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.     

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game. This research was funded in part by National Science Foundation grants DMI-0545910 and ECCS-0621922 and AFOSR MURI subaward 2003-07688-1.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

Values for strategic games in which players cooperate

In this paper we propose a new method to associate a coalitional game with each strategic game. The method is based on the lower value of finite two-player zero-sum games. We axiomatically characterize this new method, as well as the method that was described in Von Neumann and Morgenstern (1944). As an intermediate step, we provide axiomatic characterizations of the value and the lower value of matrix games and finite two-player zero-sum games, respectively.The authors acknowledge the financial support of Ministerio de Ciencia y Tecnologia, FEDER andXunta de Galicia through projects BEC2002-04102-C02-02 and PGIDIT03PXIC20701PN.We wish to thank Professor William Thomson as well as an anonymous referee for useful comments.     

Sampled-Data Nash Equilibria in Differential Games with Impulse Controls

Journal of Optimization Theory and Applications - We study a class of deterministic two-player nonzero-sum differential games where one player uses piecewise-continuous controls to affect the...     

Weighted Shapley hierarchy levels values

A new class of values for cooperative games with level structure is introduced. We apply a multi-step proceeding to the weighted Shapley values. For characterization, two well-known axiomatizations of the weighted Shapley values are extended, the first one by efficiency and weighted balanced contributions and the second one by weighted standardness for two-player games and consistency. We get a new axiomatization of the Shapley levels value too.     

Algorithms for uniform optimal strategies in two-player zero-sum stochastic games with perfect information

We deal with zero-sum two-player stochastic games with perfect information. We propose two algorithms to find the uniform optimal strategies and one method to compute the optimality range of discount factors. We prove the convergence in finite time for one algorithm. The uniform optimal strategies are also optimal for the long run average criterion and, in transient games, for the undiscounted criterion as well.     

When is tit-for-tat unbeatable?

We characterize the class of symmetric two-player games in which tit-for-tat cannot be beaten even by very sophisticated opponents in a repeated game. It turns out to be the class of exact potential games. More generally, there is a class of simple imitation rules that includes tit-for-tat but also imitate-the-best and imitate-if-better. Every decision rule in this class is essentially unbeatable in exact potential games. Our results apply to many interesting games including all symmetric 2 $\times $ 2 games, and standard examples of Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.     

Two-player stochastic games II: The case of recursive games

This paper contains the second step in the proof of existence of equilibrium payoffs for two-player stochastic games. It deals with the case of positive absorbing recursive games     

On Nash equilibrium points and games of imperfect information

This paper considers a class of two-player, nonzero-sum games in which the players have only local, as opposed to global, information about the payoff functions. We study various modes of behavior and their relationship to different stability properties of the Nash equilibrium points.