Coalitional Strategic Games

Transcription

1 Coalitional Strategic Games Kazuhiro Hara New York University November 28, 2016 Job Market Paper The latest version is available here. Abstract In pursuit of games played by groups of individuals (each group itself being a player), we develop a theory of strategic games in which each player is rational in the sense of expected utility theory except that her preferences may fail to be transitive. To this end, we use the coalitional expected utility representation by Hara, Ok, and Riella (2015), and define, and then characterize, the set of Nash equilibria in terms of this representation. In particular, we provide sufficient conditions for the existence of equilibrium. For instance, it turns out that an equilibrium is sure to exist if each player possesses two pure strategies (and may have cyclic preferences across pure and mixed strategy profiles), without any further qualifications. We also study rationalizability in such games (without transitivity), as well as some equilibrium refinements, and compare our findings with those of standard game theory. Our investigation is meant to be a step toward understanding the nature of strategic interaction across groups of individuals, and clarifying the role of transitivity in game theory. 1 Introduction Game Theory is at the core of modern economic theory. Since the foundational contribution by von Neumann and Morgenstern (1944), followed by the seminal work of Nash and many others, it has been developed and found to have countless applications inside and outside economics. A strategic game, the simplest form of noncooperative games, consists of a set of players and a set of strategies and preference relation for each player. It is assumed that each player of a game is rational. In particular, they are assumed to satisfy the axioms specified in the expected utility theorem. That is, the preference relation of a player satisfies the I am grateful to Efe Ok for his suggestions and encouragement. I thank Ariel Rubinstein, David Pearce, Lukiel Christopher Levy-Moore, and seminar participants at New York University for helpful comments. Department of Economics, New York University, 19th West 4th Street, New York, NY kh1401@nyu.edu 1

2 completeness, independence, transitivity, and continuity axioms. Little is assumed on the set of strategies or the identity of the players beyond rationality. A player of a game can be an individual, a firm, or an animal etc as long as it meets the assumption of rationality. Many experiments have found that the axioms of expected utility are violated by individual decision makers. This led decision theorists to investigate what would happen to the expected utility theorem if some of the axioms are weakened or dropped. In particular, motivated by Allais Paradox, generalizations of expected utility that replace the independence axiom with weaker conditions have been extensively studied and flourished as the non-expected utility theory. Hausner and Wendel (1952) studied expected utility theory without the continuity axiom. Aumann (1962), followed by Dubra, Maccheroni, and Ok (2004) studied expected utility theory without the completeness axiom. Recently, Hara, Ok, and Riella (2015) conducted a comprehensive analysis of the axiomatic system of expected utility. They introduced the coalitional expected utility representation that subsumes the expected utility representation as a special case. In particular, they investigated what would happen if the transitivity axiom is dropped from the expected utility theorem. Game theory has followed this development of the expected utility theory. For example, equilibrium notions suitable for the non-expected utility theory have been proposed (See for example Crawford (1990), and Dekel, Safra, and Segal (1990)). Fishburn (1972) studied games without assuming continuity. Bade (2005) and Evren (2014) studied games without imposing completeness. However, no satisfactory analysis has been done on games without transitivity. The purpose of this paper is to provide a comprehensive analysis of games without transitivity. That is, we study games in which each player is fully rational in the sense of expected utility except that their preference relations may fail to be transitive. Our motivation extends beyond a mere exercise as we elaborate below. The first motivation of this work is to understand the nature of the strategic interaction of groups of individuals. A nontransitive binary relation can be viewed as a preference relation of a group of individuals. A typical example of group preference relations is one generated by majority voting. If individual preference relations that satisfy the axioms of expected utility are aggregated through majority voting, the relation obtained satisfies all the axioms of expected utility except transitivity. Majority voting is just an example of decision making procedures that are used in practice. In general, when we talk about a group of individuals, it can be a family, a firm, a country, or something else. Needless to say, different groups use different decision making procedures. For some groups, their decision making procedures are explicitly stated (e.g. countries) while for some other groups, their decision making procedures are either in the black box or hard to state explicitly (e.g. families). At the bottom line, the types of groups are diverse and their decision making processes are also different. Our approach allows us to abstract away from the complex nature of group decision making by taking the group s preference relation as primitives. A more traditional way of studying the strategic interaction of groups is to consider solution concepts that allow group deviation in the standard noncooperative framework. In this approach, groups are either exogenously given as a partition of players or formed at an equilibrium as a result of the strategic consideration of rational individual players. The solution concepts that have been proposed in this approach include strong equilibrium by Aumann (1960), social coalitional equilibrium by Ichiishi (1981), coalition proof Nash equilibrium by Bernheim, Peleg, and Whinston (1987), coalitional equilibrium by Laraki 2

