Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

In Pang and Fukushima (Comput Manage Sci 2:21–56, 2005), a sequential penalty approach was presented for a quasi-variational inequality (QVI) with particular application to the generalized Nash game. To test the computational performance of the penalty method, numerical results were reported with an example from a multi-leader-follower game in an electric power market. However, due to an inverted sign in the penalty term in the example and some missing terms in the derivatives of the firms’ Lagrangian functions, the reported numerical results in Pang and Fukushima (Comput Manage Sci 2:21–56, 2005) are incorrect. Since the numerical examples of this kind are scarce in the literature and this particular example may be useful in the future research, we report the corrected results. The online version of the original article can be found under doi:.     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Directed graphical structure,Nash equilibrium,and potential games

This paper considers the directed graphical structure of a game, called influence structure, where a directed edge from player $i$ to player $j$ indicates that player $i$ may be able to affect $j$’s payoff via his unilateral change of strategies. We give a necessary and sufficient condition for the existence of pure-strategy Nash equilibrium of games having a directed graph in terms of the structure of that graph. We also discuss the relationship between the structure of graphs and potential games.     

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.     

On Nash equilibria inN-person games over reflexive Banach spaces

In this note, an extended version of the Ky Fan inequality is obtained for reflexive Banach spaces. As an application of the result, it is proved that Nash equilibrium points exist under weaker conditions than those of Williams (Ref. 1).     

可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

Informational uniqueness of closed-loop Nash equilibria for a class of nonstandard dynamic games

This paper discusses an extension of the currently available theory of noncooperative dynamic games to game models whose state equations are of order higher than one. In a discrete-time framework, it first elucidates the reasons why the theory developed for first-order systems is not applicable to higher-order systems, and then presents a general procedure to obtain an informationally unique Nash equilibrium solution in the presence of random disturbances. A numerical example solved in the paper illustrates the general approach.Dedicated to G. LeitmannResearch that led to this paper was supported in part by the Office of Naval Research under Contract No N00014-82-K-0469 and in part by the U.S. Air Force under Grant No. AFOSR-84-0054.     

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

Nash equilibria of Cauchy-random zero-sum and coordination matrix games

We consider zero-sum games (A,  − A) and coordination games (A,A), where A is an m-by-n matrix with entries chosen independently with respect to the Cauchy distribution. In each case, we give an exact formula for the expected number of Nash equilibria with a given support size and payoffs in a given range, and also asymptotic simplications for matrices of a fixed shape and increasing size. We carefully compare our results with recent results of McLennan and Berg on Gaussian random bimatrix games (A,B), and describe how the three situations together shed light on random bimatrix games in general.     

On purification of equilibrium in Bayesian games and expost Nash equilibrium

Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also expost stable. Through examples we demonstrate the ‘bounds obtained on the approximation’ are tight. The main results of this paper first appeared in University of Warwick Department of Economics Discussion Paper #710. 2004.     

State transformations and the derivation of Nash closed-loop equilibria for non-zero-sum differential games

In this paper the usefulness of state transformations in differential games is demonstrated. It is shown that different (admissible) state transformations give rise to different closed-loop Nash equilibrium candidates, which may all be found by solving systems of ordinary differential equations. A linear-quadratic duopoly differential game is solved to illustrate the results.     

Bounding the inefficiency of Nash equilibria in games with finitely many players

This paper derives bounds on the gap between optimal performance and the performance of Nash equilibria in n-person games with continuous action sets. Specific interesting expressions are obtained for the average efficiency per player in congestion games.     

A class of differential games for which the closed-loop and open-loop Nash equilibria coincide

It is well known that, in general, Nash equilibria in open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate, in the sense that it depends on time only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies.The helpful comments of Professors Y. C. Ho, G. Leitmann, H. Y. Wan, Jr., and an anonymous referee are gratefully acknowledged.     

Pure strategy Nash equilibria and the probabilistic prospects of Stackelberg players

We consider the set of all m×n bimatrix games with ordinal payoffs. We show that on the subset E of such games possessing at least one pure strategy Nash equilibrium, both players prefer the role of leader to that of follower in the corresponding Stackelberg games. This preference is in the sense of first-degree stochastic dominance by leader payoffs of follower payoffs. It follows easily that on the complement of E, the follower’s role is preferred in the same sense. Thus we see a tendency for leadership preference to obtain in the presence of multiple pure strategy Nash equilibria in the underlying game.     

Topological degree and number of Nash equilibrium points of bimatrix games

In this paper, using the topological degree, we give a new proof of a well-known result: the number of Nash equilibrium points of a nondegenerate bimatrix game is odd. The calculation of the topological degree allows the localization of the whole set of non-degenerate equilibrium points.     

Information patterns and Nash equilibria in extensive games — II

In Part I of this paper we introduced extensive games with a non-atomic continuum of players. It was shown that the Nash plays (outcomes) are invariant of the information patterns on the game, provided that no player's unilateral change in moves can be observed by others. This led to an enormous reduction in the Nash plays of the these games, as exemplified in the anti-folk theorem. Our concern in this sequel is to develop a finite version of these results.     

Imitation,coordination and the emergence of Nash equilibrium

We consider a learning dynamic in which players imitate and better reply. Sufficient conditions are provided for Nash equilibrium play to emerge over time. The role of imitation in the learning dynamic is discussed through a series of examples. Most interestingly we demonstrate how imitation can ‘help’ the emergence of Nash equilibrium where ‘more rational’ methods do not.     

α-Well-posedness for Nash Equilibria and For Optimization Problems with Nash Equilibrium Constraints

We present the concepts of α-well-posedness for parametric noncooperative games and for optimization problems with constraints defined by parametric Nash equilibria. We investigate some classes of functions that ensure these types of well-posedness and the connections with α-well-posedness for variational inequalities and optimization problems with variational inequality constraints.