Existence of pure Nash equilibria in discontinuous and non quasiconcave games

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny’s theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence (I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper).     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

On weakly correlated and correlated equilibria in discontinuous games

This paper treats of non-zero-sum discontinuous games with compact Hausdorff strategy spaces. It is assumed that the payoff function of each player in the game is bounded, Borel measurable and is upper semicontinuous on his strategy space, for all fixed actions of the remaining players. It is shown that for each ε>0, such games possess weakly correlated ε-epuilibria introduced by Moulin and Vial as extension of correlated equilibria in the sense of Aumann. An upper semicontinuous came having weakly correlated equilibria and no correlated equilibria is discussed in details.     

Average-discounted equilibria in stochastic games

In stochastic games with finite state and action spaces, we examine existence of equilibria where player 1 uses the limiting average reward and player 2 a discounted reward for the evaluations of the respective payoff sequences. By the nature of these rewards, the far future determines player 1's reward, while player 2 is rather interested in the near future. This gives rise to a natural cooperation between the players along the course of the play. First we show the existence of stationary ε-equilibria, for all ε>0, in these games. However, besides these stationary ε-equilibria, there also exist ε-equilibria, in terms of only slightly more complex ultimately stationary strategies, which are rather in the spirit of these games because, after a large stage when the discounted game is not interesting any longer, the players cooperate to guarantee the highest feasible reward to player 1. Moreover, we analyze an interesting example demonstrating that 0-equilibria do not necessarily exist in these games, not even in terms of history dependent strategies. Finally, we examine special classes of stochastic games with specific conditions on the transition and payoff structures. Several examples are given to clarify all these issues.     

Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasiconcavity of functions and convexity or compactness of choice sets. A new condition, called “local dominatedness property”, is shown to be necessary and further, under some mild continuity condition, sufficient for the existence of equilibrium. We then apply the basic result obtained in the paper to generalize the existing theorems on the existence of saddle points, fixed points, and coincidence points without convexity or compactness assumptions. As an application, we also characterize the existence of pure strategy Nash equilibrium in games with discontinuous and non-quasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.     

On existence of undominated pure strategy Nash equilibria in anonymous nonatomic games: a generalization

In this paper, we generalize the exitence result for pure strategy Nash equilibria in anonymous nonatomic games. By working directly on integrals of pure strategies, we also generalize, for the same class of games, the existence result for undominated pure strategy Nash equilibria even though, in general, the set of pure strategy Nash equilibria may fail to be weakly compact. Received August 2001     

Pure strategy equilibria in symmetric two-player zero-sum games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.     

On the existence of equilibria in generalized games

This paper deals with the existence of equilibrium in generalized games (the so-called abstract economies) and Nash equilibrium in games with general assumptions. Preference correspondences, unlike the existing theorems in the literature, need not have open graphs or open lower sections, strategy spaces need not be compact and finite dimensional, the number of agents need not be countable, and preference relations need not be ordered. Thus, our results generalize many of the existence theorems on equilibria in generalized games, including those of Debreu (1952), Shafer and Sonnenschein (1975), Toussaint (1984), Kim and Richter (1986), and Yannelis (1987).I wish to thank the editor and an anonymous referee for helpful comments and suggestions.     

On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium.As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of -equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of -equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.     

Coordination games on graphs

We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between \(2(n-1)/(k-1) - 1\) and \(2(n-1)/(k-1)\). The latter upper bound is tight for \(k=n\) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases.     

Zero-Sum Ergodic Semi-Markov Games with Weakly Continuous Transition Probabilities

Zero-sum ergodic semi-Markov games with weakly continuous transition probabilities and lower semicontinuous, possibly unbounded, payoff functions are studied. Two payoff criteria are considered: the ratio average and the time average. The main result concerns the existence of a lower semicontinuous solution to the optimality equation and its proof is based on a fixed-point argument. Moreover, it is shown that the ratio average as well as the time average payoff stochastic games have the same value. In addition, one player possesses an ε-optimal stationary strategy (ε>0), whereas the other has an optimal stationary strategy. A. Jaśkiewicz is on leave from Institute of Mathematics and Computer Science, Wrocław University of Technology. This work is supported by MNiSW Grant 1 P03A 01030.     

