Stochastic Nash Equilibrium Problems: Sample Average Approximation and Applications

Abstract

This paper presents a Nash equilibrium model where the underlying objective functionsinvolve uncertainties and nonsmoothness. The well known sample average approximationmethod is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator obtained from sample average approximateequilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the generalized gradients and metric regularity of the set-valued mappings whichcharacterize the first order equilibrium conditions, it is shown that with probability approaching one exponentially fast with the increase of sample size, the statistical estimatorconverge to its true counterparts. Finally, the model is applied to an equilibrium problem inelectricity market.