Solution concepts in two-person multicriteria games

In this paper, we propose new solution concepts for multicriteria games and compare them with existing ones. The general setting is that of two-person finite games in normal form (matrix games) with pure and mixed strategy sets for the players. The notions of efficiency (Pareto optimality), security levels, and response strategies have all been used in defining solutions ranging from equilibrium points to Pareto saddle points. Methods for obtaining strategies that yield Pareto security levels to the players or Pareto saddle points to the game, when they exist, are presented. Finally, we study games with more than two qualitative outcomes such as combat games. Using the notion of guaranteed outcomes, we obtain saddle-point solutions in mixed strategies for a number of cases. Examples illustrating the concepts, methods, and solutions are included.     

Existence of minmax points in discontinuous strategic games

In this note we prove the existence of minmax points for strategic form games where the sets of strategies are topological spaces and the payoff functions satisfy conditions weaker than continuity. The employed tools are the class of transfer weakly upper continuous functions and the class of weakly lower pseudocontinuous functions. An example shows that our result is of minimal character.     

A new approach toN-person,nonzero-sum,linear differential games

It is shown that there exist equilibrium strategies forn-person, nonero-sum, linear differential games if the cost to each player is convex. The approach used is believed to be novel, and is based on a theorem of Fan.This research was supported by the National Research Council of Canada under Grant No. A-7790.     

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.     

New minimax inequality with applications to existence theorems of equilibrium points

A new minimax inequality is proved on a set which is the union of an increasing sequence of compact convex sets in a topological vector space. As applications, several existence theorems of equilibrium points for different games are obtained.The authors would like to thank the referees for their helpful suggestions.     

多目标博弈平衡点存在性定理的推广

以向量值KyFan不等式的推广为基础,讨论支付函数为向量形式的n人非合作多目标博弈弱Pareto-Nash平衡点存在性条件,将多目标博弈平衡点存在性定理中策略空间的紧性,支付函数的凸性等条件减弱.     

On equilibrium points of diagonaln-person games

In this paper, we compute explicitly the equilibrium points of diagonaln-person games when all of them have the same number of strategies. This number is arbitrary. A wide generalization of two-person games is immediately obtained.The author is grateful to Professor Joel Cohen who visited IMASL during the winter of 1987 and commented on the paper.     

Cone Concavity and Multiple-Payoff Constrained n-Person Games

This paper investigates special cases of abstract economies, i.e., n-person games with multiple payoff functions. Dominances with certain convex cones and interactive strategies are introduced in such game settings. Gradients of payoff functions are involved to establish certain Lagrange or Kuhn–Tucker conditions which may lead to some algorithms to actually compute an equilibrium. Sufficient and necessary conditions for such multiple payoff constrained n-person games are obtained.     

Minimarg and maximarg operators

Two operators on the set ofn-person cooperative games are introduced, the minimarg operator and the maximarg operator. These operators can be seen as dual to each other. Some nice properties of these operators are given, and classes of games for which these operators yield convex (respectively, concave) games are considered. It is shown that, if these operators are applied iteratively on a game, in the limit one will yield a convex game and the other a concave game, and these two games will be dual to each other. Furthermore, it is proved that the convex games are precisely the fixed points of the minimarg operator and that the concave games are precisely the fixed points of the maximarg operator.     

N-person differential games governed by infinite-dimensional systems

This paper discussesN-person differential games governed by infinite-dimensional systems. The minimax principle, which is a necessary condition for the existence of open-loop equilibrium strategies, is proved. For linear-quadraticN-person differential games, global necessary and sufficient conditions for the existence of open-loop and closed-loop equilibrium strategies are derived.This work was supported by the Science Fund of the Chinese Academy of Sciences and the Research Foundation of Purdue University.The problems discussed in this paper were proposed by Professor G. Chen, during the author's visit to Pensylvania State University, and were completed at Purdue University. The author would like to thank Professors L. D. Berkovitz and G. Chen for their hospitality.     

A theorem of the maximin and applications to Bayesian zero-sum games

Consider a family of zero-sum games indexed by a parameter that determines each player’s payoff function and feasible strategies. Our first main result characterizes continuity assumptions on the payoffs and the constraint correspondence such that the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on the parameter. This characterization uses two topologies in order to overcome a topological tension that arises when players’ strategy sets are infinite-dimensional. Our second main result is an application to Bayesian zero-sum games in which each player’s information is viewed as a parameter. We model each player’s information as a sub-σ-field, so that it determines her feasible strategies: those that are measurable with respect to the player’s information. We thereby characterize conditions under which the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on each player’s information.     

