . . , n;
b) v : S1 × . . .× Sn →M , where M : 2N → R, v(s1, . . . , sn)(∅) = 0; and
c) 0 ≤ αi ≤ 1, for all i = 1, . . . , n.
N = {1, . . . , n} is the set of players. Each player i chooses its strategy si from
its strategy set Si. The resulting pro�le of strategies (s1, . . . , sn) ∈ (S1 × . . . ×Sn) de�nes a transferable utility (TU) cooperative game with characteristic function
v(s1, . . . , sn) : 2N → R. v(s1, . . . , sn)(S) denotes the value created by the subset
S ⊆ N of players. (We require v(s1, . . . , sn)(∅) = 0.) Finally, the number αi denotes
player i's con�dence index. It shows player i's anticipation of the pay-o� it will
receive in the cooperative stage, i.e. the proportion of the di�erence between the
maximum and minimum core allocation achievable for player i.
Here we �nish the introduction to the general theory of biform games. In the last
section of this chapter, we show the class of market games which we employ in the
application of biform games to model innovation in markets.
2.4 Market games
Market games are formalized by Shapley and Shubik (1969). These games derive
from typical exchange market situations where all market actors have continuous
concave utility functions. Shapley and Shubik (1969) de�ne a market situation as a
4-tuple (T,G,A, U), where T is the �nite set of traders, G is a non-negative orthant
of the �nite-dimensional vector space of goods, A = {ai : i ∈ T}, A ⊂ G is an indexed
collection of points of the initial allocation of goods and U = {ui : G→ R, i ∈ T} isan indexed collection of continuous, concave utility functions. A feasible S-allocation
of a market (T,G,A, U) is an indexed collection of allocationsXS = {xi : i ∈ S} ⊂ G
such that∑i∈S
xi =∑i∈S
ai.
8
De�nition 2.4.1. Market game (Shapley and Shubik, 1969)
A game v ∈ GNM is a market game if it derives from a market situation (T,G,A, U)
i.e. N = T, v(s) = maxXS
∑i∈S
ui(xi), for all S ⊆ N , where GNM ⊆ GN denotes the class
of market games.
In words, the characteristic function of a market game can be depicted directly
from the maximization of utilities.
We claim that market games are monotonic and superadditive, i.e. bear Axioms
2.1.2 and 2.1.3.
Claim 2.4.2. All market games v ∈ GNM are monotonic and superadditive.
Market games have overwhelmingly huge literature. This is reasonable as Shap-
ley and Shubik (1969) prove a solid theorem. They show that every market game
is totally balanced, i.e. all of its subgames posses non-empty cores. Consequently,
all totally balanced games can be depicted as games deriving from market situa-
tions, at least in an abstract sense of a market. The achievement of Shapley and
Shubik (1969) results in the vivid interest of game theoreticians in market games
and in further solid theorems. Without attempting a comprehensive survey we men-
tion some. Kalai and Zemel (1982b) show that glove-market games and maximum
�ow games are equivalent and both are totally balanced. Owen (1975) observe that
linear production games imply �ow games, as concluded by Apartsin and Holzman
(2003) and amended with the observation that glove-market games are also linear
production games. Furthermore, Csóka et al. (2009) show that the class of risk al-
location games coincide with the class of totally balanced games and conclude that
totally balanced games are generated by permutation games with less than four
players (Tijs et al., 1984), generalized network problems (Kalai and Zemel, 1982a)
and controlled mathematical programming problems (Dubey and Shapley, 1984).
Now that we have introduced corresponding notations and de�ned concepts
which we refer to in the followings, we display and illustrate two models. First,
we model innovation in monopolistic markets, then we show that the de�nition of
the biform game as at (Brandenburger and Stuart, 2007) is not applicable to multi-
ple players. We provide an alternative de�nition of the biform game and �nally we
use our de�nition to model innovation in oligopolistic markets.
9
Chapter 3
Innovation in monopolistic markets
In this chapter, we discuss innovation in monopolistic markets. We assume the actors
of the market to play a biform game. In the �rst stage, one distinguished player,
the seller chooses whether to innovate its product or not. Then in the second stage,
it cooperates with the other players: the buyers. We highlight that the quality of
the product may change via innovation, not its quantity. We also emphasize that
the innovation a�ects all products of the seller; we do not allow for multiple kind of
products on the market.
First of all, we de�ne a class of games which the second-stage cooperative game
of our model belongs to. This is the class of big boss games as introduced by Muto
et al. (1988). Then we provide our model of innovation in monopolistic markets and
provide equilibrium solution of the model. Finally, we illustrate our model with a
simple example.
3.1 Big boss games
We state that there is a well-known class of games regarding monopolistic situations
in the literature. Muto et al. (1988) introduce a class of games in which there is one
distinguished player without whom no value can be created. They call this player
the big boss, and consequently they call this type of games big boss games. Referring
to player 1 as the big boss, they de�ne the big boss game as follows.
De�nition 3.1.1. Big boss game (Muto et al., 1988)
A game v ∈ GNBB is a big boss game with player 1 as big boss if it satis�es the
following two conditions:
1. v(S) = 0 if {1} /∈ S (big boss property); and
2. v(N)− v(S) ≥∑i∈N\S
(v(N)− v(N \ {i})
)if {1} ∈ S (union property),
10
where GNBB ⊆ GN denotes the class of big boss games.
Following subsequent literature, e.g. Brânzei et al. (2006), we call Condition 1
the big boss property and Condition 2 the union property. The big boss property
remarks the power of the big boss. The union property shows that although there is
no value created without the big boss, the remaining weak players have some power
in their hands as well, i.e. they can threat the big boss with withdrawing from the
coalition.
In their article, Muto et al. (1988) give the core of big boss games among other
solution concepts. They show that the core of a big boss game has a special structure,
i.e. C(v) = {x ∈ RN : v(N) = x(N), 0 ≤ xi ≤ Mi(v) for all i ∈ N \ {1}}. It means
the weak players may capture their marginal contribution at most, as it comes from
the union property. The pay-o� vector that gives all weak players their marginal
contribution is called the union point (U(v)), and the pay-o� vector that assigns the
value of the grand coalition to the big boss is called the big boss point (B(v)) in the
literature, i.e.
U(v) =
v(N)−∑
i∈N\{1}
Mi(v),M2(v), . . . ,Mn(v)
and
B(v) = (v(N), 0, . . . , 0) .
These points and the ones between them form the core of a big boss game.
Therefore the core of an n-player big boss game is in general an (n− 1)-dimensional
parallelotope, e.g. the core of a 3-player big boss game is a parallelogram as shown
in the following �gure.
2 3
1
U(v)
B(v)
Figure 3.1: The core of a 3-player big boss game.
