risk dominance

We propose a formulation of a general-sum bimatrix game as a bipartite directed graph with the objective of establishing a correspondence between the set of the relevant structures of the graph (in particular elementary cycles) and the set of the Nash equilibria of the game. We show that finding the set of elementary cycles of the graph permits the computation of the set of equilibria. For games whose graphs have a sparse adjacency matrix, this serves as a good heuristic for computing the set of equilibria. The heuristic also allows the discarding of sections of the support space that do not yield any equilibrium, thus serving as a useful preprocessing step for algorithms that compute the equilibria through support enumeration.

We propose a formulation of a general-sum bimatrix game as a bipartite directed graph with the objective of establishing a correspondence between the set of the relevant structures of the graph (in particular elementary cycles) and the set of the Nash equilibria of the game. We show that finding the set of elementary cycles of the graph permits the computation of the set of equilibria. For games whose graphs have a sparse adjacency matrix, this serves as a good heuristic for computing the set of equilibria. The heuristic also allows the discarding of sections of the support space that do not yield any equilibrium, thus serving as a useful preprocessing step for algorithms that compute the equilibria through support enumeration.

We consider the class of symmetric two-player games that have the property that for any mixed strategy of the opponent, a player's best responses are included in the support of this mixed strategy-the total bandwagon or coordination property (CP). We show that for any number of pure strategies n, a symmetric two-player game has CP if and only if the game has 2 n -1 symmetric Nash equilibria. In view of the importance of 1/2-dominance as an equilibrium selection criterion, we show that if in addition to CP a game is supermodular, it will always have a 1/2-dominant equilibrium, and the 1/2-dominant equilibrium will be either the lowest or the highest strategy profile. Furthermore we show that if a game with CP has a unique potential maximizer, it will be equivalent to a 1/2-dominant equilibrium. As a specific application, we consider the minimum-effort game and reexamine the experimental equilibrium selection result of to compare it with the theoretical prediction. Our results allow us to give a new insight into an aspect of the experiment that so far could not be explained.

In a classic model of tax competition, we show that the level of public good provision and taxation in a Nash equilibrium can be efficient or inefficient with either too much, or too little public good provision. The key is whether there exists a unilateral incentive to deviate from the efficient state and, if so, whether this entails raising or lowering taxes. A priori, there is no reason to suppose the incentive is in either one direction or the other. In addition, we demonstrate conditions ensuring existence of an asymmetric Nash equilibrium with efficient public good provision. As in prior literature, local amenities enhance capital's productivity. Prior literature, however, focuses on under-provision of public goods.

Individuals in many social networks imperfectly monitor other individuals' network relationships. This paper shows that, in a model of a communication network, imperfect monitoring leads to the existence of many ine cient equilibria. Reasonable restrictions on actions or on beliefs about others' actions can, however, eliminate many of these ine cient equilibria even with imperfect monitoring. Star networks, known to be e cient in many settings, are shown to have desirable monitoring characteristics. More generally, this paper provides a formal framework in which to study incorrect perceptions as an equilibrium phenomenon in social networks.

We consider a standard coalitional bargaining game where once a coalition forms it exits as in Okada (2011), however, instead of alternating o¤ers, we have simultaneous payo¤ demands. We focus in the producer game he studies. Each player is chosen with equal probability. If that is the case, she can choose any coalition she belongs to. However, a coalition can form if an only if payo¤ demands are feasible as in the Nash (1953) demand game. After smoothing the game (as in Van Damme (1991)), when the noise vanishes, when the discount factor is close to 1, and as in Okada´s ( ), the coalitional Nash bargaining solution is the unique stationary subgameperfect equilibrium.

The focus of this paper is on developing easily veri…able suf-…cient conditions for the existence of a mixed strategy Nash equilibrium for both diagonally transfer continuous and better-reply secure games. First, we show that employing the concept of diagonal transfer continuity in place of better-reply security might be advantageous when the existence of a mixed strategy Nash equilibrium is concerned. Then, we study equilibrium existence in better-reply secure games possessing a payo¤ secure mixed extension. With the aid of an example we show that such games need not have mixed strategy Nash equilibria. We provide geometric conditions for the mixed extension of a two-person game that is reciprocally upper semicontinuous and uniformly payo¤ secure to be better-reply secure.

