Two players engage in a repeated game with incomplete information on one side, where the underlying stage-games are zero-sum. In the case where players evaluate their stage-payoffs by using different discount factors, the payoffs of the infinitely repeated game are typically non zero-sum. However, if players grow infinitely patient, then the equilibrium payoffs will sometimes approach the zero-...

In the previous lecture we saw that there always exists a Nash equilibrium in two-player zero-sum games. Moreover, the equilibrium enjoys several attractive properties such as polynomial-time tractability, convexity of the equilibrium set, and uniqueness of players’ payoffs in all equilibria. In this lecture we explore whether we can generalize this theorem to multi-player zero-sum games. Befor...

Zero-sum stochastic games provide a formalism to study competitive sequential interactions between two agents with diametrically opposing goals and evolving state. A solution to such games with discrete state was presented by Littman (Littman, 1994). The continuous state version of this game remains unsolved. In many instances continuous state solutions require nonlinear optimisation, a problem...

We show 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. We point out that the class of symmetric two-player zero-sum games coincides with ...

These are two person non-zero(constant)-sum games in which each player has finitely many pure strategies. We call these non-zero(constant)sum games because the interests of the players is not required to be exactly opposed to each other. Of course, these include zero(constant)-sum games and are a true generalization of zero(constant)-sum games. But the methods used to analyze them are different...

W present a diagrammatic method that allows the determination of all Nash equilibria of 2×M nonzero sum games, extending thus the well known diagrammatic techniques for 2 × M zero sum and 2 × 2 nonzero sum games. We show its appropriateness for teaching purposes by analyzing modified versions of the prisoners’ dilemma, the battle of the sexes, as well as of the zero sum game of matching pennies...

We study optimal stochastic control problems under model uncertainty. We rewrite such problems as (zero-sum) stochastic differential games of forward-backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the non-zero sum games (finding conditions for Nash equilibria). We...

While the idea of a matching pennies game may seem contrived, it is merely the simplest example of a general class of zero-sum games, where the total payoff of the players is constant regardless of the outcome. Consequently gains for one player can only come from losses of the other. For this reason, zero-sum games will rarely have a pure strategy Nash equilibrium. Examples would be chess, or m...