Nash equilibria equal competitive equilibria

With strictly monotone preferences and a continuum of traders, there is a game in which the set of Nash equilibria is exactly the same as the set of competitive equilibria.     

Policy rule Nash equilibria

Abstract.  We study how different types of instrument rules affect the outcome of a monetary policy game between the central bank and the private sector. Policy rules can be independent of output and a shock, functions of the shock, or functions of output. We rank the Nash equilibria generated by different types of policy rules according to the central bank's ex ante expected loss. If both players can condition on output, then the following is true: no equilibrium exists if the central bank cares much about output stability and little about price stability, and infinitely many equilibria exist otherwise. JEL classification: E52, E61
Equilibres à la Nash des règles de politique.  L'auteur étudie comment le choix de divers types de règles instrumentales a un effet sur le résultat d'un jeu de politique monétaire entre la banque centrale et le secteur privé. Les règles de politique peuvent être indépendantes de la production ou d'un choc, être fonction d'un choc ou de la production. On établit un ordre des équilibres à la Nash engendrés par ces divers types de règles de politique selon la taille des pertes anticipées ex ante par la banque centrale. Si les deux joueurs fondent leur comportement sur la production, il s'ensuit que — aucun équilibre n'existe si la banque centrale est très préoccupée par la stabilité du niveau de production et peu préoccupée de la stabilité des prix, et autrement, une infinité d'équilibres existent.     

The effects of risk preferences in mixed-strategy equilibria of games

We consider the effects of risk preferences in mixed-strategy equilibria of 2×2 games, provided such equilibria exist. We identify sufficient conditions under which the expected payoff in the mixed equilibrium increases or decreases with the degree of risk aversion. We find that (at least moderate degrees of) risk aversion will frequently be beneficial in mixed equilibria.     

The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition

Abstract. We analyze the Nash equilibria of a standard Bertrand model. We show that in addition to the marginal-cost pricing equilibrium there is a possibility for mixed-strategy equilibria yielding positive profit levels. We characterize these equilibria and find that having unbounded revenues is the necessary and sufficient condition for their existence. Hence, we demonstrate that under realistic assumptions the only equilibrium is marginal-cost pricing.     

Discontinuous stable games and efficient Nash equilibria

In the recent paper by the author [Scalzo, V., 2010. Pareto efficient Nash equilibria in discontinuous games. Economics Letters 107, 364-365], a class of discontinuous games where efficient Nash equilibria exist has been defined. In the present paper, we complete the previous investigation and recognize a class of discontinuous games where the efficient Nash equilibria are stable with respect to perturbations of the characteristics of players.     

Double implementation in Nash and undominated Nash equilibria in social choice environments

Summary Jackson [1] and Yamato [12] constructed game forms which implement social choice correspondences in Nash and undominated Nash equilibria simultaneously in exchange economies. In this paper, I deal with social choice environments and construct a game form which implements social choice correspondences satisfying monotonicity, no veto power and having at least three agents in Nash and undominated Nash equilibria under the existence of an alternative called a holocaust. The game form constructed in this paper includes an integer game but satisfies the boundedness condition introduced by Jackson [1].I am grateful to Professors Tatsuyoshi Saijo and Stephen Turnbull, and to an anonymous referee for their helpful comments and suggestions. However all remaining errors are mine. An earlier version of this paper was presented at the 1991 Annual Meeting of the Japan Association of Economics and Econometrics at Hokkaido University.After this paper was submitted toEconomic Theory I became aware of the work of Jackson, Palfrey, and Srivastava [2] who have obtained a similar result to mine as a by-product of their main result.     

New complexity results about Nash equilibria

We provide a single reduction that demonstrates that in normal-form games: (1) it is -complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [Gilboa, I., Zemel, E., 1989. Nash and correlated equilibria: Some complexity considerations. Games Econ. Behav. 1, 80–93]), (2) more significantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless ), and (3) it is -hard to count the Nash equilibria. We also show that determining whether a pure-strategy Bayes–Nash equilibrium exists in a Bayesian game is -complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is -hard even if the game is unobserved (and that this remains -hard if the game has finite length). All of our hardness results hold even if there are only two players and the game is symmetric.     

On the uniqueness of Nash equilibria

Although the use of Nash equilibria in economic models has been increasing, little attention has been paid to the uniqueness of these equilibria. This paper presents relatively weak conditions that are sufficient for the existence of a globally unique Nash equilibrium.     

More strategies, more Nash equilibria

We show in this paper that for the class of two-player games with compact real intervals as strategy spaces and continuous and strictly quasi-concave payoff functions there exists a monotone relation between the size of strategy spaces and the number of Nash equilibria. These sufficient conditions for our theorem to hold are shown to be tight.     

Comparative statics for integrable Nash equilibria

This paper derives comparative statics results for stable Nash equilibria in integrable, concave orthogonal games. Application of these results to cost curve shifts in the asymmetric Cournot oligopoly immediately uncovers some apparently new comparative statics results.     

Enumeration of Nash equilibria for two-player games

This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.     

Mixed-strategy equilibria in the Nash Demand Game

In the Nash Demand Game, each of the two players announces the share he demands of an amount of money that may be split between them. If the demands can be satisfied, they are; otherwise, neither player receives any money. This game has many pure-strategy equilibria. This paper characterizes mixed-strategy equilibria. The condition critical for an equilibrium is that players’ sets of possible demands be balanced. Two sets of demands are balanced if each demand in one set can be matched with a demand in the other set such that they sum to one. For Nash’s original game, a complete characterization is given of the equilibria in which both players’ expected payoffs are strictly positive. The findings are applied to the private provision of a discrete public good.     

Topological upper limits of mixed Nash equilibria

Our purpose in this article is to prove that given any integer n ≥ 2 and any non-empty compact Polish spaces S ₁, ..., S _n , if for any u ∈ C( S ₁ × ... × S _n , R) ⁿ , we denote by MNE(u) the set of mixed Nash equilibria of (S ₁, ..., S _n , u), then MNE(u) is a non-empty compact subset of P(S ₁) × ... × P(S _n ) and if u ^k → u in C(S ₁ × ... × S _n , R) ⁿ as k → ∞, then lim sup _{k → ∞} MNE (u ^k ) MNE(u). The author would like to thank the referee for offering critical comments on this paper.     

Undecidability of the existence of pure Nash equilibria

Summary. The purpose of this paper is to show that for any positive integer n, there exists no algorithm which decides for each non-cooperative n-person game in strategic form with partially computable payoff functions whether it has a pure Nash equilibrium or not.Received: 20 March 2002, Revised: 23 April 2003, JEL Classification Numbers: C72.The author thanks the Referee and the Associate Editor for offering critical comments on this paper, which is dedicated to his brother Konstantinos Efstathiou Sofronidis.