Embarking from the concept of uniform payo¤ security (Monteiro P.K., Page F.H, J Econ Theory 134: 566-575, 2007), we introduce two other uniform conditions and then study the existence of mixed strategy Nash equilibria in games where the sum of the payo¤ functions is not necessarily upper semicontinuous.

We construct a infinite-horizon political game where the production of a public good is delegated to a politician. The politician is controlled by finitely many citizens who, on the other hand, trade commodities and pay taxes on a voluntary basis. We provide conditions in terms of heterogenous beliefs under which one single commodity is used both as a universal means of exchange and a means to pay taxes. These provide an analytical framework for the understanding of money as originating both from the private and the public sector simultaneously.

describes the development of cooperation within exchange relationships as a slow process, starting with minor transactions in which little trust is required because little risk is involved, and eventually expanding into much riskier situations that require significantly higher levels of trust to generate cooperative behavior. If this sort of trust-building works, it does so because a counterpart's behavior in the less risky situation serves as a precedent for his or her behavior in the riskier situation. Since organizations are complex webs of exchange relationships , these precedent effects are important in organizational environments. In this paper, we use game theory to formalize how cooperative relationships develop through precedents. Then we run experiments to see whether a precedent of efficient play in a coordination game will raise the level of cooperative behavior in a finite horizon prisoner's dilemma (PD) game. We first illustrate our idea with a simple example, and later present the more complex games used in the experiments. Consider the PD game and the coordination game displayed in Tables and. (The upper left payoff in each cell is the row player's payoff; the lower right payoff is the column player's payoff.) The "efficient" outcome is the one which is unambiguously better for both players. In both games efficient outcome occurs when both players select C (cooperate). However, in the PD game it is a dominant strategy to play D (defect); individually rational players who are trying to maximize their own payoffs will both choose D, leaving both players worse off. In the coordination game, in contrast, choosing C is a best response to an expectation of C; so if one player expects the other to cooperate (with sufficiently high probability) then she will cooperate, and both player are better off. In the language of game theory, the efficient (C,C)

This paper presents results from a series of experiments designed to test the impact on subject behavior of changes in the risk dominance and payoff dominance characteristics of two player coordination games. The main finding is that changes in risk dominance significantly affect play of the subjects, whereas changes in the level of payoff dominance do not. Observed history of play also has an important influence on subject behavior, both when subjects are randomly rematched after each game and when they remain matched with the same individual for a sequence of games.

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large. * The authors thank J. J. Kline, A. Okada, J. Wako and two referees of this journal for helpful comments on earlier drafts. I ) The role of a disagreement point has been extensively studied in axiomatic bargaining theory with the assumption of an exogenously given disagreement point, see Chun and Thomson (1990) and Peters and Van Damme (1991). -235 -0 Japan Association of Economics and Econometrics 1996. Published by Blackwell Publishers. 108 Cowley Road Oxford OX4 IJF. UK.

