On the characterization of solution sets of smooth and nonsmooth stochastic Nash games

Abstract

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous strategy sets. We examine whether similar statements, articularly pertaining to existence and uniqueness may be made, when player objectives are given by expectations of either smooth or nonsmooth functions. In general, a direct application f deterministic results is difficult since the expectation operation results in a less tractable nonlinear function. Our interest is in developing an analytical framework that only requires the analysis of the integrands of the expectations. Accordingly, in both the smooth and nonsmooth settings, we how that if an appropriate coercivity result holds in an almost-sure fashion, then the existence of an equilibrium to the original stochastic Nash game may be claimed. In the smooth setting, a corresponding sufficiency condition for uniqueness is also provided. We illustrate the utility of our framework by examining a class of stochastic Nash-Cournot games in which nonsmoothness arises from the use of a risk measure.