Marginal Values of a Stochastic Game

Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general. Funding: This work was supported by Fondation CFM pour la Recherche; the European Research Council Grant ERC-CoG-863818 (ForM-SMArt); and Agence Nationale de la Recherche Grant ANR-21-CE40-0020.