Nash Equilibria in Pure Strategies

We consider an n ‐person non‐zero‐sum non‐cooperative game in normal form, where the strategy sets are some closed intervals of the real line. It is shown that if the pay‐off functions are continuous on the whole space and if for each pay‐off function the smallest local maximum in the strategy variable is a global maximum, then the game possesses a pure strategy Nash equilibrium.     

Nash Equilibria of Repeated Games with Observable Payoff Vectors

We study a model of repeated games with imperfect monitoring where the payoff vector is observable. In this situation, any profitable deviation is detectable by all the players but the identity of the deviator may be unknown. We design collective punishments directed against the set of potential deviators. A particular class of signals is studied for which a characterization of the set of equilibrium payoffs is obtained. Journal of Economic Literature Classification Numbers: C73.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

Stable Provision vs. Cournot–Nash Equilibria in Pure Public Good Economies

Shitovitz and Spiegel (1998, Journal of Economic Theory 83 , pp. 1–18) demonstrated that in pure public good economies, for some consumers the Lindahl consumption bundles can be inferior (utility-wise) to their Cournot–Nash allocation. In this paper, we prove that in any finite pure public good economy there exists a core allocation that is unanimously preferred, utility-wise, by all consumers over their Cournot–Nash consumption bundles.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

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.     

Demonstrating the Possibility of Pareto Inferior Nash Equilibria

This study reviews the status of student outcomes assessment in departments of economics at colleges and universities across the country.     

On the Frequency of Interior Cournot–Nash Equilibria in a Public Good Economy

In a public good economy the distribution of initial income is an important determinant of how many individuals contribute to the public good in Cournot–Nash equilibrium. In this paper, first a simple formula is derived that provides a measure for the size of the set of income distributions leading to an interior Cournot–Nash equilibrium in which all individuals contribute to the public good. Furthermore, we give an estimate for the frequency that all members of a certain subgroup of the population are contributors.     

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 Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Markov Equilibria of Stochastic Games with Complementarities

The existence of Markov equilibria for stochastic games with a continuum of states is a complex issue for which no general result holds as yet. In this article, the problem is solved for a class of stochastic games that satisfy assumptions of complementarity and monotonicity. The proof of existence relies on results from lattice programming. In the Markov equilibria singled out by the Theorem of Existence, the policies and continuation values are increasing and Lipschitz continuous functions of the state variable.Journal of Economic LiteratureClassification Numbers: C62, C73.     

Perfect and Proper Equilibria of Large Games

This paper deals with the existence and other related issues of perfect and proper equilibria of games with a continuum of players. A sufficient condition for the existence of a perfect (proper) equilibrium as an almost everywhere limit of a sequence of ε-perfect (ε-proper) equilibria is given. An example shows that almost everywhere convergence need not obtain if the condition is violated. Extension to the case where the set of actions available to the players can differ is discussed.Journal of Economic LiteratureClassification Number: C79.     

Credible Equilibria in Games with Utilities Changing during the Play

Whenever one deals with an interactive decision situation of long duration, one has to take into account that priorities of the participants may change during the conflict. In this paper we propose an extensive-form game model to handle such situations and suggest and study a solution concept, called credible equilibrium, which generalizes the concept of Nash equilibrium. We also discuss possible variants to this concept and applications of the model to other types of games. American Mathematical Society Classification Numbers: 90A06, 90A07, 90A43, 90A56, 90D06, 90D10, 90D35, 90D40, 90D80. Journal of Economic Literature Classification Numbers: C70, C72, D10, D11, D80, D83.     

Cost Structures and Nash Play in Repeated Cournot Games

This paper reports an experiment designed to assess the effects of a rotation in the marginal cost curve on convergence in a repeated Cournot triopoly. Increasing the cost curve's slope both reduces the serially-undominated set to the Nash prediction, and increases the peakedness of earnings. We observe higher rates of Nash equilibrium play in the design with the steeper marginal cost schedule, but only when participants are also rematched after each decision. Examination of response patterns suggests that the treatment with a steeper marginal cost curve and with a re-matching of participants across periods induces the selection of Nash Consistent responses.     

Comparing Open-loop With Markov Equilibria in a Class of Differential Games

We consider a class of differential games with transition equations that are homogeneous of degree one. For any game G with a discount rate r , consider a Markov-perfect equilibrium (MPE) with strategies that are linear in the state variables. We show that the time paths of the control variables of this equilibrium constitute an open-loop equilibrium of a corresponding game G , which differs from G only in that its rate of discount r is equal to r plus a suitably chosen constant. In the context of a resource depletion game, this implies that the open-loop solution is more conservationist.
JEL Classification Nos.: C72, C73, Q30.     

Public Information, Private Information, and the Multiplicity of Equilibria in Coordination Games

I study coordination games with incomplete public and private information and relate equilibrium convergence to convergence of higher-order beliefs. As the players' signals become more and more precise, the equilibrium manifold converges to the correspondence of common knowledge equilibria, whenever the variance of the public signal converges to 0 at a rate faster than one half the rate of convergence of the variance of private signals. The same condition also determines the convergence of common p-belief to common knowledge, which leads to a simple intuition for its origin and an immediate generalization of the former results about equilibrium convergence. Journal of Economic Literature Classification Numbers: C72, D82.     

Bayesian Estimation of Entry Games with Multiple Players and Multiple Equilibria

Entry game models are often used to study the nature of firms' profits and the nature of competition among firms in empirical studies. However, when there are multiple players in an oligopoly market, the resulting multiple equilibria have made it difficult in previous studies to estimate the payoff functions of players in complete information, static and discrete games without using unreasonable assumptions. To overcome this difficulty, the present paper proposes a practical estimation method for an entry game with three players using a Bayesian approach. Some mild assumptions are imposed on the payoff function, and the average competitive effect is used to capture the entry effect of the number of firms. Our proposed methodology is applied to Japanese airline data from the year 2000, when there were three major airline companies, ANA, JAL and JAS. The model comparison is conducted to investigate the nature of strategic interaction among these Japanese airline companies.