11
3.2 The model
Now that we have shown the big boss game, we introduce our model. We depict
innovation in monopolistic markets as a biform game of which the second-stage
cooperative games are big boss games.
Suppose a market of one seller (indexed 1) and n buyers. The seller supplies the
market with one kind of product. Each buyer i wants to buy one product and has a
willingness-to-pay wi ≥ 0 for it.
The seller has the option to innovate its product, increasing the buyers' willing-
nesses-to-pay to w′i. (0 ≤ wi ≤ w′i for all i = 2, . . . , n+1.) The innovation has a �xed
cost c ≥ 0. The question is whether it is worth for the �rm to innovate its product
or not.
We can state that this game is a biform game (S1, . . . , Sn+1; v;α1, . . . , αn+1). We
see instantly that the strategy set of the seller consists of two strategies, maintains
status quo (SQ) and innovates (I) namely, i.e. S1 = (SQ, I). The strategy sets of
the buyers are empty as they have no strategic decisions in the �rst non-cooperative
stage, i.e. Si = ∅ for all players i = 2, . . . , n+ 1.
The corresponding second-stage cooperative games are big boss games, where
the big boss is the seller. Clearly, no value is created when the seller is not in the
coalition, as it comes from the big boss property. Also, since we model a market, the
singleton coalition of the seller evidently cannot create any value. When there is at
least one buyer in the coalition beside the seller, value is created. The value of any
coalition of this kind can be decomposed to two factors. The buyers' willingnesses-
to-pay (wi ≥ 0 for all i ∈ N \ {1}) add up and, in addition, a surplus (εS\{1} for all
S ⊆ N where {1} ∈ S and |S| ≥ 2) adds up to this value. Naturally, the surplus is
zero when there is only one buyer in the coalition beside the seller, i.e. ε1i = 0 for
all i ∈ N \ {1}. This can be interpreted as forming coalitions is not neutral for the
players: they can pro�t upon these coalition forming decisions or they can acquire
the same value as before, but they cannot lose any value.
We de�ne the characteristic function of that second-stage cooperative game
which consequences from the status quo as follows:
v(SQ)(S) =
0 if {1} /∈ S or S = {1},∑i∈S\{1}
wi + εS\{1} if {1} ∈ S and |S| ≥ 2.
If the seller innovates its product, the corresponding cooperative game is as
follows:
12
v(I)(S) =
0 if {1} /∈ S or S = {1},∑i∈S\{1}
w′i + ε′S\{1} if {1} ∈ S and |S| ≥ 2.
Both characteristic functions can be found in Appendix C in tabular form.
Referring to Claim 2.4.2, we state that the surplus a coalition achieves when they
cooperate cannot decrease when merging additional players. To be more precise,
since v(S) + v(T ) ≤ v(S ∪ T ) in superadditive games, if {1} ∈ S and |S| ≥ 2 then∑i∈S\{1}
wi + εS\{1} + wj ≤∑
k∈T\{1}
wk + εT\{1}, i.e.
εS\{1} ≤ εT\{1}
for all i ∈ S, j /∈ S, S + {j} = T, k ∈ T, S ⊂ N, T ⊆ N .
In order to solve the biform game, we calculate the core of the corresponding
second-stage cooperative games �rst. To be more exact, we need the minimum and
maximum core pay-o�s of the decisive player, the seller in this case. These core
points are the union and big boss points, so we can easily calculate their con�dence
index-weighted averages. The maximum pay-o� of the seller is at the big boss point
and is v(N) = w2 + . . . + wn+1 + εN\{1}. Its minimum pay-o� is a bit more di�-
cult to calculate, though. For this, we �rst remark that in this game the marginal
contribution of player i is Mi(v) = wi + εN\{1}− εN\{1}\{i}.1 Then, we can write that
U1(v) = v(N) −∑
i∈N\{1}
Mi(v) = w2 + . . .+ wn+1 + εN\{1} −
−∑
i∈N\{1}
(wi + εN\{1} − εN\{1}\{i}
)=
∑i∈N\{1}
(εN\{1}\{i}
)− (|N \ {1}| − 1) · εN\{1}.
Note that in the union point, the buyers' individual willingnesses-to-pay do not
play any role.
Now we can solve the game. When the seller maintains status quo, it anticipates
to capture
α(w2 + . . .+wn + εN\{1}) + (1−α)
∑i∈N\{1}
(εN\{1}\{i}
)− (|N \ {1}| − 1) · εN\{1}
.
(3.1)
1I.e. Mi(v) = w2 + . . .+ wn+1 + εN\{1} − (w2 + . . .+ wi−1 + wi+1 + . . .+ wn+1 + εN\{1}\{i}).
13
When it decides to innovate, it anticipates to capture
α(w′2+. . .+w′n+ε′N\{1})+(1−α)
∑i∈N\{1}
(ε′N\{1}\{i}
)− (|N \ {1}| − 1) · ε′N\{1}
−c.(3.2)
Thus, in the �rst non-cooperative stage the seller faces the following decision:
SellerSQ Expression 3.1
I Expression 3.2
Figure 3.2: The innovation game on a monopolistic market in normal form.
The decision of the seller depends only on the values of parameters α;wi, εN\{1},
εN\{1}\{i};w′i, ε′N\{1}, ε
′N\{1}\{i}; c for all i ∈ N \ {1}.
We illustrate the innovation game with a simple example.
Example 3.2.1. Three research institutes are considering to purchase supercomput-
ers, one piece each. Such supercomputers are able to obtain bigger processing powers
when connected to each other than their cumulated individual processing powers,
due to the synergy between them. Only one IT solution company is on the mar-
ket to satisfy their demand and o�ers them two di�erent options. If the institutes
choose the �rst option, the company faces no costs as this is standard routine in
their course of business. The resulting game is presented in Table 3.1. The second
option smooths the individual surpluses of coalitions due to synergy. However, if the
institutes opt this o�er, the company has to test the system and face a �xed cost
c. This game is presented in Table 3.1 as well. What is that �xed cost c subject to
the con�dence index α of the company at which it is worth it to o�er the second
option?
v(12) v(13) v(14) v(123) v(124) v(134) v(1234) otherwise v(S)
Option 1 100 85 78 200 190 170 280 0
Option 2 100 90 90 210 210 200 300 0
Table 3.1: The second-stage cooperative games of Example 3.2.1.
14
We depict the coalitional surpluses of both cooperative games for coalitions that
consist of the company and at least two research institutes in the following table.
ε23 ε24 ε34 ε234
Option 1200− 100− 190− 100− 170− 85− 280− 100−−85 = 15 −78 = 12 −78 = 7 −85− 78 = 17
Option 2210− 100− 210− 100− 200− 90− 300− 100−−90 = 20 −90 = 20 −90 = 20 −90− 90 = 20
Table 3.2: The coalitional surpluses of the second-stage cooperative games of Exam-
ple 3.2.1.