3 (2009). The present work has little to add to this approach. Our work rather complements it by providing an alternative perspective for the strategic interaction of groups. A nontransitive binary relation can be viewed not only as preference relations of groups but as preference relations of individuals. There are, by now, plenty of experimental studies that have shown that individual decision makers violate transitivity. This brings us to the second motivation of our work, that is, studying games played by individuals who are irrational to the extent that their preference relations are nontransitive. An individual may possess a nontransitive preference for various reasons such as perception difficulties, similarity considerations or regret. Outside game theory, the demand theory and general equilibrium theory have been studied with nontransitive preferences. The present work brings attention to the strategic game. Having said that, there is vast literature on the abstract economy which includes the strategic game as a special case where each individual has nontransitive preferences. For the present work, Yannelis and Prabhakar (1983) is relevant. Indeed, our existence theorem of Nash equilibrium relies on their existence theorem. While the primary focus of this literature is to establish a general existence theorem with the fewest assumptions as possible for the sake of generality, our focus is beyond the existence. With the full force of the axioms of expected utility except for transitivity, we can conclude far beyond the existence of Nash equilibrium. As such, we study Nash equilibrium and its existence, rationalizability, and equilibrium refinements. The third motivation is to clarify the role of transitivity in the standard game theory. As we retain all the axioms of expected utility other than transitivity, all the results we obtain diverge from the standard theory only due to lack of transitivity. By comparing our results with the standard theory, the role of transitivity will be fleshed out. For example, Nash equilibrium does not exist in general in our environment. In the standard theory, which corresponds to the case where every player satisfies transitivity, Nash s existence theorem guarantees the existence. We provide a sufficient condition for the existence that is weaker than transitivity. The condition relates the size of cycles that each player can have in its preference relation and the existence of Nash equilibrium. In particular, our existence theorem implies that any game in which each player has only two pure strategies has an equilibrium. Some other results have implications even beyond mathematical interest. For example, in the standard theory, the way mixed strategies are interpreted, either as objective randomization or as beliefs, has no significance beyond interpretation. This is no longer the case when transitivity is not assumed. While the first interpretation leads us to the standard definition of Nash equilibrium in mixed strategies, the second interpretation leads to the notion of equilibrium in beliefs. We will see that Nash equilibrium is an equilibrium in beliefs but not vice versa. 1 The coalitional minmax expected utility representation introduced by Hara, Ok, and Riella (2015) plays a significant role in our work. The representation consists of a collection of sets of expected utility functions. It represents the relation in the sense that a lottery p is preferred to q if and only if for every set in the collection, there is one function in the set with which the expected utility with respect to p is higher than that with respect to q. Their main finding is that any binary relation over lotteries that satisfies the independence axiom admits such a representation and the axioms of expected utility are understood in terms of 1 The same phenomenon is observed in games with non-expected utility theory. See Crawford (1990). 3

4 different properties on the representation. The representation suggests an interpretation of a nontransitive relation that satisfies the independence axiom as a preference relation of a group of individuals. That is, each function in each set in the representation corresponds to a rational individual. Then, the nontransitive relation is understood as a nonblocking relation in the sense that a lottery p is preferred to q if and only if no group of individuals blocks p in favor of q. The usefulness of this representation, however, goes beyond this interpretation. We use the coalitional minmax expected utility representation as a main mathematical tool to deal with games with nontransitive preferences. It is as useful as the expected utility representation for the standard theory. For example, we provide characterizations of the set of Nash equilibria and the best response correspondence in terms of this representation. The characterizations relate a game with nontransitive preferences with standard games with transitive preferences. This makes it possible to apply standard tools that have been developed for solving standard games to our environment. This, in turn, allows us to solve games with nontransitive preferences explicitly. This paper is organized as follows. In Section 2, we first introduce the preliminary notation and terminology. Then, we provide an overview of results regarding the coalitional minmax expected utility representation that are relevant to this work. In Section 3, we define games with nontransitive preferences and two equilibrium concepts, Nash equilibrium and the equilibrium in beliefs. We provide sufficient conditions for the existence of the two solution concepts. The results in this section are demonstrated in examples. Our examples include a nontransitive version of the Prisoner s Dilemma (Gang s Dilemma), and a nontransitive version of Cournot Duopoly. In Section 4, we study rationalizability without imposing transitivity. We will see that without transitivity, two approaches by Bernheim (1984) and Pearce (1984) yield different predictions. In Section 5, we look at two important equilibrium refinements, the perfect and proper equilibrium. We show that Selten (1975) s two formulations of the perfection are equivalent even without transitivity, but Myerson (1978) s formulation is not. We will see that Selten s definition corresponds to Nash equilibrium while Myerson s definition corresponds to the equilibrium in beliefs. Section 6 conclude this paper with comments on some unsolved problems and future work. Appendix contains the proofs of main results. 2 Preliminaries The purpose of this section is to introduce the coalitional minmax expected utility representation by Hara, Ok, and Riella (2015) which plays a major role in the analysis in the later sections. We first introduce basic notation and terminology. We then present the coalitional minmax expected utility representation and discuss how the axioms of expected utility are understood through this representation. As a prominent example of nontransitive relations that satisfies the axioms of expected utility except transitivity, we introduce binary relations generated by generalized majority voting. 4

5 2.1 Notation and terminology Let Y be a nonempty set. We denote its cardinality by Y. When Y is a subset of a vector space, coy denotes its convex hull. A binary relation on Y is a nonempty subset R of Y Y. We write arb to mean (a, b) R. The asymmetric part of this relation is denoted by R >. For any a Y and S Y, we write ars to mean arb for every b S. For any S Y, a Y maximizes R on S if a S and ars. The set of all maximizers of R on S is denoted by max(s, R), that is max(s, R) := {a S ars}. We say R is reflexive if ara for each a Y, complete if arb or bra holds for each a, b Y, transitive if arb and brc imply arc for each a, b, c Y, and quasitransitive if R > is transitive. We say R has k-cycle if there are k distinct elements a 1,..., a k of Y such that a 1 R >... R > a k R > a 1. We say R is acyclic if R has no k-cycle for any k N. When Y is a metric space, we say R is continuous if for any two convergent sequences (x m ) and (y m ) in Y with x m Ry m for every m, lim x m R lim y m. A lottery on Y is a Borel probability measure on Y. The support of a lottery p on Y, denoted by supp(p), is the smallest closed subset S of Y such that p(s) = 1. We denote by δ a the degenerate lottery on Y whose support is {a}. We often identity δ a as a itself and treat Y as a subset of (Y ). The set of all lotteries on Y is denoted by (Y ). This set is endowed with the topology of weak convergence. For a subset Y of Y, we sometimes identify a lottery p over Y as the lottery p in (Y ) such that the restriction of p on Y coincides with p. The set of all continuous functions on Y and the set of all continuous and bounded functions on Y are denoted by C(Y ) and C b (Y ) respectively. We view the latter set as a metric space with respect to the sup-metric. Note when Y is a compact metric space, C(Y ) and C b (Y ) coincide. For any (u, p) C b (Y ) (Y ), the expectation of u with respect to p is denoted by E(u, p). That is, E(u, p) := udp. 2.2 Coalitional Minmax Expected Utility Representation Let Y be a metric space. A binary relation R on (Y ) satisfies the independence axiom, if prq iff λp + (1 λ)rrλq + (1 λ)r for every p, q, r (Y ) and λ (0, 1]. We call a reflexive binary relation that satisfies the independence axiom an affine relation. The independence axiom is the central axiom of expected utility theory. The expected utility theorem states that a binary relation on lotteries satisfies the completeness, independence, transitivity and continuity axioms, if and only if it admits an expected utility representation. The Expected Utility Theorem. Let Y be a separable metric space and R a binary relation on (Y ). Then, R satisfies the completeness, independence, transitivity, and continuity axioms, if and only if there is a u C b (Y ) such that prq iff E(u, p) E(u, q) 5 Y

