Discrete-time Zero-Sum Games for Markov chains with risk-sensitive average cost criterion

TL;DR

This paper investigates zero-sum stochastic games for controlled Markov chains with a risk-sensitive average cost, establishing the existence of a value and saddle point equilibria under stability conditions, and characterizing optimal strategies.

Contribution

It introduces a framework for risk-sensitive zero-sum games on countable state spaces, proving existence of solutions and characterizing stationary saddle point strategies.

Findings

Existence of a value and saddle point equilibrium under Lyapunov stability.

Complete characterization of stationary saddle point strategies.

Analysis of an illustrative example demonstrating the theoretical results.

Abstract

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded. Under a certain Lyapunov stability assumption on the dynamics, we establish the existence of a value and saddle point equilibrium. Further we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Finally, we present and analyze an illustrative example.