Psychological game theory encompasses formal theories designed to remedy game-theoretic indeterminacy and to predict strategic interaction more accurately. Its theoretical plurality entails second-order indeterminacy, but this seems unavoidable. Orthodox game theory cannot solve payoff-dominance problems, and remedies based on interval-valued beliefs or payoff transformations are inadequate. Evolutionary game theory applies only to repeated interactions, and behavioral ecology is powerless to explain cooperation between genetically unrelated strangers in isolated interactions. Punishment of defectors elucidates cooperation in social dilemmas but leaves punishing behavior unexplained. Team reasoning solves problems of coordination and cooperation, but aggregation of individual preferences is problematic. I am grateful to commentators for their thoughtful and often challenging contributions to this debate. The commentaries come from eight different countries and an unusually wide range of disciplines, including psychology, economics, philosophy, biology, psychiatry, anthropology, and mathematics. The interdisciplinary character of game theory and experimental games is illustrated in Lazarus's tabulation of more than a dozen disciplines studying cooperation. The richness and fertility of game theory and experimental games owe much to the diversity of disciplines that have contributed to their development from their earliest days. The primary goal of the target article is to argue that the standard interpretation of instrumental rationality as expected utility maximization generates problems and anomalies when applied to interactive decisions and fails to explain certain empirical evidence. A secondary goal is to outline some examples of psychological game theory, designed to solve these problems. Camerer suggests that psychological and behavioral game theory are virtually synonymous, and I agree that there is no pressing need to distinguish them. The examples of psychological game theory discussed in the target article use formal methods to model reasoning processes in order to explain powerful intuitions and empirical observations that orthodox theory fails to explain. The general aim is to broaden the scope and increase the explanatory power of game theory, retaining its rigor without being bound by its specific assumptions and constraints. Rationality demands different standards in different domains. For example, criteria for evaluating formal arguments and empirical evidence are different from standards of rational decision making (Manktelow & Over 1993; Nozick 1993). For rational decision making, expected utility maximization is an appealing principle but, even when it is combined with consistency requirements, it does not appear to provide complete and intuitively convincing prescriptions for rational conduct in all situations of strategic interdependence. This means that we must either accept that rationality is radically and permanently limited and riddled with holes, or try to plug the holes by discovering and testing novel principles. In everyday life, and in experimental laboratories, when orthodox game theory offers no prescriptions for choice, people do not become transfixed like Buridan's ass. There are even circumstances in which people reliably solve problems of coordination and cooperation that are insoluble with the tools of orthodox game theory. From this we may infer that strategic interaction is governed by psychological game-theoretic principles that we can, in principle, discover and understand. These principles need to be made explicit and shown to meet minimal standards of coherence, both internally and in relation to other plausible standards of rational behavior. Wherever possible, we should test them experimentally. In the paragraphs that follow, I focus chiefly on the most challenging and critical issues raised by commentators. I scrutinize the logic behind several attempts to show that the problems discussed in the target article are spurious or that they can be solved within the orthodox theoretical framework, and I accept criticisms that appear to be valid. The commentaries also contain many supportive and elaborative observations that speak for themselves and indicate broad agreement with many of the ideas expressed in the target article. I am grateful to Hausman for introducing the important issue of rational beliefs into the debate. He argues that games can be satisfactorily understood without any new interpretation of rationality, and that the anomalies and problems that arise in interactive decisions can be eliminated by requiring players not only to choose rational strategies but also to hold rational beliefs. The only requirement is that subjective probabilities "must conform to the calculus of probabilities." Rational beliefs play an important role in Bayesian decision theory. Kreps and Wilson (1982b) incorporated them into a refinement of Nash equilibrium that they called perfect Bayesian equilibrium, defining game-theoretic equilibrium for the first time in terms of strategies and beliefs. In perfect Bayesian equilibrium, strategies are best replies to one another, as in standard Nash equilibrium, and beliefs are sequentially rational in the sense of specifying actions that are optimal for the players, given those beliefs. Kreps and Wilson defined these notions precisely using the conceptual apparatus of Bayesian decision theory, including belief updating according to Bayes' rule. These ideas prepared the ground for theories of rationalizability (Bernheim 1984; Pearce 1984), discussed briefly in section 6.5 of the target article, and the psychological games of Geanakoplos et al. (1989), to which I shall return in section R7 below. Hausman invokes rational beliefs in a plausible -though I believe ultimately unsuccessful -attempt to solve the payoff-dominance problem illustrated in the Hi-Lo Matching game (Fig. in the target article). He acknowledges that a player cannot justify choosing H by assigning particular probabilities to the co-player's actions, because this leads to a contradiction (as explained in section 5.6 of the target article). 1 He therefore offers the following suggestion: "If one does not require that the players have point priors, then Player I can believe that the probability that Player II will play H is not less than one-half, and also believe that Player II believes the same of Player I. Player I can then reason that Player II will definitely play H, update his or her subjective probability accordingly, and play H." This involves the use of interval-valued (or set-valued) probabilities, tending to undermine Hausman's claim that it "does not need a new theory of rationality." Intervalvalued probabilities have been axiomatized and studied ), but they are problematic, partly because stochastic independence, on which the whole edifice of probability theory is built, cannot be satisfactorily defined for them, and partly because technical problems arise when Bayesian updating is applied to interval-valued priors. Leaving these problems aside, the proposed solution cleverly eliminates the contradiction that arises when a player starts by specifying a point probability,

Psychological game theory encompasses formal theories designed to remedy game-theoretic indeterminacy and to predict strategic interaction more accurately. Its theoretical plurality entails second-order indeterminacy, but this seems unavoidable. Orthodox game theory cannot solve payoff-dominance problems, and remedies based on interval-valued beliefs or payoff transformations are inadequate. Evolutionary game theory applies only to repeated interactions, and behavioral ecology is powerless to explain cooperation between genetically unrelated strangers in isolated interactions. Punishment of defectors elucidates cooperation in social dilemmas but leaves punishing behavior unexplained. Team reasoning solves problems of coordination and cooperation, but aggregation of individual preferences is problematic.

