Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Dynamically stable sets in infinite strategy spaces

Evolutionary game theory has largely focused on finite games. Dynamic stability is harder to attain in infinite strategy spaces; Bomze [Bomze, I., 1990. Dynamical aspects of evolutionary stability. Monatsh. Math. 110, 189–206] and Oechssler and Riedel [Oechssler, J., Riedel, F., 2001. Evolutionary dynamics on infinite strategy spaces. Econ. Theory 17, 141–162] provide conditions for the stability of rest points under the replicator dynamics. Here, conditions are given for the stability of sets of strategies under this process.     

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.     

Extensive Form Games with Uncertainty Averse Players

Nash equilibrium presumes that the beliefs of a player are represented by a probability measure. Motivated by the Ellsberg Paradox and relevant experimental findings demonstrating that this representation of beliefs may be unrealistic, this paper generalizes Nash equilibrium in finite extensive form games to allow for preferences conforming to the multiple priors model developed by Gilboa and Schmeidler [Journal of Mathematical Economics, 18 (1989), 141–153]. The implications of this generalization for strategy choices and welfare are studied. Journal of Economic Literature Classification Numbers: C72, D81.     

Pure Strategy Nash Equilibrium in a Group Formation Game with Positive Externalities

This paper identifies a domain of payoff functions inno spillovernoncooperative games withPositive externalitywhich admit a pure strategy Nash equilibrium. Since in general a Nash equilibrium may fail to exist, in order to guarantee the existence of an equilibrium, we impose two additional assumptions,AnonymityandOrder preservation. The proof of our main result is carried out by constructing, for a given gameG, a potential function Ψ over the set of strategy profiles in such a way that the maximum of Ψ yields a Nash equilibrium in pure strategies ofG.Journal of Economics LiteratureClassification Numbers: C72, D62, H73.     

Nash equilibria in ∞-dimensional spaces: an approximation theorem

Summary. We show, by employing a density result for probability measures, that in games with a finite number of players and ∞-dimensional pure strategy spaces Nash equilibria can be approximated by finite mixed strategies. Given ε>0, each player receives an expected utility payoff ε/2 close to his Nash payoff and no player could change his strategy unilaterally and do better than ε. Received: July 15, 1997; revised version: February 6, 1998     

When is reputation bad?

In traditional reputation models, the ability to build a reputation is good for the long-run player. In [Ely, J., Valimaki, J., 2003. Bad reputation. NAJ Econ. 4, 2; http://www.najecon.org/v4.htm. Quart. J. Econ. 118 (2003) 785–814], Ely and Valimaki give an example in which reputation is unambiguously bad. This paper characterizes a class of games in which that insight holds. The key to bad reputation is that participation is optional for the short-run players, and that every action of the long-run player that makes the short-run players want to participate has a chance of being interpreted as a signal that the long-run player is “bad.” We allow a broad set of commitment types, allowing many types, including the “Stackelberg type” used to prove positive results on reputation. Although reputation need not be bad if the probability of the Stackelberg type is too high, the relative probability of the Stackelberg type can be high when all commitment types are unlikely.     

Perfect communication equilibria in repeated games with imperfect monitoring

This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.     

Equivalence of strong and coalition-proof Nash equilibria in games without spillovers

Summary This paper examines the conditions which guarantee that the set of coalition-proof Nash equilibria coincides with the set of strong Nash equilibria in the normal form games withoutspillovers. We find thatpopulation monotonicity properties of the payoff functions, when the payoff of a player changes monotonically when the size of the group of players choosing the same strategy increases, are crucial to obtain the equivalence of these two solution concepts. We identify the classes of games, satisfying population monotonicity properties, which yield the equivalence of the set of coalition-proof Nash equilibria and the set of strong Nash equilibria. We also provide sufficient conditions for the equivalence result even when the population monotonicity assumptions are relaxed.We wish to thank Mamoru Kaneko, Akihiko Matsui, Tomoichi Shinotsuka, Benyamin Shitoviz, Tayfun Sonmez, William Thomson, the participants of the Southeastern Economic Theory Meeting in Charlottesville and the seminars at CORE and University of Tsukuba for useful discussions and comments. Our special thanks due anonymous referee for the suggestion to add a section addressing the issue of existence of a strong Nash equilibrium.     