Perfect information stochastic games and related classes

Forn-person perfect information stochastic games and forn-person stochastic games with Additive Rewards and Additive Transitions (ARAT) we show the existence of pure limiting average equilibria. Using a similar approach we also derive the existence of limiting average ε-equilibria for two-person switching control stochastic games. The orderfield property holds for each of the classes mentioned, and algorithms to compute equilibria are pointed out.     

Optimal strategic reasoning with McNaughton functions

The aim of the paper is to explore strategic reasoning in strategic games of two players with an uncountably infinite space of strategies the payoff of which is given by McNaughton functions—functions on the unit interval which are piecewise linear with integer coefficients. McNaughton functions are of a special interest for approximate reasoning as they correspond to formulas of infinitely valued Lukasiewicz logic. The paper is focused on existence and structure of Nash equilibria and algorithms for their computation. Although the existence of mixed strategy equilibria follows from a general theorem (Glicksberg, 1952) [5], nothing is known about their structure neither the theorem provides any method for computing them. The central problem of the article is to characterize the class of strategic games with McNaughton payoffs which have a finitely supported Nash equilibrium. We give a sufficient condition for finite equilibria and we propose an algorithm for recovering the corresponding equilibrium strategies. Our result easily generalizes to n-player strategic games which don't need to be strictly competitive with a payoff functions represented by piecewise linear functions with real coefficients. Our conjecture is that every game with McNaughton payoff allows for finitely supported equilibrium strategies, however we leave proving/disproving of this conjecture for future investigations.     

Remarks on sensitive equilibria in stochastic games with additive reward and transition structure

A class of stochastic games with additive reward and transition structure is studied. For zero-sum games under some ergodicity assumptions 1-equilibria are shown to exist. They correspond to so-called sensitive optimal policies in dynamic programming. For a class of nonzero-sum stochastic games with nonatomic transitions nonrandomized Nash equilibrium points with respect to the average payoff criterion are also obtained. Included examples show that the results of this paper can not be extented to more general payoff or transition structure.     

大博弈中Nash均衡的存在性

研究了具有任意多个局中人的非合作博弈(大博弈)中Nash均衡的存在性.将1969年Ma的截口定理推广得到新的截口定理.用这个新的截口定理进一步证明了:1)大博弈中Nash均衡的存在性;2)纯策略集为紧度量空间而且支付函数为连续函数时,连续大博弈中混合策略Nash均衡的存在性.并且存在性定理推出了2010年Salonen的结果,即此研究结果较Salonen的结论更具普遍意义.     

On the existence of Nash equilibria in large games

We study the existence of Nash equilibria in games with an infinite number of players. We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are compact metric spaces and utility functions are continuous. The player set can be any nonempty set.     

Hadamard well-posedness in discontinuous non-cooperative games

We consider some recent classes of discontinuous games with Nash equilibria and we prove that such classes have the Hadamard well-posedness property. This means that given a game y, a net (y^α)_α of games converging to y and a net (x^α)_α such that x^α is a Nash equilibrium of any y^α, then at least a cluster point of (x^α)_α is a Nash equilibrium of y. In order to obtain this property, we prove that the map of Nash equilibria is upper semicontinuous. Using the pseudocontinuity, a generalization of the continuity, we improve previous results obtained with continuous functions.     

Evolutionary games and matching rules

This study considers evolutionary games with non-uniformly random matching when interaction occurs in groups of \(n\ge 2\) individuals using pure strategies from a finite strategy set. In such models, groups with different compositions of individuals generally co-exist and the reproductive success (fitness) of a specific strategy varies with the frequencies of different group types. These frequencies crucially depend on the matching process. For arbitrary matching processes (called matching rules), we study Nash equilibrium and ESS in the associated population game and show that several results that are known to hold for population games under uniform random matching carry through to our setting. In our most novel contribution, we derive results on the efficiency of the Nash equilibria of population games and show that for any (fixed) payoff structure, there always exists some matching rule leading to average fitness maximization. Finally, we provide a series of applications to commonly studied normal-form games.     

一种n人静态博弈纯策略纳什均衡存在性判别法

本首先给出了n人静态博弈纯策略纳什均衡存在的充要条件。然后给出n人静态博弈纯策略纳什均衡存在性的一种判别方法。最后在判别纯策略纳什均衡存在的条件下，给出判定该静态博弈存在多少纯策略纳什均衡以及哪些纯策略组合是纯策略纳什均衡(解)的方法。