This paper studies an evolutionary model of network formation with endogenous decay, in which agents benefit both from direct and indirect connections. In addition to forming (costly) links, agents choose actions for a coordination game that determines the level of decay of each link. We address the issues of coordination (long-run equilibrium selection) and network formation by means of stochastic stability techniques. We find that both the link cost and the trade-off between efficiency and risk-dominance play a crucial role in the longrun behavior of the system.

This paper studies evolutionary games in which players can condition their strategy choice on some observable characteristic of their opponent, a characteristic we can their type. Recently, examples have been provided in which some players discriminate in this way, causing the evolutionary process to converge on non-Nash equilibrium play. Moreover, in some cases this generalization of the standard setup of evolutionary game theory has been shown to destabilize certain inefficient Nash equilibria. We here provide a general model of evolutionary selection among discriminating behaviors, and flnd that the above examples are not robust; the elose connection between evolutionary selection and Nash equilibrium, already established for the standard setup , continues to hold, albeit in a slightly more complexform. Moreover , inefficient Nash equilibria may indeed be (weakly) stable in the evolutionary dynamics, and efficient Nash equilibria may be unstable.

We consider a tournament among four equally strong semifinalists. The players have to decide how much stamina to use in the semifinals, provided that the rest is available in the final and the third-place playoff. We investigate optimal strategies for allocating stamina to the successive matches when players' prizes (payoffs) are given according to the tournament results. From the basic assumption that the probability to win a match follows a nondecreasing function of stamina difference, we present symmetric Nash equilibria for general payoff structures. We find three different phases of the Nash equilibria in the payoff space. First, when the champion wins a much bigger payoff than the others, any pure strategy can constitute a Nash equilibrium as long as all four players adopt it in common. Second, when the first two places are much more valuable than the other two, the only Nash equilibrium is such that everyone uses a pure strategy investing all stamina in the semifinal. Thi...

Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.

We consider a linear price setting duopoly game with di®erentiated products and determine endogenously which of the players will lead and which will follow. While the follower role is most attractive for each¯rm, we show that waiting is more risky for the low cost¯rm so that, consequently, risk dominance considerations, as in Harsanyi and Selten (1988), allow the conclusion that only the high cost¯rm will choose to wait. Hence, the low cost¯rm will emerge as the endogenous price leader.

GLOBAL GAMES AND EQUILIBRIUM SELECTION' BY HANS CARLSSON AND ERIC VAN DAMME A global game is an incomplete information game where the actual payoff structure is determined by a random draw from a given class of games and where each player makes a noisy observation of the selected game. For 2 x 2 games, it is shown that, when the noise vanishes, iterated elimination of dominated strategies in the global game forces the players to conform to Harsanyi and Selten's risk dominance criterion.

We prove that, under appropriate conditions, an abstract game with quasi-Leontief payoff functions u i : n j=1 X j → R has a Nash equilibria. When all the payoff functions are globally quasi-Leontief, the existence and the characterization of efficient Nash equilibria mainly follows from the analysis carried out in part I. When the payoff functions are individually quasi-Leontief functions the matter is somewhat more complicated. We assume that all the strategy spaces are compact topological semilattices, and under appropriate continuity conditions on the payoff functions, we show that there exists an efficient Nash equilibria using the Eilenberg-Montgomery Fixed Point Theorem for acyclic valued upper semicontinuous maps defined on an absolute retract and some non trivial properties of topological semilattices. The map in question is defined on the set of Nash equilibria and its fixed points are exactly the efficient Nash equilibria.

A function u : X → R defined on a partially ordered set is quasi-Leontief if, if for all x ∈ X, the upper level set {x ′ ∈ X : u(x ′) u(x)} has a smallest element. A function u : n j=1 X j → R whose partial functions obtained by freezing n − 1 of the variables are all quasi-Leontief is an individually quasi-Leontief function; a point x of the product space is an efficient point for u if it is a minimal element of {x ′ ∈ X : u(x ′) u(x)}. Part I deals with the maximisation of quasi-Leontief functions and the existence of efficient maximizers. Part II is concerned with the existence of efficient Nash equilibria for abstract games whose payoff functions are individually quasi-Leontief. Order theoretical and algebraic arguments are dominant in the first part while, in the second part, topology is heavily involved. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars.

