Bias and Overtaking Equilibria for Zero-Sum Stochastic Differential Games

Abstract

This paper deals with zero-sum stochastic differential games with long-run average payoffs. Our main objective is to give conditions for existence and characterization of bias and overtaking optimal equilibria. To this end, first we characterize the family of optimal average payoff strategies. Then, within this family, we impose suitable conditions to determine the subfamilies of bias and overtaking equilibria. A key step to obtain these facts is to show the existence of solutions to the average payoff optimality equations. This is done by the usual "vanishing discount" approach. Finally, a zero-sum game associated to a certain manufacturing process illustrates our results.

Policy Iteration Algorithms for Zero-Sum Stochastic Differential Games with Long-Run Average Payoff Criteria

“…As for Lemma 4.1 and Theorem 4.3, we acknowledge that they are inspired in [19,Lemma 4.5] (see also [45,Proposition 4.1]) and [19,Theorem 4.3] (see also Theorem 4.2 in [45]), respectively. This work represents as well a continuation of our study of bias games (presented in [12]), in the sense that it provides a way of finding bias equilibria in terms of an ergodic game.…”

“…Now we will give an example to illustrate our results. This is an extension of the example presented in [12] which in turn was motivated by the manufacturing system in [1].…”

“…Proof A slight variation of the vanishing discount technique (see, for instance the Appendix of [12], or [27,Chaps. 5,6]) gives us that, for fixed p 1 ; p 2 ð Þ2P 1 Â P 2 , the Poisson equation (3.5) has a solutionh p 1 ;p 2 , which is a member of…”

“…We offer a complete proof based on the vanishing discount technique for control problems (cf. [4,30]) in [12,Theorem 4.1]. See also […”

“…This paper is about two-person zero-sum stochastic differential games (SDGs) with ergodic payoffs on an infinite horizon. This sort of games has been studied, for instance in [10,12,26]. Our aim is to give conditions under which the policy iteration algorithm (PIA) produces convergent sequences of values and policies for an SDG with ergodic payoffs.…”