Zero-sum risk-sensitive stochastic games with unbounded payoff functions and varying discount factors

Abstract

This paper is an attempt of studying zero-sum risk-sensitive stochastic games with unbounded payoff functions and varying discount factors. First, we obtain a logarithm growth condition to guarantee the finiteness of the expected risk-sensitive discounted payoffs, which is an extension of the bounded reward/cost functions in the existing literature. Second, an example with unbounded payoff functions is given, where the solution to the (associated) Shapley equation (SE) no longer has the boundaries as in the existing research. This motivates us to replace the boundaries of the SE with new ones. Third, using the new boundaries and novel arguments, we not only establish the existence of a solution to the SE, but also prove the existence of the value of the game and a Nash equilibrium. Furthermore, we develop an iteration algorithm for computing (at least approximating) the value and Nash equilibria of the games. Finally, we use an inventory system to illustrate our results and show the difference between our conditions and those in the existing articles.