We prove that, under appropriate conditions, an abstract game with quasi-Leontief payoff functions u i : n j=1 X j → R has a Nash equilibria. When all the payoff functions are globally quasi-Leontief, the existence and the characterization of efficient Nash equilibria mainly follows from the analysis carried out in part I. When the payoff functions are individually quasi-Leontief functions the matter is somewhat more complicated. We assume that all the strategy spaces are compact topological semilattices, and under appropriate continuity conditions on the payoff functions, we show that there exists an efficient Nash equilibria using the Eilenberg-Montgomery Fixed Point Theorem for acyclic valued upper semicontinuous maps defined on an absolute retract and some non trivial properties of topological semilattices. The map in question is defined on the set of Nash equilibria and its fixed points are exactly the efficient Nash equilibria.

RATIONALIZABLE STRATEGIC BEHAVIOR BY B. DOUGLAS BERNHEIMI This paper examines the nature of rational choice in strategic games. Although there are many reasons why an agent might select a Nash equilibrium strategy in a particular game, rationality alone does not require him to do so. A natural extension of widely accepted axioms for rational choice under uncertainty to strategic environments generates an alternative class of strategies, labelled "rationalizable." It is argued that no rationalizable strategy can be discarded on the basis of rationality alone, and that all rationally justifiable strategies are members of the rationalizable set. The properties of rationalizable strategies are studied, and refinements are considered.

RATIONALIZABLE STRATEGIC BEHAVIOR This paper examines the nature of rational choice in strategic games. Although there are many reasons why an agent might select a Nash equilibrium strategy in a particular game, rationality alone does not require him to do so. A natural extension of widely accepted axioms for rational choice under uncertainty to strategic environments generates an alternative class of strategies, labelled "rationalizable." It is argued that no rationalizable strategy can be discarded on the basis of rationality alone, and that all rationally justifiable strategies are members of the rationalizable set. The properties of rationalizable strategies are studied, and refinements are considered.

The concept of the Cournot-Stackelberg-Nash equilibrium is proposed in which the leader's payoff is no less than its payoff in the case of the Cournot-Nash equilibrium and the follower's payoff is no less than its payoff in the case of the Stackelberg-Nash equilibrium. The generalized equilibrium coincides with the Cournot-Nash and Stackelberg-Nash equilibria for extreme values of a parameter.

We consider the class of symmetric two-player games that have the property that for any mixed strategy of the opponent, a player's best responses are included in the support of this mixed strategy-the total bandwagon or coordination property (CP). We show that for any number of pure strategies n, a symmetric two-player game has CP if and only if the game has 2 n − 1 symmetric Nash equilibria. In view of the importance of 1/2-dominance as an equilibrium selection criterion, we show that if in addition to CP a game is supermodular, it will always have a 1/2-dominant equilibrium, and the 1/2-dominant equilibrium will be either the lowest or the highest strategy profile. Furthermore we show that if a game with CP has a unique potential maximizer, it will be equivalent to a 1/2-dominant equilibrium. As a specific application, we consider the minimum-effort game and reexamine the experimental equilibrium selection result of Van Huyck, Battalio and Beil (1990) to compare it with the theoretical prediction. Our results allow us to give a new insight into an aspect of the experiment that so far could not be explained.

Routing games are amongst the most studied classes of games. Their two most well-known properties are that learning dynamics converge to equilibria and that all equilibria are approximately optimal. In this work, we perform a stress test for these classic results by studying the ubiquitous dynamics, Multiplicative Weights Update, in different classes of congestion games, uncovering intricate non-equilibrium phenomena. As the system demand increases, the learning dynamics go through period-doubling bifurcations, leading to instabilities, chaos and large inefficiencies even in the simplest case of non-atomic routing games with two paths of linear cost where the Price of Anarchy is equal to one. Starting with this simple class, we show that every system has a carrying capacity, above which it becomes unstable. If the equilibrium flow is a symmetric 50 − 50% split, the system exhibits one period-doubling bifurcation. A single periodic attractor of period two replaces the attracting fixed point. Although the Price of Anarchy is equal to one, in the large population limit the time-average social cost for all but a zero measure set of initial conditions converges to its worst possible value. For asymmetric equilibrium flows, increasing the demand eventually forces the system into Li-Yorke chaos with positive topological entropy and periodic orbits of all possible periods. Remarkably, in all non-equilibrating regimes, the time-average flows on the paths converge exactly to the equilibrium flows, a property akin to no-regret learning in zero-sum games. These results are robust. We extend them to routing games with arbitrarily many strategies, polynomial cost functions, non-atomic as well as atomic routing games and heteregenous users. Our results are also applicable to any sequence of shrinking learning rates, e.g., 1/ √ T , by allowing for a dynamically increasing population size.

