Zero-sum Risk-Sensitive Stochastic Games

TL;DR

This paper investigates two-player zero-sum stochastic games with risk-sensitive criteria, establishing the existence of game values and optimal strategies across various reward frameworks, extending previous results and addressing open questions.

Contribution

It generalizes prior work by proving the existence of game values and optimal strategies in risk-sensitive stochastic games under broad conditions, including ergodic cases.

Findings

Existence of game value and optimal strategies in risk-sensitive settings.

Solution of the Shapley and Poisson equations for the game.

Extension of previous results and resolution of open questions.

Abstract

In this paper we consider two-person zero-sum risk-sensitive stochastic dynamic games with Borel state and action spaces and bounded reward. The term risk-sensitive refers to the fact that instead of the usual risk neutral optimization criterion we consider the exponential certainty equivalent. The discounted reward case on a finite and an infinite time horizon is considered, as well as the ergodic reward case. Under continuity and compactness conditions we prove that the value of the game exists and solves the Shapley equation and we show the existence of optimal (non-stationary) strategies. In the ergodic reward case we work with a local minorization property and a Lyapunov condition and show that the value of the game solves the Poisson equation. Moreover, we prove the existence of optimal stationary strategies. A simple example highlights the influence of the risk-sensitivity parameter. Our results generalize findings in Basu/Ghosh 2014 and answer an open question posed there.