多目标大博弈中弱Pareto-Berge均衡的存在性

研究了具有任意多个局中人的非合作多目标博弈(多目标大博弈).基于一般非合作博弈中的Berge均衡概念,定义多目标大博弈中的弱Pareto-Berge均衡.进一步推广了截口定理,得到新的截口定理,并且利用这个新的截口定理证明多目标大博弈中弱Pareto-Berge均衡的存在性.多目标大博弈中弱Pareto-Nash均衡的存在性结论可作为弱Pareto-Berge均衡存在性的特例给出.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

Oddness of the number of equilibrium points: A new proof

A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.     

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.     

Strictly perfect equilibrium points of bimatrix games

We consider the existence of strictly perfect equilibrium points for bimatrix games. We prove that an isolated and quasi-strong equilibrium point is strictly perfect. Our result shows that in a nondegenerate bimatrix game all equilibrium points are strictly perfect. Our proof is based on the labeling theory ofShapley [1974] for bimatrix games.     

Variational convergence of bivariate functions: lopsided convergence

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, of solutions to inclusions, the primal and dual solutions of convex optimization problems and of zero-sum games. These applications will be dealt with in a couple of accompanying papers.     

A note on polymatrix games

This work is concerned with the class ofn-person games called polymatrix games (Yanovskaya (1968)). The structure of the set of Nash equilibrium points in a polymatrix game is studied and characterizations of these games are given.     

On equilibrium points in bimatrix games

We discuss sensitivity of equilibrium points in bimatrix games depending on small variances (perturbations) of data. Applying implicit function theorem to a linear complementarity problem which is equivalent to the bimatrix game, we investigate sensitivity of equilibrium points with respect to the perturbation of parameters in the game. Namely, we provide the calculation of equilibrium points derivatives with respect to the parameters.     

On the structure of the set of degenerate finite noncooperative games

It is shown that the set of games having infinitely many equilibrium points is a subset of a proper algebraic subvariety within the set of noncooperative games of a particular size. This statement is true for games over an arbitrary order field.     

A remark on the Harsanyi-Selten theory of equilibrium selection

We consider a class of 3-person games in normal form with two pure strategies for each player and two strict equilibrium points. To select one of these two strict equilibrium points as the solution, the equilibrium selection theory of Harsanyi and Selten is applied. The games are constructed in such a way that the a priori probabilities reflect somewhat poorly the risk situation of the players. It is argued and illustrated by examples that this might yield unreasonable results. The a priori probabilities would describe the risk situation of the players more completely if their definition were not based on the expectation of correlated decision behavior.     

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.     

Sufficient conditions for Nash equilibria inN-person games over reflexive Banach spaces

Sufficient conditions are obtained for the existence of Nash equilibrium points inN-person games when the strategy sets are closed, convex subsets of reflexive Banach spaces. These conditions require that each player's cost functional is convex in that player's strategy, weakly continuous in the strategies of the other players, weakly lower semicontinuous in all strategies, and furthermore satisfies a coercivity condition if any of the strategy sets is unbounded. The result is applied to a class of linear-quadratic differential games with no information, to prove that equilibrium points exist when the duration of these games is sufficiently small.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

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.     

Ordered events games on graphs admitting chance moves

The games on finite graphs having chance moves are considered. The preferences are given by the partial ordering at the set of plays. The existence of equilibrium points of these games is proved.     

超级模数博弈的存在性

本文定义了一类在有序Banach空间上的超模博弈,并利用著名的Birkhoff不动点定理证明了有序Banach空间上超模博弈Nash均衡的存在性.     

Equilibrium points of a class of many-person games

This paper deals with the competitive equilibrium points of a class of (n+1)-person games, the players beingn sellers and a single buyer. Each seller offers a price for his product; depending upon the prices offered, the buyer then determines the amount he wishes to buy from each seller. Two types of equilibrium points in this class of games are discussed.The authors are indebted to Professor G. Leitmann for his valuable comments on an earlier version of this paper.     

有限理性与一类多主从博弈问题的良定性

利用Ky Fan不等式证明了一类多主从博弈平衡点的存在性,并且定义了此类多主从博弈的有限理性函数.在非线性问题的良定性的框架下,使用有限理性证明了此类多主从博弈问题是广义Had-amard良定的和广义Tykhonov良定的.     

On the existence of equilibrium points of vector functions

The equilibrium problem concerning vector functions is formulated and some results on the existence of equilibrium points for this problem are proved and applied to vector optimizations , Nash equilibrium in noncooperative games, variational inequalities etc. concerning vector functions     

Inequalities and Nash equilibria

This paper concentrates on the problem of the existence of equilibrium points for non-cooperative generalized N-person games, N-person games of normal form and their related inequalities. We utilize the K-K-M lemma to obtain a theorem and then use it to obtain a new Fan-type inequality and minimax theorems. Various new equilibrium point theorems are derived, with the necessary and sufficient conditions and with strategy spaces with no fixed point property. Examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down.