Best response equivalence

Two games are best-response equivalent if they have the same best-response correspondence. We provide a characterization of when two games are best-response equivalent. The characterizations exploit a dual relationship between payoff differences and beliefs. Some “potential game” arguments [Games Econ. Behav. 14 (1996) 124] rely only on the property that potential games are best-response equivalent to identical interest games. Our results show that a large class of games are best-response equivalent to identical interest games, but are not potential games. Thus we show how some existing potential game arguments can be extended.     

Purification in the infinitely-repeated prisoners' dilemma

This paper investigates the Harsanyi [Harsanyi, J.C., 1973. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory 2 (1), 1–23]-purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of one-period memory mixed strategy equilibria used by Ely and Välimäki [Ely, J.C., Välimäki, J., 2002. A robust folk theorem for the prisoner's dilemma. Journal of Economic Theory 102 (1), 84–105] in their study of the repeated prisoners' dilemma with private monitoring. We find that any such strategy profile is not the limit of one-period memory equilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable.     

A game of timing and visibility

We consider the following abstraction of competing publications. There are n players in the game. Each player i chooses a point x_i in the interval [0,1], and a player's payoff is the distance from its point x_i to the next larger point, or to 1 if x_i is the largest. For this game, we give a complete characterization of the Nash equilibrium for the two-player game, and, more important, we give an efficient approximation algorithm to compute numerically the symmetric Nash equilibrium for the n-player game. The approximation is computed via a discrete version of the game. In both cases, we show that the (symmetric) equilibrium is unique. Our algorithmic approach to the n-player game is non-standard in that it does not involve solving a system of differential equations. We believe that our techniques can be useful in the analysis of other timing games.     

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

Understanding some recent existence results for discontinuous games

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (Econometrica 67:1029–1056, 1999) and Carmona (J Econ Theory 144:1333–1340, 2009) and that it is equivalent to a recent result of Barelli and Soza (On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, University of Rochester, Rochester, 2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.     

Cycling in a stochastic learning algorithm for normal form games

In this paper we study a stochastic learning model for 2×2 normal form games that are played repeatedly. The main emphasis is put on the emergence of cycles. We assume that the players have neither information about the payoff matrix of their opponent nor about their own. At every round each player can only observe his or her action and the payoff he or she receives. We prove that the learning algorithm, which is modeled by an urn scheme proposed by Arthur (1993), leads with positive probability to a cycling of strategy profiles if the game has a mixed Nash equilibrium. In case there are strict Nash equilibria, the learning process converges a.s. to the set of Nash equilibria.     

Cheap talk in games with incomplete information

The paper studies Bayesian games which are extended by adding pre-play communication. Let Γ be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q, that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of Γ, provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in Γ such that for each type t_i of each player i the expected payoff of t_i in q is larger than the expected payoff of t_i in s.     

Generalised weakened fictitious play

A general class of adaptive processes in games is developed, which significantly generalises weakened fictitious play [Van der Genugten, B., 2000. A weakened form of fictitious play in two-person zero-sum games. Int. Game Theory Rev. 2, 307–328] and includes several interesting fictitious-play-like processes as special cases. The general model is rigorously analysed using the best response differential inclusion, and shown to converge in games with the fictitious play property. Furthermore, a new actor–critic process is introduced, in which the only information given to a player is the reward received as a result of selecting an action—a player need not even know they are playing a game. It is shown that this results in a generalised weakened fictitious play process, and can therefore be considered as a first step towards explaining how players might learn to play Nash equilibrium strategies without having any knowledge of the game, or even that they are playing a game.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.