To answer the question by solving the biform game, we have to calculate the
union and big boss points of both games �rst. Start with the cooperative game
consequencing Option 1. In this case, B1(v) = v(1234) = 280 and U1(v) = ε23 +
ε24 + ε34 − 2 · ε234 = 15 + 12 + 7− 2 · 17 = 0. Thus the company anticipates it will
receive α · B1(v) + (1− α) · U1(v) = α · 280.
Now get to the cooperative game corresponding from Option 2. Here B1(v) = 300
and U1(v) = 20 + 20 + 20 − 2 · 20 = 20. Therefore the company's anticipation of
performing in the cooperative stage is α ·300 +(1−α) ·20 = α ·280 +20. This result
means that whatever the magnitude of α is, the company will always make an o�er
whenever the cost of the innovation does not exceed 20.
We note that we could have got deterministic results with respect to the strategic
move of the seller. If the beforehand-expected core pay-o�s of the seller di�er, it
would choose that strategy which assigns it greater value.
In the next chapter, we show that the de�nition of biform games in its current
form is not applicable for multiple players. We provide an alternative de�nition to
apply in the model of innovation in oligopolistic markets.
15
Chapter 4
An alternative de�nition of the
biform game
In this chapter, we show that the de�nition of the biform game in the work of Bran-
denburger and Stuart (2007) is heavily inconsistent. The application of con�dence
indices of the players which provide the link between the non-cooperative and co-
operative stages results in a "synthetic" pay-o� that usually does not belong to
the core. We show an example to illustrate this phenomenon, then we rede�ne the
biform game.
Recall the original de�nition of the biform game, as displayed in De�nition 2.3.1.
Brandenburger and Stuart (2007) illustrate the biform game by the example of
innovation. Their example is as follows.
4.1 An example of the original biform game
Example 4.1.1. Consider the market of two �rms and three buyers. Each �rm pro-
duces the same product and has the capacity of two units. They do not face any
variable costs. Both �rms have the option to innovate their product; the innovation
requires a �xed cost of 5$. Each buyer is interested in one unit of the product. Their
willingnesses-to-pay for the original product are uniformly 4$ per unit, and 7$ per
unit for the innovated product. The higher willingness-to-pay for the innovated prod-
uct represents that the buyers prefer this product against the original one. Note that
we do not specify the con�dence indices; we show that the magnitude of con�dence
indices is arbitrary in this game.
Denoting the strategies maintains status quo and innovates of both �rms by SQ
and I respectively, we depict the game in the following table.
16
Firm 1
Firm 2
SQ I
SQ 4 pcs: (4, 4, 4) 2 pcs: (7, 7, 7), 2 pcs: (4, 4, 4)
I 2 pcs: (4, 4, 4), 2 pcs: (7, 7, 7) 4 pcs: (7, 7, 7)
Table 4.1: The original example of the biform game.
Each cell of the table contains two pieces of information. They show what the
buyers' ratings are for each kind of product, and how many products of such ratings
are available on the market.
It is easy to acknowledge that this game is a biform game. The �rms have strate-
gic moves in the �rst stage, then all players cooperate in the second stage that is
formed by the initial decision. Thus we can solve this game and provide equilibrium
via backward induction. For this, we have to calculate the cores of all possible co-
operative games. The characteristic functions of the cooperative games are shown
in Appendix A.
The cores of all corresponding cooperative games consist of one point. When
none of the �rms innovate, the corresponding core is {(0, 0, 4, 4, 4)}. When both
�rms innovate, the core is {(0, 0, 7, 7, 7)}. The interesting corresponding games are
when only one �rm innovates. In these cases, the core pay-o� of the innovator �rm
is 6$, while the non-innovator �rm captures no value, i.e. the cores consist of the
imputations {(6, 0, 4, 4, 4)} or {(0, 6, 4, 4, 4)}, when Firm 1 or Firm 2 innovates,
respectively. Since all the cores of the corresponding cooperative games consist of
one point, there is no need to apply con�dence indices.
These results can be explained by intuition as well. It can easily be proved
that in those cases when both �rms choose the same strategy, all the social excess is
captured by the buyers as there is over-supply on the market. In those cases although,
when the �rms choose di�erent strategies, the non-innovator �rm cannot capture any
value. This is because of the constant threat that the buyer who cooperates with the
non-innovator �rm can break the coalition and form another with the innovator �rm.
Alongside this, the innovator �rm captures the di�erence between the willingness-
to-pays of buyers for the non-innovated and innovated product, i.e. 2·(7$−4$) = 6$.
Since there is over-demand for the innovated product, the innovator �rm can do so.
Consequently, we can write the �rst-stage non-cooperative game in normal form.
We remind the Reader that the innovation has a �xed cost of 5$. The reason why we
do not present this cost in the corresponding cooperative games is the well-known
17
property of cooperative games and their solutions, namely the strategic equivalence.
If we decrease the value of all coalitions containing player i by a constant, pay-o�s of
players besides i provided by certain solutions do not decrease, but player i's pay-o�
decreases by the constant. We call these games strategically equivalent and de�ne
as follows.
De�nition 4.1.2. Strategic equivalence
Games v, w ∈ GN are strategically equivalent if there exist such α > 0, β1, . . . , βn ∈ Rthat w(S) = αv(S) +
∑i∈S
βi, for all S.
We say the cores of games v and w are covariant with respect to their strategically
equivalent transformation, i.e. C(w) = αC(v) + β = {αx + β : x ∈ C(v)}, whereα ∈ R, β, x ∈ Rn. Since here we use two extrema of the core, it is su�cient to
decrease the pay-o�s of the innovator �rms by the �xed cost of innovation instead
of presenting it in the second-stage cooperative games.
Firm 1
Firm 2
SQ I
SQ 0, 0 0, 1
I 1, 0 −5,−5
Figure 4.1: The original game in normal form.
This �rst-stage non-cooperative game resembles a well-known bimatrix game,
the battle of the sexes. The di�erence is that the sum of pay-o�s is higher in case
of dissimilar decisions. Even so, similarly to the battle of the sexes, this game has
two pure strategy Nash equilibria: when one �rm innovates and the other doesn't.
The equilibria pay-o�s are boxed in Figure 4.1. This means none of the �rms are
interested in choosing a di�erent strategy by themselves.
