Nonzero-sum risk-sensitive stochastic games on a countable state space

Abstract

The infinite horizon risk-sensitive discounted-cost and ergodic-cost nonzero-sum stochastic games for controlled Markov chains with countably many states are analyzed. For the discounted-cost game, we prove the existence of Nash equilibrium strategies in the class of Markov strategies under fairly general conditions. Under an additional weak geometric ergodicity condition and a small cost criterion, the existence of Nash equilibrium strategies in the class of stationary Markov strategies is proved for the ergodic-cost game. The key nontrivial contributions in the ergodic part are to prove the existence of a particular form of a (relative) value function solution to a player’s Bellman equation and the continuity of this solution with respect to the opponent’s strategies.