Zero-sum stochastic games with the average-value-at-risk criterion

Abstract

<p>This paper introduces an average-value-at-risk (AVaR) criterion for discrete-time zero-sum stochastic games with varying discount factors. The state space is a Borel space, the action space is denumerable, and the payoff function is allowed to be unbounded. We first transform the AVaR game problem into a bi-level optimization-game problem in which the outer optimization problem is a problem of minimizing a function of a single variable and the inner game problem has been shown to be equivalent to a so-called expected-discounted-positive-deviation (EDPD) game for discrete-time stochastic game. We solve the EDPD game problem in advance. More precisely, under suitable conditions, we not only establish the Shapley equation, the existence of the value of the game, and saddle points, but also prove that the saddle points can be computed by introducing a primal linear program and a dual linear program. Then, we show that the outer problem can be settled by solving the EDPD game problem. Furthermore, we provide an algorithm for computing (or at least approximating) the value of the game and the saddle points for the AVaR game problem. Finally, as an application, we apply our main results to an inventory-production system with numerical experiments.</p>