6 for every p and q in (Y ). When all or some of the axioms of expected utility other than the independence axiom are dropped from the above theorem, a version of the expected utility representation called the coalitional minmax expected utility representation is obtained. To introduce it, let Y be a compact metric space and R an affine relation on (Y ). A coalitional minmax expected utility representation of R is a collection U of nonempty convex subsets of C(Y ) such that prq iff for every U U there is a u U such that E(u, p) E(u, q) (1) for each p and q in (Y ). The representation subsumes the expected utility representation. Indeed, the case U = {{u}} for some u C(Y ) corresponds to the expected utility representation. On the other hand, when U is a collection of singletons, it becomes the expected multi-utility representation of Dubra, Maccheroni, and Ok (2004). For each p and q in (Y ), we say q blocks p if there is a U U such that E(u, q) > E(u, q) for all u U. Notice that under this representation, prq holds if and only if there is no U U such that E(u, q) > E(u, q) for all u U. That is, p is weakly preferred to q if and only q does not block p. This suggests an interpretation of an affine relation as a preference relation of a group. Each u U can be viewed as a fully rational individual whose preference relation admits an expected utility representation u. Then, U is understood as a coalitional structure over these individuals and the relation it represents is a blocking relation of these individuals given the coalitional structure. The coalitional minmax representation was introduced by Hara, Ok, and Riella (2015). They showed that any affine relation admits such a representation. Moreover, they showed that each axiom of expected utility corresponds to different conditions on the collection U. The completeness axiom imposes a consistency condition that requires any two sets in U to have nonempty intersection. This condition is called coherence. The continuity axiom corresponds to the compactness of elements of U. The following theorem provides a characterization of a complete and continuous affine relation in terms of the coalitional minmax expected utility representation. Theorem 1 (Hara, Ok, and Riella (2015)). Let Y be a compact metric space and R a binary relation on (Y ). Then, R is a complete and continuous affine relation if, and only if, there is a coherent and compact 2 collection U of nonempty compact convex subsets of C(Y ) such that (1) holds for every p and q in (Y ). Hara, Ok, and Riella (2015) demonstrated that the coalitional minmax expected utility representation is suitable to deal with weaker transitivity conditions. For example, if an affine relation is quasitransitive and satisfies the axioms of expected utility theory except for transitivity, the collection U becomes a singleton. Theorem 2 (Evren (2014), Hara, Ok, and Riella (2015)). Let Y be a compact metric space and R a binary relation on (Y ). Then, R is a complete, continuous and quasitransitive 2 Here, the compactness is with respect to the Hausdorff metric on the set of all nonempty compact subsets of C(Y ). 6

7 affine relation if, and only if, there is a nonempty compact convex subset U of C(Y ) such that prq iff E(u, p) E(u, q) for some u U for every p and q in (Y ). From optimization theoretic point of view, acyclicity is another important relaxation of transitivity as it guarantees the existence of maximal elements under suitable assumptions. In terms of the coalitional minmax expected utility representation, acyclicity corresponds to a stronger coherence condition. We say that a collection A of sets is k-coherent if any k sets in the collection have nonempty intersection. It is strongly coherent if it has nonempty intersection, that is A. Theorem 3 (Hara, Ok, and Riella (2016)). Let Y be a compact metric space and R a binary relation on (Y ). Then, R is a complete, continuous and acyclic affine relation if, and only if, there is a strongly coherent and compact collection U of nonempty compact convex subsets of C(Y ) such that (1) holds for every p and q in (Y ). Let U be a strongly coherent representation. When we view it as a collection of coalitions of individuals, an individual that belongs to U has a veto power in the sense that q does not block p unless E(u, q) > E(u, p). Acyclicity is implied by this observation. Indeed, if p 1 R >... p k 1 R > p k then E(u, p 1 ) >... E(u, p k 1 ) > E(u, p k ) and hence p 1 and p k must be distinct. More generally, if R admits a k-coherent coalitional minmax representation U, then the relation does not have a k-cycle. 3 In fact, suppose p 1 R >... p k R > p k+1. Then for each j = 1,..., k, there is U j U such that E(u, p j ) > E(u, p j+1 ) for all u U j. By k-coherence, there is a u that belongs to U j for all j = 1,..., k. Therefore, E(u, p 1 ) >... E(u, p k ) > E(u, p k+1 ) and hence p 1 p k+1. Voting. An important example of nontransitive relations is one generated by voting. Let Y be a compact metric space, I a finite set of voters. Each i I is a fully rational individual in the sense of expected utility. In particular, his preference relation on (Y ) admits an expected utility representation u i C(Y ). A generalized majority voting is a pair ((u i ) i I, W) where W is a collection of subsets of I that satisfies the following conditions: 4 1. W and I W, 2. S W and S T imply T W, 3. S W implies I \ S W. A coalition S I is called winning if it belongs to W. Given a generalized majority voting ((u i ) i I, W), a binary relation R W is defined as follows: for each p, q (X), pr W q iff {i I E(u i, q) > E(u i, p)} W. 3 The converse is also true. That is, complete and continuous affine relation does not have a k-cycle if and only if it admits a k-coherent coalitional minmax expected utility representation. 4 A collection W that satisfies these conditions is called simple game in social choice and cooperative game theory. We call it generalized majority voting instead as we use the term game for different purpose later. 7