We can also calculate the probabilities with which the mixed Nash equilibria
occur. For this, we write the expected pay-o� of Firm 1. We denote the probability
of choosing the strategy maintains status quo for Firm 1 by ξ and for Firm 2 by η
in the mixed Nash equilibrium.
maxξxe1(ξ, η) = ξη · 0 + ξ(1− η) · 0 + (1− ξ)η · 1− (1− ξ)(1− η) · 5
∂xe1∂ξ
= −η + (1− η) · 5 .= 0
18
η =5
6
As the game is symmetric, ξ = 56as well.
This result can be interpreted as follows: mixed Nash equilibria occur when both
players opt to innovate with the probability of 16and to remain at the original
technology with the probability of 56. In this case, the mixed Nash equilibria are the
strategy pairs (maintains status quo, innovates) and (innovates, maintains status
quo).
Now we modify the original example of Brandenburger and Stuart (2007) in
order to highlight the inconsistency of their model.
4.2 The modi�ed example
We provide an updated innovation game. The game is similar to the one in Example
4.1.1, the only modi�cation is the willingnesses-to-pay of the buyers.
Example 4.2.1. (This game is taken from (Gyetvai and Török, 2012).) Take Example
4.1.1. Suppose the buyers have a willingness-to-pay of 4$ for the original product,
but their ratings di�er for the innovated product, i.e. the innovated product is worth
9$ for Buyer 1, 8$ for Buyer 2 and 7$ for Buyer 3.
As before, we depict the game in the following table.
Firm 1
Firm 2
SQ I
SQ 4 pcs: (4, 4, 4) 2 pcs: (9, 8, 7), 2 pcs: (4, 4, 4)
I 2 pcs: (4, 4, 4), 2 pcs: (9, 8, 7) 4 pcs: (9, 8, 7)
Table 4.2: The modi�ed example of the biform game.
Following Brandenburger and Stuart (2007), we calculate the cores of the corre-
sponding second-stage cooperative games. The calculations are shown in Appendix
B. The cores of the corresponding cooperative games are similar in those cases when
both �rms choose the same strategy, i.e. {(0, 0, 4, 4, 4)} and {(0, 0, 9, 8, 7)}. How-
ever, when only one of the �rms innovate, the cores of the corresponding cooperative
games are not single-valued but set-valued. The projection of the core to the plane
pay-o�s of Firm 1 and Firm 2 are shown in the following �gure.
19
x2
x16 7 8 9
1
Figure 4.2: Projection of the core of the game v(I, SQ)(S) in the modi�ed example
to the plane of pay-o�s of the decisive players.
The core of this game is represented by the dark-grey trapezoidal-shaped area
in the �gure.
The results can be explained by intuition as well. Similarly to Example 4.1.1,
when both �rms choose the same strategy, there is over-supply of the same product
on the market, thus all the value is captured by the buyers. However, when only
one �rm innovates, none of the sellers can supply their own market. As the core
consists of more than one point in this case, we have to reduce this set to one
point. Following Brandenburger and Stuart (2007), we apply the con�dence indices.
E.g. suppose α1 = 20% and α2 = 60%, thus x1 = 0.2 · 9$ + 0.8 · 6$ = 6.6$ and
x2 = 0.6 · 1$ + 0.4 · 0$ = 0.6$. It can be explained as follows: Firm 1 anticipates it
can capture 20% of the di�erence between its maximum and minimum core pay-o�s,
while Firm 2 anticipates 60%. In other words, Firm 1 is not that con�dent about its
future performance during the bargainings, while Firm 2 is a bit more optimistic.
Note that when the willingness-to-pay of the buyers for the innovated product
di�er, the non-innovator �rm can capture some value as well. This occurs because
the threat of the buyer who cooperates with the non-innovator �rm to break the
coalition is not credible anymore. That is to say, since it is commonly known that
the innovator �rm will cooperate with the buyers with the two highest willingness-
to-pays for its product, the remaining buyer will be left out from the coalition and
is forced to cooperate with the non-innovator �rm.
In this case, we can provide the �rst-stage non-cooperative game in normal form.
20
Firm 1
Firm 2
SQ I
SQ 0, 0 0.6, 1.6
I 1.6, 0.6 −5,−5
Figure 4.3: The modi�ed innovation game in normal form.
Similarly to Example 4.1.1, the Nash equilibria are boxed in the normal form of
the game.
Now we show the inconsistency of the application of con�dence indices in the orig-
inal model of Brandenburger and Stuart (2007). We remark that there are such con�-
dence index-pairs by which the original anticipation of the players are not achievable.
The area in which such pairs occur are bordered with blue lines on Figure 4.4. E.g.
in Example 4.2.1, the pay-o� pair under the constellation of con�dence indices falls
outside the core, as shown in Figure 4.4.
x2
x16 7 8 9
1(6.6, 1.6)
Figure 4.4: Inconsistency of con�dence indices in Example 4.2.1.
The area bordered by blue lines remarks pay-o�s outside the core that originate
from certain con�dence indices. This is because con�dence indices may not be con-
sistent with each other, and may result in such non-equilibrium pay-o�s. Thus, it is
not rational for players to decide upon such expected pay-o�s in the non-cooperative
stage of the biform game. Although con�dence indices are commonly known in the
model of Brandenburger and Stuart (2007), this knowledge does not deter players
from anticipating such pay-o�s.
21
4.3 The alternative biform game
To resolve this inconsistency, we provide another link between the non-cooperative
and cooperative stages of the biform game then the con�dence indices. We think
this link should somehow originate from the second-stage cooperative games in a
way that it gets hold of the essence of such games. Such link de�nitely should be
a kind of equilibrium pay-o�, like the con�dence index-weighted core pay-o�s at
Brandenburger and Stuart (2007) because players decide upon their anticipation
of their �nal pay-o�s. We list three conditions of this equilibrium pay-o� that we
consider necessary for providing a su�cient link. Such link should
1. Always exist,
2. Belong to the core, whenever that is non-empty,
3. Be a single-valued solution of the resulting cooperative game.
Following these criteria, we give an alternative de�nition of the biform game.
De�nition 4.3.1. Biform game (Gyetvai and Török, 2012)
An n-player biform game is a collection (S1, . . . , Sn; v;ψ), where
a) Si is a �nite set, for all i = 1, . . . , n;
b) v : S1 × . . .× Sn →M , where M : 2N → R, v(s1, . . . , sn)(∅) = 0; and
c) ψ : 2N → RN such that ψ(v) ∈ C(v) whenever C(v) 6= ∅.
Similarly to De�nition 2.3.1, the biform game is given by the set of strategies and
the corresponding cooperative games. The di�erence between De�nitions 2.3.1 and
4.3.1 is that the link between the non-cooperative and cooperative stages is provided
by a single-valued solution that belongs to the core. Thus we have dissolved the
inconsistency of the original model of Brandenburger and Stuart (2007).