Polytope Games

Starting from the definition of a bimatrix game, we restrict the pair of strategy sets jointly, not independently. Thus, we have a set , which is the set of all feasible strategy pairs. We pose the question of whether a Nash equilibrium exists, in that no player can obtain a higher payoff by deviating. We answer this question affirmatively for a very general case, imposing a minimum of conditions on the restricted sets and the payoff. Next, we concentrate on a special class of restricted games, the polytope bimatrix game, where the restrictions are linear and the payoff functions are bilinear. Further, we show how the polytope bimatrix game is a generalization of the bimatrix game. We give an algorithm for solving such a polytope bimatrix game; finally, we discuss refinements to the equilibrium point concept where we generalize results from the theory of bimatrix games.     

Equilibrium points of nonatomic games over a nonreflexive Banach space

Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean n-space are generalized to nonatomic games with stategy sets in a separable Banach space whose dual possesses the Radon-Nikodým property.     

Equilibrium with strategy-dependent trembles

Trembling-hand arguments, introduced by Selten, where players make errors in implementing a strategy, are a successful and intuitively plausible approach to eliminating some equilibria. Typically, these errors are independent of the strategy the player aims to implement. In many contexts, however, when a player aims at one strategy, some errors are more likely than if a different strategy is the target. We introduce the notion of strategy-dependent trembles where each strategy intended by a player implies a different probability vector of errors over other strategies. This seemingly minor but plausible change makes a major difference. The focus becomes a player's own errors rather than the errors of others. A weakly dominated strategy is not necessarily eliminated if less costly errors arise when aiming at it than at the dominating strategy. Almost every Nash equilibrium in pure strategies is a strategy-dependent trembling-hand perfect equilibrium. We thank the associate editor, two referees, James Friedman, and audiences at the Public Choice Society Meetings and Southeastern Economic Theory Meetings for comments and the Public Utility Research Center and the College of Business Administration at the University of Florida for financial support.     

Robust game theory

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our ``robust game' model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call ``robust-optimization equilibrium,' to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results. The research of the author was partially supported by a National Science Foundation Graduate Research Fellowship and by the Singapore-MIT Alliance. The research of the author was partially supported by the Singapore-MIT Alliance.     

EQUILIBIA OF NONCOMPACT GENERALIZED GAMES WITH CORRESPONDENCES WITHOUT OPEN LOWER SECTIONS

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Limit Consistent Solutions in Noncooperative Games

Strong and limit consistency in finite noncooperative games are studied. A solution is called strongly consistent if it is both consistent and conversely consistent (Ref. 1). We provide sufficient conditions on one-person behavior such that a strongly consistent solution is nonempty. We introduce limit consistency for normal form games and extensive form games. Roughly, this means that the solution can be approximated by strongly consistent solutions. We then show that the perfect and proper equilibrium correspondences in normal form games, as well as the weakly perfect and sequential equilibrium correspondences for extensive form games, are limit consistent.     

Equilibrium in semimonotone market games

This paper analyzes the existence of equilibrium for a class of market games in which agents are allowed to follow different patterns of behaviour, including cases where the strategy sets are neither compact nor convex. Agent’s behaviour is modelled in terms of “inverse reply correspondences” (mappings that associate to each agent’s strategy those outcomes that she finds acceptable). Sufficient conditions for an equilibrium to exist are provided     

On constructing games with a convex set of equilibrium strategies

Necessary and sufficient conditions on a convex setC (of strategy pairs) are given for the existence of a 2×n bimatrix game with equilibrium setC. This is done with the use of a geometric-combinatorial solution method for 2×n bimatrix games.

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Zusammenfassung Es werden notwendige und hinreichende Bedingungen an die konvexe MengeC der Strategiepaare für die Existenz eines 2×n Bimatrix Spieles mit GleichgewichtsmengeC aufgegeben. Dies wird durch eine geometrisch-kombinatorische Lösungsmethode für 2×n Bimatrix Spiele erreicht.

     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.