8 The relation R W is the group s preference relation expressed through voting in a sense that p is weakly preferred to q if and only if the coalition of individuals who strictly prefer q to p is not winning. In particular, p is strictly preferred to q if and only if the coalition of individuals who strictly prefer p to q is winning. The following proposition guarantees the relation R W defined this way is a complete and continuous affine relation. It also relates the collection of winning coalitions and a coalitional minmax representation of R W. Proposition 4. Let Y be a compact metric space and ((u i ) i I, W) be a generalized majority voting. Then, 1. R W is a complete and continuous affine relation on (Y ), and 2. the collection U W := {co{u i i S} S MIN(W, )} 5 is a coherent coalitional minmax expected utility representation of R W. Moreover, U W is k-coherent (strongly coherent) if W is k-coherent (strongly coherent). 3 Nash Equilibrium In this section, we first define a game and Nash equilibrium. Our definition of the game is exactly the standard definition of a game in that each player is fully rational in the sense of expected utility theory except his preference relation may fail to be transitive. Nash equilibrium is then defined as a mixed strategy profile from which no player deviates. We provide a characterization of the set of Nash equilibria in terms of the coalitional minmax expected utility representation. We also characterize the best response correspondence in terms of the representation and study its basic properties. When mixed strategies are viewed as beliefs held by players, an alternative equilibrium notion, the equilibrium in beliefs, is defined. An equilibrium in beliefs is a mixed strategy profile in which any action in the support of each player s equilibrium strategies is a best response. Nash equilibrium and the equilibrium in beliefs are the same concepts for games where each player satisfies transitivity. This is no longer the case in our environment. We discuss how two equilibrium concepts are related. We provide sufficient conditions for the existence of two equilibrium notions. Main results in this section will be demonstrated in terms of examples. 3.1 Games with Nontransitive Preferences In the conventional game theory, a game consists of a finite set of players, a set of actions and payoff function for each player. The payoff function is assumed to be the vonneumann- Morgenstern utility function and each player evaluates any mixed strategy profile by its expected value of payoff. This is nothing but assuming that each player of a game is a fully rational individual in the sense that his preference relation satisfies the axioms of expected utility, namely the completeness, transitivity, independence, and continuity axioms. Our definition of a game is exactly the conventional definition except it does not require the preference relation of each player to be transitive. 5 MIN(W, ) := {S W T S implies T W} 8

9 Definition. Let N be a nonempty finite set of players. A game is a collection (X i, R i ) i N of a set of actions X i and a preference relation R i for each i N that satisfy the following conditions: 1. X i is a compact metric space, and 2. R i is a complete and continuous affine relation on (X) where X := j N X j. For each player i, X i is the set of pure strategies. We allow mixed strategies and interpret them as objective mixtures over pure strategies. Therefore, player i s choice object is a lottery over X i or an element of (X i ). As usual, for a mixed strategy profile p = (p i ) i N, we write p i to mean (p j ) j i. Similarly, for a collection of sets (M i ) i N, we write M i to mean j i M i. For any two probability measures p and q, p q denotes their product measure and for any mixed strategy profile p = (p i ) i N, p is its product measure. When each player of a game satisfies transitivity, we say the game is transitive. If a game (X i, R i ) i N is transitive, each R i satisfies transitivity and then by the expected utility theorem, there is a vonneumann-morgenstern utility function u i such that pr i q iff E(u i, p) E(u i, q), for every p and q in (X). Thus, the transitive game is nothing but the standard game where each player is endowed with a payoff function. We denote the game by (X i, u i ) i N in this case. Following is an example of a game played by groups of individuals that fit this definition of a game while it cannot be analyzed in the standard framework precisely because the preference relation fails to be transitive. Example 1 (Gang s Dilemma). There are two gang groups P 1 and P 2 competing against each other over a territory. Each group has two actions (e) expand and (y) yield (X 1 = X 2 := {e, y}) and can perform any mixture of the two actions. If both gangs yield, they maintain their territories. If only one of them expands, the expanding group gains the territory. If both gangs expand, there will be an armed confrontation between them. The first gang group has three members, a boss, a fighter, and a collector. All of them are fully rational individuals whose von Neumann-Morgenstern utility functions over the four possible outcomes are given as follows respectively; P 2 u 1 e y e 1 3 P 1 y 0 2 P 2 u 2 e y e 0 3 P 1 y 1 2 P 2 u 3 e y e 1 2 P 1 y 0 3 For example, when P 1 expands and P 2 yields, the boss gets the payoff u 1 (e, y) = 3. The boss prefers to expand when the opponent yields (u 1 (e, y) > u 1 (y, y)) and is willing to fight when the opponent expands (u 1 (e, e) > u 1 (y, e)). The fighter, on the other hand, prefers expansion without confrontation (u 2 (e, y) > u 2 (y, y)) but does not want to confront (u 2 (y, e) > u 2 (e, e)) because he has to fight in that case. The collector does not want further expansion (u 3 (y, y) > u 3 (e, y)) because otherwise he has to work more, but at the same 9

