Single-Controller Chance-Constrained Stochastic Games

Strategic decision-making problems involving multiple players competing over an infinite horizon have been extensively studied in the literature using a stochastic game framework. It assumes that the model parameters, namely, running costs and transition probabilities are exactly known by ignoring the fact that, in reality, these parameters are estimated or learned from historical data. In this paper, we consider a two-player single-controller discounted stochastic game where the dynamics defined by transition probabilities of a Markov chain depends only on the actions of player 2. We consider the case where the running costs are defined by the random variables and the transition probabilities are exactly known. The players are risk-averse and are interested in expected discounted costs incurred with a given confidence level. To model this game theoretic situation, we formulate the expected discounted costs using chance constraints and call the game a single-controller chance-constrained stochastic game. When the running cost vector of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium in the class of stationary strategies. To compute the Nash equilibria, we propose an equivalent mathematical program and a system of nonlinear constraints whose global maximum and feasible solution, respectively, give a Nash equilibrium of the game. We illustrate our theoretical results by considering randomly generated instances.