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].     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.     

Undominated Nash Implementation with Collusion and Renegotiation

This article looks at implementation in economic environments when agents have perfect information about the state of the world, but cannot commit not to renegotiate bad outcomes or to collude against each other. If renegotiation satisfies a weak condition of disagreement point monotonicity, then any Pareto-efficient social choice function can be implemented if there are at least three agents who play undominated Nash equilibria. The mechanism does not use modulo or integer games, has no bad mixed strategy equilibria, and is “bounded.”Journal of Economic LiteratureClassification Number: D71.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

Evolutionary Stability of Pure-Strategy Equilibria in Finite Games

Sufficient conditions for pure-strategy Nash equilibria of finite games to be (Lyapunov) stable under a large class of evolutionary dynamics, the regular monotonic selection dynamics, are discussed. In particular, it is shown that in almost all finite extensive-form games, all the pure-strategy equilibria are stable. In such games, all mixed-strategy equilibria close to pure-strategy equilibria are also stable.Journal of Economic LiteratureClassification Numbers: C70, C72.     

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.     

Congestion games with malicious players

We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counterintuitive phenomenon which we demonstrate is the “windfall of malice”: paradoxically, when a myopically malicious player gains control of a fraction of the flow, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium.     

Bounding the inefficiency of equilibria in nonatomic congestion games

Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a “best possible” outcome. We study a nonatomic version of the congestion games defined by Rosenthal [Int. J. Game Theory 2 (1973) 65], and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players—games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games.     

On ‘Informationally Robust Equilibria’ for Bimatrix Games

Informationally robust equilibria (IRE) are introduced in Robson (Games Econ Behav 7: 233–245, 1994) as a refinement of Nash equilibria for strategic games. Such equilibria are limits of a sequence of (subgame perfect) Nash equilibria in perturbed games where with small probability information about the strategic behavior is revealed to other players (information leakage). Focusing on bimatrix games, we consider a type of informationally robust equilibria and derive a number of properties they form a non-empty and closed subset of the Nash equilibria. Moreover, IRE is a strict concept in the sense that the IRE are independent of the exact sequence of probabilities with which information is leaked. The set of IRE, like the set of Nash equilibria, is the finite union of polytopes. In potential games, there is an IRE in pure strategies. In zero-sum games, the set of IRE has a product structure and its elements can be computed efficiently by using linear programming. We also discuss extensions to games with infinite strategy spaces and more than two players. The authors would like to thank Marieke Quant for her helpful comments.     

The evolution of cooperation through imitation

We study evolutionarily stable outcomes for a class of games that admit cooperation and conflict as possible Nash equilibria. We make use of two ideas: existing strategies are more likely to be imitated than new strategies are to be introduced; players are able to identify opponents' behavior prior to interaction. The long-run evolutionary limit is efficient for the case of perfect recognition of opponents' behavior. For the case of imperfect recognition, efficiency is not achieved and long-run outcomes are more efficient the more accurate is the information. Strategies that emerge in the long run are those where players reward opponents who are likely to play the same way, and punish opponents who are likely to play differently.     

Competitive behavior in market games: Evidence and theory

We explore whether competitive outcomes arise in an experimental implementation of a market game, introduced by Shubik (1973) [21]. Market games obtain Pareto inferior (strict) Nash equilibria, in which some or possibly all markets are closed. We find that subjects do not coordinate on autarkic Nash equilibria, but favor more efficient Nash equilibria in which all markets are open. As the number of subjects participating in the market game increases, the Nash equilibrium they achieve approximates the associated competitive equilibrium of the underlying economy. Motivated by these findings, we provide a theoretical argument for why evolutionary forces can lead to competitive outcomes in market games.     

Evolution and Mixed Strategies

Selten (1980, J. Theoret. Biol., 84, 93–101) showed that no mixed equilibria are evolutionarily stable when players can condition their strategies on their roles in a game. Alternatively, Harsanyi's (1973, Int. J. Game Theory, 2, 1–23) purification argument implies that all mixed equilibria are approximations of strict, and hence evolutionarily stable, equilibria of games with slightly perturbed payoffs. This paper reconciles these results: Approximations of mixed equilibria have high invasion barriers, and hence are likely to persist, when payoff perturbations are relatively important and role identification is relatively noisy. When payoff perturbations are unimportant and role identification is precise, approximations of mixed equilibria will have small invasion barriers and are unlikely to persist. Journal of Economic Literature Classification Numbers: C70, C78.     

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games.     

Coherent behavior in noncooperative games

A new concept of mutually expected rationality in noncooperative games is proposed: joint coherence. This is an extension of the “no arbitrage opportunities” axiom that underlies subjective probability theory and a variety of economic models. It sheds light on the controversy over the strategies that can reasonably be recommended to or expected to arise among Bayesian rational players. Joint coherence is shown to support Aumann's position in favor of objective correlated equilibrium, although the common prior assumption is weakened and viewed as a theorem rather than an axiom. An elementary proof of the existence of correlated equilibria is given, and relationships with other solution concepts (Nash equilibrium, independent and correlated rationalizability) are also discussed.     

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.     

Minimum-Effort Coordination Games: Stochastic Potential and Logit Equilibrium

This paper revisits the minimum-effort coordination game with a continuum of Pareto-ranked Nash equilibria. Noise is introduced via a logit probabilistic choice function. The resulting logit equilibrium distribution of decisions is unique and maximizes a stochastic potential function. In the limit as the noise vanishes, the distribution converges to an outcome that is analogous to the risk-dominant outcome for 2 × 2 games. In accordance with experimental evidence, logit equilibrium efforts decrease with increases in effort costs and the number of players, even though these parameters do not affect the Nash equilibria. Journal of Economic Literature Classification Numbers: C72, C92.     

Beyond chopsticks: Symmetric equilibria in majority auction games

Majority auction games are simultaneous sealed-bid auctions of identical objects among identical bidders who each want to win a specified fraction (more than a half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in first-price, second-price and all-pay versions of such games when the number of bidders is sufficiently large. This contrasts with earlier results for the two-bidder “pure chopstick” majority auction games where the only known equilibria are more complicated.