10 time he does not want to lose the territory (u 3 (e, e) > u 3 (y, e)). Each member i of the group evaluates any objective uncertainty p over the outcomes {e, y} 2 by its expected payoff E(u i, p). The group makes its decision by majority voting. For any two objective uncertainty p and q over outcomes, they prefer p to q if and only if more than half of the members prefer p to q. This defines a preference relation of P 1 denoted by R 1. That is pr 1 q iff {i {1, 2, 3} E(u i, p) E(u i, q)} 2. In particular, given the opponents mixed strategy q, the group prefers a strategy p over p if and only if {i {1, 2, 3} E(u i, p q) E(u i, p q)} 2. The second gang is a mirror image of the first. It has three members, each corresponding to the following payoff functions, respectively, and makes its decision by voting. P 2 v 1 e y e 1 0 P 1 y 3 2 P 2 v 2 e y e 0 1 P 1 y 3 2 P 2 v 3 e y e 1 0 P 1 y 2 3 Their preference relation R 2 is defined by majority voting as follows. For any p and q, pr 2 q iff {i {1, 2, 3} E(v i, p) E(v i, q)} 2. As R 1 and R 2 are complete and continuous affine relations on (X 1 X 2 ) (See Proposition 4), this defines a game. Note R 1 and R 2 are not transitive. However, they are transitive over pure strategy profiles. Moreover, this game is a Prisoner s Dilemma over pure strategies. Indeed, one can check that (δ e, δ y )R > 1 (δ y, δ y )R > 1 (δ e, δ e )R > 1 (δ y, δ e ), and (δ y, δ e )R > 2 (δ y, δ y )R > 2 (δ e, δ e )R > 2 (δ e, δ y ). In the above example, each group aggregates preference relations of its members by using majority voting. In real life where a group of individuals makes a collective decision, its decision making procedure or aggregation rule can be either obscure or far more complicated than majority voting. Our definition of a game allows us to study the strategic interaction of groups of individuals without specifying the detail of decision making procedure employed by each group. Example 2. (Cournot Duopoly) There are two firms i = 1, 2 producing a homogeneous good with a constant marginal cost C where 1 > C > 0. Each firm chooses the quantity Q i [0, 1]. The set of pure strategies for each player is [0, 1] and hence the set of mixed strategies is [0, 1]. The inverse demand function is given by P = 2 (Q 1 + Q 2 ). Each firm i consists of three individuals: a profit maximizer, a sales maximizer, and a cost minimizer. The profit maximizer maximizes the expected value of profit π i (Q 1, Q 2 ) = (P C)Q 1. The sales maximizer maximizes the expected value of sales s i (Q 1, Q 2 ) = P Q i. The cost minimizer minimizes the expected value of cost c i (Q 1, Q 2 ) = CQ i. Each firm s decision rule is given by majority voting. That is, firm i s preference relation on ([0, 1] 2 ) is defined by µr i ρ iff {k {1, 2, 3} E(f k i, ρ) > E(f k i, µ)} W, where f 1 i = π i, f 2 i = s i, f 3 i = c i and W := {S {1, 2, 3} S 2}. 10

11 3.2 Equilibrium sets For transitive games, Nash equilibrium is defined as a mixed strategy profile in which every player maximizes his payoff function given the opponents strategy. In order to define Nash equilibrium in our environment, we take this definition as it is except that we use preference relations instead of payoff functions. Formally, we define Nash equilibrium in mixed strategies as follows. Definition. Let G = (X i, R i ) i N be a game. A Nash equilibrium of G is a mixed strategy profile p such that for each i N, pr i (p i, p i ) for every p i (X i ). The set of all Nash equilibria of G is denoted by NE(G). When the game G is transitive, or G = (X i, u i ) i N, this definition of Nash equilibrium clearly coincides with the standard definition. We denote the set of all Nash equilibria of a transitive game (X i, u i ) i N by NE(u 1,..., u N ). Our first result in this section is a characterization of equilibrium sets in terms of the coalitional minmax expected utility representation. Theorem 5. Let G = (X i, R i ) i N be a game and for each i N, U i a coalitional minmax expected utility representation of R i. Then, NE(G) = NE(u). (2) u U i (U i ) U i This characterization provides a way of solving the game, though it can be tedious in general. That is, given a game G = (X i, R i ) i N, take a coalitional minmax expected utility representation U i for each player i. For any u i N U i for each (U i ) i N U i, consider the transitive game (X i, u i ) i N. Then, compute the equilibrium set NE(u) of this game. Now, take a union of these equilibrium sets over i N U i for each (U i ) i N U i. The equilibrium set NE(G) is obtained as the intersection of these unions over i N U i. The characterization also suggests that methods of solving transitive games are still useful in solving nontransitive games and in fact that are all we need. Remark 1. If R i is quasitransitive for each i N, our setup is exactly that of Evren (2014). He studied games where each player has an incomplete preference relation that is asymmetric, transitive, affine and open continuous. 6 In the statement of Theorem 5, if R i is quasitransitive for each i N, then for each i N, there is a convex compact subset U i of C(X) which represents R i in the sense of Theorem 2. In this case, the characterization in Theorem 5 becomes NE(G) = NE(u 1,..., u N ). u U i This characterization corresponds to Corollary 2 in Evren (2014). A similar characterization of equilibrium sets of games with incomplete preferences appears in Bade (2005). 6 A binary relation on a metric space is open continuous if it is an open set. 11