We remark a potential drawback of our model. In the original model of Bran-
denburger and Stuart (2007), the application of con�dence indices represents the
belief of the players about their own performance during the bargainings. However,
if we derive the link between the stages directly from the cooperative stage, we un-
wittingly withdraw the assumption of unbound beliefs. A solution of this contrast
may be to represent beliefs as well in the model in a way that it harmonizes with
Conditions 4.3. This achievement is subject to further research.
In the following chapter, we use our de�nition of the biform game to illustrate
innovation in oligopolistic markets. First, we make a restrictive but necessary as-
sumption regarding the second-stage cooperative games. Then we present our model
22
of innovation in oligopolistic markets. Finally, we illustrate our model by a simple
example.
23
Chapter 5
Innovation in oligopolistic markets
In this chapter, we analyze innovation in oligopolistic markets with biform games.
Since there are multiple decisive players (the sellers) on such a market, we need
to feature multiple players in the �rst non-cooperative stage of the biform game.
Thus, our illustration is based upon the alternative de�nition of the biform game
(De�nition 4.3.1). First, we display basic notations we use throughout this chapter.
Then we show a class of games which we assume the market actors play in the
second stage, the class of transportation games. Thereafter we present our model of
innovation in oligopolistic markets, and we illustrate it by a simple example.
We distribute the actors on an oligopolistic market into two disjunct set. Denot-
ing the set of sellers by P and the set of buyers by Q, we can state that this two set
forms a partition of the set of players N , i.e. P ∪Q = N and P ∩Q = ∅. Note that|N | = |P |+ |Q|, i.e. n = m+(n−m), where m is the number of sellers and (n−m) is
the number of buyers on the market.1 Taking for instance a game with 2 sellers and
3 buyers, v(1) and v(2) denote the values of the singleton coalitions of the sellers,
v(3), v(4) and v(5) the values of the singleton coalitions of the buyers, and v(134)
the value of the coalition of Seller 1 and Buyers 1 and 2. v(134) is interpreted as the
value which the coalition of Seller 1 and Buyers 1 and 2 captures when there are no
other trades on the market then theirs.
5.1 Transportation games
The class of linear production games have been introduced by Owen (1975). The
characteristic function of these games derives from optimal solutions of linear pro-
1Unlike in Chapter 3, where we suppose 1 seller and n buyers, adding up a total of n+1 players,
here we suppose only n players. We consider the latter notation correct, but we have ignored this
rule in Chapter 3 for convenience.
24
gramming problems.
One of the linear production issues is the transportation situation. The formal-
ization of this situation and the corresponding cooperative game is introduced by
Sánchez-Soriano et al. (2001). A transportation situation is a 5-tuple (P,Q,B, p, q)
where P denotes the set of origin players (sellers in this instance), Q denotes the
set of destination players (here buyers), B is the m × (n − m) matrix of surplus
the players achieve by an appropriate assignment, p is the m-dimensional vector of
available units of the sellers and q is the (n−m)-dimensional vector of demands of
the buyers.
We highlight that the name of the class of games may be misleading. That is,
we mean transportation in an abstract sense; we depict transportation as a method
in which certain goods originating from the sellers are assigned to the buyers.
We can de�ne the consequent linear program for every transportation situation
(P,Q,B, p, q) and every coalition S ⊆ N.= P ∪Q with origin players SP
.= S ∩ P ,
SP 6= ∅ and destination players SQ.= S ∩Q, SQ 6= ∅, i.e.
T (S): max∑i∈SP
∑j∈SQ
bijxij
s.t.∑j∈SQ
xij ≤ pi, i ∈ SP∑i∈SP
xij ≤ qj, j ∈ SQ
xij ≥ 0, (i, j) ∈ SP × SQ
Denoting the optimal value of the linear program T (S) by OS(T ), we can de�ne
the cooperative game derived from a transportation situation. We use the nota-
tion W instead of B for denoting the matrix of surplus as we discuss markets. W
represents the willingnesses-to-pay of the buyers for the products of the sellers.
De�nition 5.1.1. Transportation game (Sánchez-Soriano et al., 2001)
A v ∈ GNT game is a transportation game if it derives from a transportation situtation
(P,Q,W, p, q), i.e.
v(S) =
{0 if SP = ∅ or SQ = ∅,OS(T ) otherwise,
where GNT ⊆ GN denotes the class of transportation games.
For better understanding, the characteristic function of a 5-player transportation
game is shown in tabular form in Appendix D. We note that the transportation game
is a generalization of the assignment game (Shapley and Shubik, 1971), i.e. the sellers
supply only one product and the buyers demand only one product as well.
25
5.2 The model
Now that we have displayed the transportation game, we introduce a model of in-
novation in an oligopolistic market. We model this situation with a biform game of
which the second-stage cooperative games are transportation games. We highlight
that this assumption is di�erent from the one we have applied to the monopolis-
tic market model. Namely, here we assume that the coalitions cannot capture any
surplus over their individual value. However, we remark that we still assume that
the second-stage cooperative games are market games, and consequently they are
monotonic and superadditive (Claim 2.1.4). Thus, the coalitional surplus equals to
zero for all coalitions.
Suppose there are m sellers and n−m buyers on the market of a certain product.
Each seller i has a supply of pi pieces of the product, each buyer j has a demand of qj
pieces of the product. The individual supplies and demands are concluded in vectors
p ∈ Rm and q ∈ Rn−m. The buyers' willingnesses-to-pay for the products of the
sellers are concluded in matrix W ∈ Rm×(n−m), i.e. wij represents the willingness-
to-pay of buyer j for one product of seller i. We require wij ≥ 0. Thus the market
can be depicted by a transportation situation (P,Q,W, p, q).
Now we suppose that before they step up the market, all sellers have the option
to innovate their product, i.e. all m sellers have two strategic moves: to maintain
status quo (SQ) or to innovate (I). This results in the sets of strategies of the sellers,
i.e. the strategy set of seller i is Si such that |Si| = 2·2m−1 = 2m. The innovation has
a �xed cost c ≥ 0 for all sellers.2 The main driver of innovation is the increase of the
sellers' pro�t as the buyers' willingnesses-to-pay increase for the innovated product,
i.e. w′ij ≥ wij ≥ 0 if seller i innovates its product, for all i ∈ P, j ∈ Q. Every pro�le ofstrategies results in a di�erent transportation game. We assume that the innovation
of seller i does not change the buyers' willingnesses-to-pay for the products of sellers
P \ {i}. Consequently, we note that the di�erent matrices W of willingnesses-to-pay
of di�erent second-stage transportation games di�er row-wise only.
