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Stochastic Nash Equilibrium Problems: Sample Average Approximation and Applications

Huifu Xu (h.xu***at***soton.ac.uk)
Zhang Dali (zhangdl***at***soton.ac.uk)

Abstract: This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well known sample average approximation method 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 (a Nash equilibrium or a Nash-C-stationary point) obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the Clarke generalized derivatives, it is shown that with probability approaching one exponentially fast by increasing sample size, the Nash-C-stationary point converges to a weak Nash-C-stationary point of the true problem. Finally, the model is applied to a stochastic Nash equilibrium problem in the electricity market.

Keywords: Stochastic Nash equilibrium, exponential convergence, H-calmness, Nash-C-stationary point.

Category 1: Stochastic Programming

Category 2: Other Topics (Game Theory )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 05/22/2009
Entry Accepted: 05/22/2009
Entry Last Modified: 01/22/2010

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