12 In the standard theory of games, the best response correspondence plays a crucial role. It is used in the proof of equilibrium existence and it provides a method of solving games. Some properties of Nash equilibrium are direct consequence of that of the best response correspondence. It is still a useful tool in our environment to learn properties of Nash equilibrium and solving games. Definition. Let G = (X i, R i ) i N be a game. For each i N, the best response correspondence B i is defined by B i (p i ) = {p i (X i ) (p i, p i )R i (p i, p i ) for all p i (X i )} for each p i j i (X j). A strategy p i is a best response to p i if p i B i (p i ). More generally, for any nonempty subset M i of (X i ), we say a strategy p i is a best response to p i in M i if (p i, p i )R i (p i, p i ) for all p i M i. We write the set of all best responses to p i in M i by B M i i (p i ). This defines a correspondence B M i i. Clearly, B (X i) i = B i. The following statement provides a characterization of the best response correspondence in terms of the coalitional minmax expected utility representation and some basic properties of the best response correspondence. Proposition 6. Let G = (X i, R i ) i N be a game, i N and M i a nonempty closed convex subset of (X i ). Then the following holds. 1. B i has closed graph, i.e. {(p i, p i ) p i j i (X j) and p i B i (p i )} is closed. 2. If U i is a coalitional minmax expected utility representation of R i, then B M i i (p i ) = B M i i (p i, u) (3) U U i u U for each p i j i (X j) where B M i i (p i, u) = arg max pi M i E(u, (p i, p i )). 3. For any p i (X i ) and p i j i (X j), if p i B i (p i ) then δ a B i (p i ) for every a supp(p i ). This characterization provides a way of computing the best response correspondence. First, compute the best response in M i with respect to u or B M i i (p i, u) for each u U i for every U i U i. Then, take the union of these sets over each U i and then take the intersection of these unions across U i. We will demonstrate the usefulness of this characterization through examples later in this section. When R i is transitive, the converse of the third part of the statement holds, that is supp(p i ) B i (p i ) implies p i B i (p i ). Moreover, B i is convex valued. However, this is not necessarily the case when R i is not transitive. The following example, when viewed as a game with one player, shows the converse may fail even when R i is quasitransitive. 12

13 Example 3. Let X := {a, b, c} and v, w be given as follows: v w a 1 0 b 1 1 c 0 1 Define a binary relation R as follows: for each p and q in (X), prq iff E(u, p) E(u, q) for some u co{v, w}. By Theorem 2, R is a complete, continuous and quasitransitive affine relation on (X). For each u co{v, w}, u is given as follows for some t [0, 1]: Then, By Proposition 6, therefore, arg max E(u, p) = p (X) u a t b 1 c 1 t {a, b} t = 1 {δ b } t (0, 1) {b, c} t = 0 max( (X), R) = {a, b} {b, c}. Clearly, 1 2 δ a δ c is not in max( (X), R), although δ a and δ c are. Mixed strategies are sometimes interpreted as beliefs held by players regarding other players actions. Under this interpretation, an equilibrium is understood as a profile of beliefs p such that for each i N, any action in the support of p i is a best response against p i where p i is a belief held by the other players regarding i s action. We call this alternative equilibrium notion the equilibrium in beliefs. Definition. Let G = (X i, R i ) i N be a finite game. An equilibrium in beliefs of G is a mixed strategy profile p such that for each i N, δ a B i (p i ) for every a supp(p i ). The set of all equilibria in beliefs of G is denoted by NE b (G). For transitive games, Nash equilibrium and the equilibrium in beliefs coincide. That is, for any game G of the form (X i, u i ) i N, NE(G) = NE b (G). This need not hold in general however. Example 3 shows that this does not hold even when quasitransitivity is satisfied. In view of the third claim of Proposition 6, however, the following inclusion holds in general. 13

14 Proposition 7. Let G = (X i, R i ) i N be a game. Then, NE(G) NE b (G). Remark 2. Crawford (1990) introduced the equilibrium in beliefs as a solution concept for a game played by individuals whose preference relations confirm with non-expected utility theories. The way he defined the equilibrium in beliefs is different from the way we define it here. However, they are equivalent in our environment. He showed that Nash equilibrium is an equilibrium in beliefs. 3.3 Existence of Equilibrium In this section, we are concerned with the existence of equilibrium. When all the players of a game are fully rational in the sense of expected utility theory, the existence of Nash equilibrium is guaranteed by Nash (1950) s existence theorem. When the preference relation of a player is not transitive, on the other hand, existence is not guaranteed even when all the players satisfy axioms of expected utility except transitivity. This point is clear from the following example. Example 4. Let X := {a, b, c}, I := {1, 2, 3}, and for each i I, u i given as follows: u 1 u 2 u 3 a b c Define a binary relation R by prq iff {i I E(u i, p) E(u i, q)} 2 for each p and q in (X). Note that R is not acyclic. Indeed, δ b R > 1 2 δ b δ cr > δ a R > δ b is a 3-cycle. Let W := {S I S 2}. Then, ((u i ) i I, W) is a generalized majority voting. Observe that R = R W. In view of Proposition 4, R is a complete and continuous affine relation on (X). Let U 1 := co{u 1, u 2 }, U 2 := co{u 2, u 3 }, U 3 := co{u 1, u 3 }, and U := {U 1, U 2, U 3 }. Then, U is a coalitional minmax representation of R. By Proposition 6, max( (X), R) = arg max E(u, p) (4) p (X) i=1,2,3 u U i The right hand side can be computed as follows. For each u U 1, u is given as follows for some t [0, 1]: 14