We study the dynamics of simple congestion games with two resources where a continuum of agents behaves according to a version of Experience-Weighted Attraction (EWA) algorithm. The dynamics is characterized by two parameters: the (population) intensity of choice a > 0 capturing the economic rationality of the total population of agents and a discount factor σ ∈ [0, 1] capturing a type of memory loss where past outcomes matter exponentially less than the recent ones. Finally, our system adds a third parameter b ∈ (0, 1), which captures the asymmetry of the cost functions of the two resources. It is the proportion of the agents using the first resource at Nash equilibrium, with b = 1/2 capturing a symmetric network. Within this simple framework, we show a plethora of bifurcation phenomena where behavioral dynamics destabilize from global convergence to equilibrium, to limit cycles or even (formally proven) chaos as a function of the parameters a, b and σ. Specifically, we show that for any discount factor σ the system will be destabilized for a sufficiently large intensity of choice a. Although for discount factor σ = 0 almost always (i.e., b = 1/2) the system will become chaotic, as σ increases the chaotic regime will give place to the attracting periodic orbit of period 2. Therefore, memory loss can simplify game dynamics and make the system predictable. We complement our theoretical analysis with simulations and several bifurcation diagrams that showcase the unyielding complexity of the population dynamics (e.g., attracting periodic orbits of different lengths) even in the simplest possible potential games.

Routing games are amongst the most studied classes of games. Their two most well-known properties are that learning dynamics converge to equilibria and that all equilibria are approximately optimal. In this work, we perform a stress test for these classic results by studying the ubiquitous dynamics, Multiplicative Weights Update, in different classes of congestion games, uncovering intricate non-equilibrium phenomena. As the system demand increases, the learning dynamics go through period-doubling bifurcations, leading to instabilities, chaos and large inefficiencies even in the simplest case of non-atomic routing games with two paths of linear cost where the Price of Anarchy is equal to one. Starting with this simple class, we show that every system has a carrying capacity, above which it becomes unstable. If the equilibrium flow is a symmetric 50 − 50% split, the system exhibits one period-doubling bifurcation. A single periodic attractor of period two replaces the attracting fixed point. Although the Price of Anarchy is equal to one, in the large population limit the time-average social cost for all but a zero measure set of initial conditions converges to its worst possible value. For asymmetric equilibrium flows, increasing the demand eventually forces the system into Li-Yorke chaos with positive topological entropy and periodic orbits of all possible periods. Remarkably, in all non-equilibrating regimes, the time-average flows on the paths converge exactly to the equilibrium flows, a property akin to no-regret learning in zero-sum games. These results are robust. We extend them to routing games with arbitrarily many strategies, polynomial cost functions, non-atomic as well as atomic routing games and heteregenous users. Our results are also applicable to any sequence of shrinking learning rates, e.g., 1/ √ T , by allowing for a dynamically increasing population size.

We perform an experimental test of a modification of the controversial canonical mechanism for Nash implementation, using three subjects in non-repeated groups, as well as three outcomes, states of nature, and integer choices. We find that this mechanism successfully implements the desired outcome a large majority of the time, providing empirical evidence for the feasibility of such implementation. In addition, the performance is further improved by imposing a fine on a dissident, so that the mechanism implements strict Nash equilibria. While our environment is stylized, our results offer hope that experiments can identify reasonable features for practical implementation mechanisms.

International di¤erences in fuel taxation are huge, and may be jus-ti…ed by di¤erent local negative externalities that taxes must correct, as well as by di¤erent preferences for public spending. In this context, should a worldwide unique carbon tax be added to these local taxes to correct the global warming externality? We address this question in a second best framework à la Ramsey, where public goods have to be …nanced through distortionary taxation and the cost of public funds has to be weighted against the utility of public goods. We show that when lump-sum transfers between countries are allowed for, the second best tax on the polluting good may be decomposed into three parts: one, country speci…c, dealing with the local negative externality, a second one, country speci…c, dealing with the cost of public funds, and a third one, global, dealing with the global externality and which can be interpreted as the carbon price. Our main contribution is to show that the uniqueness of the carbon price should still hold in this second best framework. Nevertheless, if lump-sum transfers between governments are impossible to implement, international differentiation of the carbon price is the only way to take care of equity concerns. 1 Documents de Travail du Centre d'Economie de la Sorbonne-2011.76 1 We thank Julien Daubanes, Pierre Lasserre, Stéphane Gauthier, and especially Roger Guesnerie for useful discussions. A. d'Autume and K. Schubert acknowledge the support of the French National Research Agency (ANR) under the CLEANER project (ANR_NT09_505778).