Finally, we suppose that the sellers anticipate to capture their nucleoli pay-o�s of
all second-stage transportation games. Schmeidler (1969) proves that the nucleolus
(which we have shown in De�nition 2.1.9) satis�es all Conditions 4.3. The usage of
the nucleolus can be interpreted as all sellers aspire to reach the most egalitarian
allocation of the products and assume that all other sellers do the same. They act as
"super" negotiators during the bargainings and are perfectly ready to compromise.
2Although the buyers' willingnesses-to-pay di�er for the product depending on from which seller
they buy it, the sellers face the same c �xed cost.
26
Even if we suspect this assumption is exaggerate, we may assume that the sellers
decide upon their nucleoli pay-o�s in lack of certain pay-o�s.
Unfortunately, we cannot go further than giving the model of innovation in
oligopolistic markets. We lack the exact expression or the algorithm that gives the
nucleolus of transportation games. Solymosi and Raghavan (1994) show an algorithm
to �nd the nucleolus of assignment games, but we show by the following example
that their algorithm is not applicable to transportation games.
Example 5.2.1. There are three farms in the countryside near each other. Gas
pipeline facilities are not installed, thus they have to use gas cylinders. There are
two nearby gas stations where they can purchase the cylinders. Both stations have
the capacity to supply two farms, but not three. Since the farms have no access to
pipelines, the local government forces the gas stations to supply the farms with gas
cylinders below market price. The three farmers and the sales managers of both gas
stations are about to start to bargain on prices for a long-term contract.
The stations are considering whether to o�er the farms to deliver them the cylinders.
To deliver the cylinders safely, they have to purchase a specially designed truck that
costs them 3 units. The farmers have higher willingnesses-to-pay for this service
than the original. The resulting matrices of willingnesses-to-pay are shown in Table
5.1 with respect to the o�ers of the gas stations.
Gas Station 2
SQ D
Gas Station 1
SQ20 15 12 20 15 12
7 16 13 12 19 17
D22 17 15 22 17 15
7 16 13 12 19 17
Table 5.1: An example of the model of oligopolistic markets.
Which option will the gas stations o�er for the farmers?
We can state this game is a 5-player biform game (S1, . . . , S5; v;N). Two play-
ers, Gas Station 1 and 2 have two strategic moves in the �rst non-cooperative stage:
they can either maintain status quo (SQ) as the farmers have to purchase the cylin-
ders at the gas stations or deliver them (D). More precisely, they have non-empty
strategy sets, i.e. Si = {(SQ if P \ {i} SQ), (SQ if P \ {i} D), (D if P \ {i}SQ), (D if P \ {i} D)} for all i ∈ P where P = {1, 2}. The farms' strategy sets
27
are empty, i.e. Sj = ∅ for all j ∈ Q where Q = {3, 4, 5}. The pro�les of strate-
gies result in four di�erent transportation situations, i.e. (P,Q,WSQ,SQ, p, q)SQ,SQ,
(P,Q,WSQ,D, p, q)SQ,D, (P,Q,WD,SQ, p, q)D,SQ, (P,Q,WD,D, p, q)D,D where sets P
and Q are as given above, matrices Wk,l for all k ∈ S1, l ∈ S2 are as given in Figure
5.1, p = (2, 2) and q = (1, 1, 1). The characteristic functions of the second-stage
transportation games can be found in Appendix E. N denotes the nucleoli of all
four transportation games.
We solve this biform game via backward induction. First, we calculate the nucleoli
of all four second-stage transportation games.3 We get
N(v(SQ, SQ)) = (6 3/4, 1, 13 1/4, 15 1/2, 12 1/2),
N(v(SQ,D)) = (5, 4 1/2, 15, 17, 14 1/2),
N(v(D,SQ)) = (9 5/6, 1/3, 13 5/6, 15 2/3, 13 1/3) and
N(v(D,D)) = (5 1/2, 2, 16 1/2, 18, 16).
Then we can give the �rst-stage non-cooperative game in normal form. We re-
mind the Reader that the innovation has a �xed cost of 3 that we have to subtract
from the regarding nucleoli pay-o�s.
Gas Station 1
Gas Station 2
SQ D
SQ 6 3/4, 1 5, 1 1/2
D 6 5/6, 1/3 2 1/2,−1
Figure 5.1: The innovation game on an oligopolistic market in normal form.
This game has two pure strategy Nash equilibria, boxed in Figure 5.1. We calcu-
late the probabilities with which the mixed Nash equilibria occur as well. We denote
the probability of choosing the strategy SQ for Gas Station 1 by ξ and for Gas
Station 2 by η in the mixed Nash equilibrium.
3We calculate the nucleoli of TU games with a version of TUGlab we have modi�ed. TUGlab
is the work of Mirás Calvo and Sánchez Rodríguez (2007): this toolbox is a teaching complement
for cooperative game theory. It is able to calculate certain geometric properties and solutions of 3-
and 4-player TU games. We have modi�ed TUGlab in order to calculate the nucleoli of 5-player
TU games as well. The original package and its extension can be downloaded from the Downloads
website of the Department of Operations Research and Actuarial Sciences, Corvinus University of
Budapest.
28
maxξxe1(ξ, η) = ξη · 6 3/4 + ξ(1− η) · 5 + (1− ξ)η · 6 5/6 + (1− ξ)(1− η) · 2 1/2
∂xe1(ξ, η)
∂ξ= η · 6 3/4 + (1− η) · 5− η · 6 5/6− (1− η) · 2 1/2
.= 0
η =30
31
maxηxe2(ξ, η) = ξη · 1 + (1− ξ)η · 1/3 + ξ(1− η) · 1 1/2− (1− ξ)(1− η) · 1
∂xe1(ξ, η)
∂ξ= η · 6 3/4 + (1− η) · 5− η · 6 5/6− (1− η) · 2 1/2
.= 0
η =30
31
Now we show that the second-stage transportation games written in assignment
game form do not result in the same nucleoli. For that, we rede�ne the transportation
games, e.g. we write v(SQ, SQ)(S) in a 7-player assignment game form (four sellers
and three buyers). We consider the two-two products of Gas Station 1 and 2 as four
di�erent products of four di�erent sellers. Then the buyers' willingnesses-to-pay can
be written in matrix form.
Buyer 1 Buyer 2 Buyer 3
Gas Station 1 20 15 12
Gas Station 1 20 15 12
Gas Station 2 7 16 13
Gas Station 2 7 16 13
Figure 5.2: The transportation game in assignment game form.
The nucleolus of this game is N(v(SQ, SQ)) = (0, 0, 1/2, 1/2, 20, 15 1/2, 12 1/2).4
Adding up the nucleolus pay-o�s of Player 1-2 and 3-4 (in order to get the pay-o�s
of the sellers of the original transportation game), we get the imputation I(v) =
(0, 1, 20, 15 1/2, 12 1/2). This clearly does not coincide with the nucleolus of the origi-
nal transportation game N(v) = (6 3/4, 1, 13 1/4, 15 1/2, 12 1/2). When considering the
sellers as multiple sellers, Gas Station 1 bargains against itself, thus its pay-o� zeros
while Buyer 3 receives all 20 units. However, what really happens is that Gas Station
withdraws its second gas cylinder in order to acquire some value, 6 3/4 namely.