15 Then, Hence, u a 1 + t b 4 3t c 0 arg max E(u, p) = p (X) {δ b } t < 3 4 {a, b} t = {δ a } < t 4 arg max E(u, p) = {a, b}. p (X) u U 1 By the similar computation, one can obtain arg max E(u, p) = {b, c}, and p (X) u U 2 arg max E(u, p) = {a, c}. p (X) u U 3 The right hand side of (4) is obtained as intersection of these and hence, max( (X), R) = {a, b} {b, c} {a, c} =. With full transitivity, we have Nash s existence theorem. On the other hand, when transitivity is completely dropped, we have nonexistence. Our goal here is to provide a sufficient condition for the existence of Nash equilibrium in terms of the primitives of the game. Our sufficient condition requires that each player has an acyclic preference relation over his mixed strategies given other players strategies. To state it formally, let (X i, R i ) i N be a game. For each i N and p i j i (X j), define a binary relation R p i i by p i R p i i q i iff (p i, p i )R i (q i, p i ), for each p i and q i in (X i ). We call R p i i the conditional preference relation of player i given p i. It is a preference relation of player i over his strategies induced from R i when other players play p i. We say R i satisfies conditional acyclicity if R p i i is acyclic for every p i j i (X j). Equivalently, R i is conditionally acyclic if it has no cycle on { (p i, p i ) p i (X i )} for every p i j i (X j). Conditional acyclicity is clearly a weaker requirement than transitivity. However, it implies certain properties that we usually take as given when transitivity is assumed. For example, if R i is conditionally acyclic, the conditional preference relation has a maximum over any nonempty closed set of strategies. In particular, player i s best response correspondence is nonempty. The following existence theorem states that conditional acyclicity is enough to guarantee the existence of Nash equilibrium. 15

16 Theorem 8. Let G = (X i, R i ) i N be a game. If for each i N, R i is conditionally acyclic, then there is a Nash equilibrium of G. Recall that Nash proved the equilibrium existence theorem appealing to Kakutani s fixed point theorem. Note that the best response correspondence in our environment does not necessarily satisfy Kakutani s conditions. In particular, it fails either to be nonempty or to be convex valued. When conditional acyclicity is assumed, it is nonempty, but still, it can be nonconvex valued. Thus, one cannot apply Kakutani s fixed point theorem in our environment. Instead, we appeal to an abstract existence theorem by Yannelis and Pranhakar (1983). It turns out conditional acyclicity, when combined with the independence axiom, allows us to invoke their existence theorem. It is clear that acyclicity implies conditional acyclicity. Therefore, the following existence theorem by Hara, Ok, and Riella (2015) is obtained as a corollary of Theorem 8. Corollary 9 (Hara, Ok, and Riella (2015)). Let G = (X i, R i ) i N i N, R i is acyclic, then there is a Nash equilibrium of G. be a game. If for each Note that conditional acyclicity is weaker than acyclicity. The following is an example of a game in which the preference relation of a player is conditionally acyclic but not acyclic. Example 5. Let X 1 = {a, b} and X 2 := {A, B}. The preference relation of player 1 is a complete and continuous affine relation R 1 that admits a coalitional minmax expected utility representation U = {co{u 1, u 2, u 3 }, co{u 1, u 2, u 4 }, co{u 1, u 3, u 4 }, co{u 2, u 3, u 4 }, } where u 1, u 2, u 3, and u 4 are given below. u 1 A B a 3 2 b 0 1 u 2 A B a 0 3 b 2 1 u 3 A B a 1 0 b 2 3 u 4 A B a 2 1 b 3 0 On the other hand, the preference relation of the second player is represented by the following payoff function: v A B a 1 0 b 0 1 Observe that R 1 has the following cycle: (a, A)R > 1 (a, B)R > 1 (b, B)R > 1 (b, A)R > 1 (a, A). For finite games, conditional acyclicity is equivalent to a simpler condition. Proposition 10. Let G = (X i, R i ) i N be a finite game and i N. Then, R i is conditionally acyclic if and only if R p i i has no X i -cycle for every p i j i (X j). The following statement is an immediate consequence of this statement. Corollary 11. Let G = (X i, R i ) i N be a finite game and i N. If any of the following conditions is satisfied, R i is conditionally acyclic. 16

17 1. R i has no X i -cycle, 2. R i admits a X i -coherent coalitional minmax expected utility representation, 3. X i = 2. If each player of a finite game satisfies one of the conditions in this statement, then by Theorem 8 there is an equilibrium of the game. For example, if each player of a finite game has no cycle shorter than the number of his pure strategies, there is an equilibrium. In application, one might prefer to give each player an explicit representation in terms of the coalitional minmax expected utility representation as primitives of a game. In that case, the second condition may be useful for determining the existence of an equilibrium. Due to the third condition, any game where each player has at most two pure strategies has an equilibrium. 7 Proposition 10 has the following implication for games where each player s preference relation is generated by generalized majority voting. Corollary 12. Let G = (X i, R i ) i N be a finite game and for each i N, R i = R Wi for some generalized majority voting W i. If for each i N, W i is X i -coherent, then there is a Nash equilibrium of G. By Proposition 7, an equilibrium in beliefs exists whenever Nash equilibrium exists. In particular, it exists when each player satisfies conditional acyclicity. It turns out, however, the existence of an equilibrium in beliefs is guaranteed under a weaker condition that the best response correspondence is nonempty valued. Theorem 13. Let G = (X i, R i ) i N be a game. If the best response correspondence is nonempty valued for each player, then there is an equilibrium in beliefs of G. In contrast to Theorem 8, this is proved by invoking a generalization of Kakutani s fixed point theorem. Indeed, an equilibrium in beliefs is obtained as a fixed point of the closed convex hull of the best response correspondence. 3.4 Examples We demonstrate some of the results we have developed so far through examples. Example 1 (Gang s dilemma conti). Since this game is a 2 2 game, there is an equilibrium. In what follows, we identify any mixed strategy p of player 1 as its probability of action e. Similarly, any mixed strategy q of player 2 is identified by its probability of action e. The set of Nash equilibria of this game is obtained as follows. Let U 1 := co{u 1, u 2 }, U 2 := co{u 2, u 3 }, and U 3 := co{u 1, u 3 }. By Proposition 4, U := {U 1, U 2, U 3 } is a coalitional minmax expected utility representation of R 1. Then, by Proposition 6, the best response correspondence of player 1 can be written as B 1 (q) = max E(u, p q) arg p {y,e} j=1,2,3 u U j 7 One can prove this statement using Kakutani s fixed point theorem. Indeed, in this case conditional relations are transitive and hence the best response correspondence is convex valued. 17