<C-AB>Abstract: Even if game theory is broadened to encompass otherregarding preferences, it cannot adequately model all aspects of interactive decision making. Payoff dominance is an example of a phenomenon that can be adequately modeled only by departing radically from standard assumptions of decision theory and game theory-either the unit of agency or the nature of rationality. <C-text begins> Gintis rests his attempt to unify the behavioral sciences on a claim that "if decision theory and game theory are broadened to encompass other-regarding preferences, they become capable of modeling all aspects of decision making"

Schaffer (1988) proposed a concept of evolutionary stability for finite-population models that has interesting implications in economic models of evolutionary learning, since it is related to perfectly competitive equilibrium. The present paper explores the relation of this concept to Nash equilibrium in particular classes of games, including constant-sum games, games with weak payoff externalities, and games where imitative decision rules are individually improving. An illustration of the latter is provided in the context of Bertrand oligopoly with homogeneous product which allows for a characterization of the set of evolutionarily stable prices.

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after having observed the irrevocable commitment of the leader. Based on a result by von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in non-degenerate bimatrix games (a) pure commitment optimal strategies together with the follower's best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero sum games, the relationship between the maximin value of the leader's payoff matrix, the Nash equilibrium payoff and the commitment optimal value is discussed. For the Traveler's Dilemma, the commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable as a solution than the unique Nash equilibrium. Finally, the relationship between commitment optimal strategies and Nash equilibria in 2 × 2 bimatrix games is thoroughly examined and in addition, necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership game than at any Nash equilibrium of the simultaneous move game are provided.

Starting from Schelling (1960), several game theorists have conjectured that payoff equity might facilitate coordination in normal-form games with multiple equilibria-the more equitable equilibrium might be selected either because fairness makes it focal or because many individuals dislike payoff inequities, as abundant experimental evidence suggests. In this line, we propose a selection principle called Equity (EQ), which selects the equilibrium in pure strategies minimizing the difference between the highest and smallest money payoff, if only one such equilibrium exists. Using a within-subjects experimental design, furthermore, we study the relative performance of EQ in twelve simple 2x2 coordination games. In many of these games, we find that EQ explains individual behavior better than a large range of alternative theories, including theories of bounded rationality and several other equilibrium selection principles. Yet we also observe that the frequency of EQ play depends on the payoff structure of the game. For instance, EQ play diminishes when the alternative equilibrium is socially efficient and not very unfair (compared with the EQ equilibrium). Our data suggests that equilibrium selection is affected by several factors and that subjects are heterogeneous in this respect, but also that equity is often a crucial factor to understand coordination.

Goeree and Holt (2001) experimentally study a number of games. In each case they initially …nd strong support for Nash equilibrium, however by changing an apparently irrelevant parameter they …nd results which contradict Nash equilibrium. In this paper, we study the …ve normal form games from Goeree and Holt (2001). We argue that their results may be explained by the hypothesis that subjects view their opponents' behaviour as ambiguous. Ambiguity-aversion causes players to avoid strategies which give low out of equilibrium pay-o¤s. Similarly ambiguity-preference can make strategies with high pay-o¤s more attractive.

In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is St-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the " utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC~ of highly parallelizable problems.

In this paper we show that some decision problems regarding the computation of Nash equilibria are to be considered particularly hard. Most decision problems regarding Nash equilibria have been shown to be NP-complete. While some NP-complete problems can find an alternative to tractability with the tools of Parameterized Complexity Theory, it is also the case that some classes of problems do not seem to have fixed-parameter tractable algorithms. We show that k-Uniform Nash and k-Minimal Nash support are W [2]-hard. Given a game G=(A,B) and a nonnegative integer k, the k-Uniform Nash problem asks whether G has a uniform Nash equilibrium of size k. The k-Minimal Nash support asks whether G has Nash equilibrium such that the support of each player's Nash strategy has size equal to or less than k. First, we show that k-Uniform Nash (with k as the parameter) is W [2]-hard even when we have 2 players, or fewer than 4 different integer values in the matrices. Second, we illustrate that even in zerosum games k-Minimal Nash support is W [2]-hard (a sample Nash equilibrium in a zero-sum 2-player game can be found in polynomial time (von Stengel 2002)). Thus, it must be the case that other more general decision problems are also W [2]-hard. Therefore, the possible parameters for fixed parameter tractability in those decision problems regarding Nash equilibria seem elusive.