To conclude, we do not have an algorithm for �nding the nucleolus of a trans-
portation game. That is subject to further research.
4Credit to dr. Tamás Solymosi for computing the nucleolus of this assignment game.
29
Chapter 6
Summary
Our paper was devoted to the application of biform games in an industrial organi-
zation topic, namely innovation. We exhaustively introduced the theory of biform
games and its components, cooperative and non-cooperative game theory.
In Chapter 2, we introduced the concepts we referred to in latter chapters. First,
we displayed elements of cooperative game theory. We declared the notations, then
we de�ned the transferable utility (TU) game. We showed two signi�cant axioms
regarding TU games and claimed that they relate to each other. We introduced
furthermore two widely accepted solution concepts, namely the nucleolus and the
core, and de�ned marginal contribution. Second, we came to non-cooperative game
theory. We declared these notations as well, then we de�ned the non-cooperative
game in normal form. We de�ned the mixed strategy, and eventually we introduced
the most known solution concept of game theory, the (pure and mixed strategy)
Nash equilibrium. Third, we provided a synthesis of cooperative and non-cooperative
game theory, the biform game. We de�ned it and detailed the original model of
Brandenburger and Stuart (2007). We also traced a future course of research to
elaborate an appropriate system of notations of biform games. Fourth, following the
introduction to the general theory of cooperative, non-cooperative and biform games,
we introduced the class of market games as the second-stage cooperative games in
our models belong to this class. We formalized the market situation in an abstract
sense, then we de�ned the deriving market game. We claimed that market games
bear some remarkable characteristics, and �nally we reviewed the vast literature of
this class of games.
In Chapter 3, we applied biform games to model innovation in monopolistic mar-
kets. We introduced the class of big boss games which the second-stage cooperative
games in our model belongs to. We remarked the special structure of the core of
the big boss game. Thereafter we introduced our model, an (n + 1)-player biform
30
game. In the model, the monopoly chooses in a non-cooperative manner whether to
innovate its product or not in the �rst non-cooperative stage, then it cooperates with
the buyers in the second cooperative stage. We provided a parametric equilibrium
solution of the model, and �nally we illustrated it with a simple example.
In Chapter 4, we showed that the original de�nition of the biform game is not ap-
plicable to multiple decisive players in the �rst non-cooperative stage. We provided
an example to show that the strategic decisions of the players are based on non-core
pay-o�s, which is to say, inconsistent with the principles of game theory. Thereafter
we rede�ned the biform game in a much more general sense, but avoiding this con-
tradiction. We declared three conditions which the link between the non-cooperative
and cooperative stages should satisfy. We highlighted a weakness of our model and
appropriated further research to avoid this weakness.
In Chapter 5, based on our alternative de�nition, we introduced a model of in-
novation in oligopolistic markets. Above all, we de�ned the class of transportation
games which the second-stage cooperative games in our model belong to and we re-
marked its relation to the class of assignment games. Then we provided our model.
We depicted the situation by an n-player biform game with m sellers and n − m
buyers. We showed that no equilibrium solution can be given in the lack of an algo-
rithm for �nding the nucleolus of transportation games. Consequently, we remarked
that an algorithm for �nding the nucleolus of transportation games is subject to
further research. Then we suggested to exploit the relationship between assignment
and transportation games as there exists such an algorithm for the preceding class,
but we brought a counterexample and rejected our conjecture.
To conclude all, we can say that the �eld of biform games are still unexploited.
We note that while we were focusing on applications, we intentionally skipped parts
of the original article of Brandenburger and Stuart (2007). We have not coped with
epistemic assumptions, nor properties and theorems regarding biform games. The
acceptable formalization of biform games is still subject to further research. Also, it
may be edifying to investigate games of which the �rst stage is a cooperative game
and the second stage is non-cooperative. Furthermore, Schwarz (2011) mentions a
path to analyze biform games in a repeated contest. Accordingly, we shortly may
experience the takeo� of the biform game theory and applications.
31
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33
Appendix A
The characteristic functions of the
game in Example 4.1.1
v(SQ, SQ)(S) v(SQ, I)(S) v(I, SQ)(S) v(I, I)
v(13) 4 4 7 7
v(14) 4 4 7 7
v(15) 4 4 7 7
v(23) 4 7 4 7
v(24) 4 7 4 7
v(25) 4 7 4 7
v(123) 4 7 7 7
v(124) 4 7 7 7
v(125) 4 7 7 7
v(134) 8 8 14 14
v(135) 8 8 14 14
v(145) 8 8 14 14
v(234) 8 14 8 14
v(235) 8 14 8 14
v(245) 8 14 8 14
v(1234) 8 14 14 14
v(1235) 8 14 14 14
v(1245) 8 14 14 14
v(1345) 8 8 14 14
v(2345) 8 14 8 14
v(12345) 12 18 18 21
otherwise v(S) 0 0 0 0
34
Appendix B
The core of v(I, SQ)(S) in Example
4.2.1
v(I, SQ)(S)
v(13) 9
v(14) 8
v(15) 7
v(23) 4
v(24) 4
v(25) 4
v(123) 9
v(124) 8
v(125) 7
v(134) 17
v(135) 16
v(145) 15
v(234) 8
v(235) 8
v(245) 8
v(1234) 17
v(1235) 16
v(1245) 15
v(1345) 17
v(2345) 8
v(12345) 21
otherwise v(S) 0
x4 ≤ 8
x3 ≤ 9
x2 = 4− x5
x4 + x5 ≤ 12
x3 + x5 ≤ 13
x3 + x4 ≤ 14
x1 = 17− x3 − x4
x4 ≥ 3
x4 ≥ 4
x3 ≥ 4
x5 ≤ 4
x4 ≤ 5
x3 ≤ 6
The equalities and inequalities can be reduced to the following inequalities:
6 ≤ x1 ≤ 9;
35
0 ≤ x2 ≤ 1;
4 ≤ x3 ≤ 6;
4 ≤ x4 ≤ 5;
3 ≤ x5 ≤ 4.
In order to calculate the core of this game, we have to �nd the connection between
x1 and x2. For this, let x1 = 6 �rst. Then x4+x5 ≥ 9 because of v(145). As 4 ≤ x4 ≤ 5
and 3 ≤ x5 ≤ 4, we get to x4 = 5 and x5 = 4. Consequently, x2 = 0.