18 for each q {y, e}. For each u U 1, u = tu 1 + (1 t)u 2 for some t [0, 1]. When t > 1 2, arg max p {e,y} E(u, p q) = {1} for every q {e, y}. On the other hand, when t 1 2, Therefore, arg max p {e,y} arg max p {e,y} u U 1 By similar computation, one can obtain max and {1} q < 1 E(u, p q) = [0, 1] q = 1 {0} q > 1 arg p {e,y} u U 2 arg max p {e,y} u U 3 By taking intersection of these, we obtain B 1 (q) = E(u, p q) = 2(1 t) 2(1 t). 2(1 t) { {1} q < 1 2 [0, 1] q 1 2 E(u, p q) = [0, 1], E(u, p q) = { {1} q 1 2 [0, 1] q = 1 2 { [0, 1] q 1 2 {1} q > 1 2 The best response correspondence of the second player can be obtained by the same procedure and { {1} p 1 2 B 2 (p) = [0, 1] p = 1 2 The following figure illustrates the graph of the best response correspondence for each player. 18

19 Finally, the set of Nash equilibria is obtained as intersection of graphs of B 1 and B 2 and hence { ( 1 NE(G) = (1, 1), 2, 1 )}. 2 This game, therefore, has two equilibrium, one in which each player expands with probability one and the other in which each player expands and yields in probability one half. Recall that this game coincides with a prisoner s dilemma when it is restricted on pure strategy profiles. In a prisoner s dilemma, there is a unique Nash equilibrium in which each player uses its strictly dominant strategy. Due to transitivity, it does not matter whether we consider mixed strategies or not. In this game, however, there is an equilibrium in completely mixed strategy ( 1 2, 1 2) due to nontransitivity. Example 2 (Cournot Duoploly conti). Let U 1 = co{π 1, s 1 }, U 2 = co{π 1, c 1 }, U 3 = co{s 1, c 1 } and U = {U 1, U 2, U 3 }. Then, U is a coalitional minmax expected utility representation of R 1. Observe that [ 2 µ2 c B 1 (µ 2, f) =, 2 µ ] f U 1 [ B 1 (µ 2, f) = 0, 2 µ ] 2 c 2 f U 2 [ B 1 (µ 2, f) = 0, 2 µ ] 2 2 f U 3 where µ 2 is the average of µ 2 [0, 1]. Thus, the best response correspondence of first firm is B 1 (µ 2 ) = 2 µ 2 c. 2 This is exactly the best response correspondence of the profit maximizer, that is B 1 (µ 2 ) = B i (µ 2, π 1 ). Since the second firm is a mirror image, its best response correspondence is also that ( of the profit maximizer. Therefore, there is a unique Nash equilibrium in this game 2 c, ) 2 c 3 3 in which each firm produces following profit maximizer s plan. This is the unique equilibrium in beliefs of this game. Suppose each firm has one more sales maximizer and uses majority voting. That is, firm i s preference relation is given by µr i ρ iff {k {1, 2, 3, 4} E(f k i, ρ) > E(f k i, µ)} W where fi 4 = s i and W = {S {1, 2, 3, 4} S 3}. By the similar computation as in the previous case, one can compute the best response correspondence of each firm as [ 2 µj c B i (µ j ) =, 2 µ ] j 2 2 The following figures visualize the graph of B 1 and B 2. 19

20 Thus the equilibrium set is obtained as NE(G) = co {w, x, y, z} where w = ( 2 c, ) ( 2 c 3 3, x = 4+2c, ) ( 2 2c 6 3, y = 2 2c, ) ( 4+2c 3 6 and z = 2, ) The following figure visualize the set of all Nash equilibria. On the other hand, (µ 1, µ 2 ) is an equilibrium in beliefs if and only if [ 2 µ2 c supp(µ 1 µ 2 ) is a subset of, 2 µ ] [ 2 2 µ1 c, 2 µ ] In particular, NE b (G) is strictly larger than NE(G) and contains strategy profiles in which each player uses a nondegenerate mixed strategy. Moreover, if (µ 1, µ 2 ) is an equilibrium in beliefs then ( µ 1, µ 2 ) NE(G). 4 Rationalizability and Iterated strict dominance In this section, we are concerned with alternative solution concepts of rationalizability and iterated elimination of strictly dominated strategies. The notion of rationalizability was introduced independently by Bernheim (1984) and Pearce (1984). We first observe by means of example that standard properties of rationalizability that are known to hold in transitive games, namely the pure strategy property and the best response property, may fail. We consider an alternative definition of rationalizability and show that it satisfies the two properties and admits characterization as the largest set of strategies that satisfies the best response property. After looking at the relationship of Nash equilibrium, equilibrium in beliefs, and 20