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

In a coarse correlated equilibrium (Moulin and Vial 1978), each player finds it optimal to commit ex ante to the future outcome from a probabilistic correlation device instead of playing any strategy of their own. In this paper, we consider a specific two-person game with unique pure Nash and correlated equilbrium and test the concept of coarse correlated equilibrium with a device which is an equally weighted lottery over three symmetric outcomes in the game including the Nash equilibrium, with higher expected payoff than the Nash payoff (as in Moulin and Vial 1978). We also test an individual choice between a lottery over the same payoffs with equal probabilities and the sure payoff as in the Nash equilibrium of the game. Subjects choose the individual lottery, however, they do not commit to the device in the game and instead coordinate to play the Nash equilibrium. We explain this behaviour as an equilibrium in the game.

We study the design of public information structures that maximize the probability of selecting a Pareto dominant equilibrium in symmetric (2 × 2) coordination games. Because the need to coordinate exposes players to strategic risk, we treat the designer as able to implement an equilibrium only if the players believe it is also risk dominant. The designer's task is therefore to pool the set of states in which the desired equilibrium is risk dominant with the largest possible set in which it is not, while keeping the desired equilibrium risk dominant in expectation. We provide a simple characterization of the optimal signal structure which holds under general conditions. We extend the analysis to related problems, and show that our intuition is robust, suggesting that our approach provides a promising way

We perform an experimental test of a modification of the controversial canonical mechanism for Nash implementation, using three subjects in non-repeated groups, as well as three outcomes, states of nature, and integer choices. We find that this mechanism successfully implements the desired outcome a large majority of the time, providing empirical evidence for the feasibility of such implementation. In addition, the performance is further improved by imposing a fine on a dissident, so that the mechanism implements strict Nash equilibria. While our environment is stylized, our results offer hope that experiments can identify reasonable features for practical implementation mechanisms.

This paper examines directed networks in which the payoff of a player depends on the total number links formed by her and the other players. After showing that these networks with global spillovers may not always have Nash equilibria in pure strategies, we introduce two additional properties for the payoff function. The first called increasing (or decreasing) difference property states that player i's payoff increases (decreases) as the number of links between the other n − 1 players increases. The second condition called the strict smaller midpoint property imposes a monotonicity restriction on the payoff function. We show that pure strategy Nash networks always exist under both conditions. The paper then characterizes these Nash equilibria showing that symmetric networks play a crucial role.

In a distributed system with attacks and defenses, an economic investment in defense mechanisms aims at increasing the degree of system protection against the attacks. We study such investments in the popular selfish setting, where both attackers and defenders are self-interested entities. In particular, we assume a reward-sharing scheme among interdependent defenders; each defender wishes to (locally) maximize its own fair share of the attackers caught due to him (and possibly due to the involvement of others). Addressed in this work is the fundamental question of determining the maximum amount of protection achievable by a number of such defenders against a number of attackers if the system is in a Nash equilibrium. As a measure of system protection, we adapt the Defense-Ratio [12], which describes the expected (inverse) proportion of attackers caught by the defenders. In a Defense-Optimal Nash equilibrium, the Defense-Ratio is optimized. We discover that the answer to this question depends in a quantitatively subtle way on the invested number of defenders. More specifically, we identify graph-theoretic thresholds for the number of defenders that determine the possibility of optimizing Defense-Ratio. In this vein, we obtain, through an extensive combinatorial analysis of Nash equilibria, a comprehensive collection of trade-off results: • When the number of defenders is either sufficiently small or sufficiently large, there are cases where the Defense-Ratio can be optimized. The corresponding optimization problem is then computationally tractable when the number of defenders is large. The problem becomes N P-complete when the number of defenders is small; the intractability is shown by reduction from a previously unconsidered combinatorial problem in Fractional Graph Theory.