Secondly, let x1 = 7. Then x3 + x4 ≥ 10, x3 + x5 ≥ 9 and x4 + x5 ≥ 8 because of
v(134), v(135) and v(145). Thus, x3 ≥ 512, x4 ≥ 41
2and x5 ≥ 31
2. As a result, x2 ≤ 1
2.
Thirdly, let x1 = 8. Then x3 + x4 ≥ 9, x3 + x5 ≥ 8 and x4 + x5 ≥ 7 because of
v(134), v(135) and v(145). Solving the inequalities, we get to x3 ≥ 5, x4 ≥ 4 and
x5 ≥ 3. Thus x2 ≤ 1, which means x2 can reach its maximum value.
Finally, we note that all the equalities and inequalities shown in this calculation
are linear, thus the ordered maximum values of x2, i.e. 0, 12and 1 may be assumed to
increase linearly while x1 ∈ [6, 8]. When x1 exceeds 8, the constraints loosen further,
but x2 cannot exceed 1. Thus, we get to the trapezoidal-shaped core projection in
Figure 4.2.
36
Appendix C
The characteristic functions of the
game in Model 3.2
v(SQ)(S) v(I)(S)
v(1) 0 0
v(2) 0 0...
......
v(n) 0 0
v(12) w2 w′2...
......
v(1(n+ 1)) wn+1 w′n+1
v(23) 0 0...
......
v(2(n+ 1)) 0 0
v(123) w2 + w3 + ε23 w′2 + w′3 + ε′23...
......
v(12(n+ 1)) w2 + wn+1 + ε2(n+1) w′2 + w′n+1 + ε′2(n+1)...
......
v(234) 0 0...
......
v(N) w2 + . . .+ wn+1 + εN\{i} w′2 + . . .+ w′n+1 + ε′N\{i}
37
Appendix D
The characteristic function of a 5-player transportation game
v(ij) wij for all i, j where i ∈ P, j ∈ Q wij
v(123) maxw13x13 + w23x23 s.t. x13 ≤ p1, x23 ≤ p2, x13 + x23 ≤ q3, x13 ≥ 0, x23 ≥ 0 O123(T )
v(124) maxw14x14 + w24x24 s.t. x14 ≤ p1, x24 ≤ p2, x14 + x24 ≤ q4, x14 ≥ 0, x24 ≥ 0 O124(T )
v(125) maxw15x15 + w25x25 s.t. x15 ≤ p1, x25 ≤ p2, x15 + x25 ≤ q5, x15 ≥ 0, x25 ≥ 0 O125(T )
v(134) maxw13x13 + w14x14 s.t. x13 + x14 ≤ p1, x13 ≤ q3, x14 ≤ q4, x13 ≥ 0, x14 ≥ 0 O134(T )
v(135) maxw13x13 + w15x15 s.t. x13 + x15 ≤ p1, x13 ≤ q3, x15 ≤ q5, x13 ≥ 0, x15 ≥ 0 O135(T )
v(145) maxw14x14 + w15x15 s.t. x14 + x15 ≤ p1, x14 ≤ q4, x15 ≤ q5, x14 ≥ 0, x15 ≥ 0 O145(T )
v(234) maxw23x23 + w24x24 s.t. x23 + x24 ≤ p2, x23 ≤ q3, x24 ≤ q4, x23 ≥ 0, x24 ≥ 0 O234(T )
v(235) maxw23x23 + w25x25 s.t. x23 + x25 ≤ p2, x23 ≤ q3, x25 ≤ q5, x23 ≥ 0, x25 ≥ 0 O235(T )
v(245) maxw24x24 + w25x25 s.t. x24 + x25 ≤ p2, x24 ≤ q4, x25 ≤ q5, x24 ≥ 0, x25 ≥ 0 O245(T )
v(1234) maxw13x13 + w23x23 + w14x14 + w24x24 s.t.x13 + x14 ≤ p1, x23 + x24 ≤ p2, x13 + x23 ≤ q3, x14 + x24 ≤ q4, O1234(T )x13 ≥ 0, x23 ≥ 0, x14 ≥ 0, x24 ≥ 0
v(1235) maxw13x13 + w23x23 + w15x15 + w25x25 s.t.x13 + x15 ≤ p1, x23 + x25 ≤ p2, x13 + x23 ≤ q3, x15 + x25 ≤ q5, O1235(T )x13 ≥ 0, x23 ≥ 0, x15 ≥ 0, x25 ≥ 0
v(1245) maxw14x14 + w24x24 + w15x15 + w25x25 s.t.x14 + x15 ≤ p1, x24 + x25 ≤ p2, x14 + x24 ≤ q4, x15 + x25 ≤ q5, O1245(T )x14 ≥ 0, x24 ≥ 0, x15 ≥ 0, x25 ≥ 0
v(1345) maxw13x13 + w14x14 + w15x15 s.t. x13+x14 + x15 ≤ p1, x13 ≤ q3, x14 ≤ q4, x15 ≤ q5, x13 ≥ 0, x14 ≥ 0, x15 ≥ 0 O1345(T )
v(2345) maxw23x23 + w24x24 + w25x25 s.t. x23+x24 + x25 ≤ p2, x23 ≤ q3, x24 ≤ q4, x25 ≤ q5, x23 ≥ 0, x24 ≥ 0, x25 ≥ 0 O2345(T )
v(12345) maxw13x13 + w14x14 + w15x15 + w23x23 + w24x24 + w25x25 s.t.x13+x14 + x15 ≤ p1, x23+x24 + x25 ≤ p2, x13 + x23 ≤ q3, x14 + x24 ≤ q4, x15 + x25 ≤ q5, O12345(T )x13 ≥ 0, x14 ≥ 0, x15 ≥ 0, x23 ≥ 0, x24 ≥ 0, x25 ≥ 0
v(S) otherwise 0
38
Appendix E
The characteristic functions of the
game in Example 5.2.1
v(SQ, SQ)(S) v(SQ,D)(S) v(D,SQ)(S) v(D,D)
v(13) 20 20 22 22
v(14) 15 15 17 17
v(15) 12 12 15 15
v(23) 7 12 7 12
v(24) 16 19 16 19
v(25) 13 17 13 17
v(123) 20 20 22 22
v(124) 16 19 17 19
v(125) 13 17 15 17
v(134) 35 35 39 39
v(135) 32 32 37 37
v(145) 27 27 32 32
v(234) 23 31 23 31
v(235) 20 29 20 29
v(245) 29 36 29 36
v(1234) 36 39 39 41
v(1235) 33 37 37 39
v(1245) 29 36 32 36
v(1345) 35 35 39 39
v(2345) 29 36 29 36
v(12345) 49 56 53 58
otherwise v(S) 0 0 0 0
39
