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Stochastic nash equilibrium problems: sample average approximation and applications

Stochastic nash equilibrium problems: sample average approximation and applications
Stochastic nash equilibrium problems: sample average approximation and applications
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.

stochastic nash equilibrium, exponential convergence, h-calmness, nash-c-stationary point
0233-1934
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Zhang, Dali
e6ceaf3b-e99f-45f9-b302-2159f9315810
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Zhang, Dali
e6ceaf3b-e99f-45f9-b302-2159f9315810

Xu, Huifu and Zhang, Dali (2009) Stochastic nash equilibrium problems: sample average approximation and applications. Optimization.

Record type: Article

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.

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More information

Published date: 22 May 2009
Keywords: stochastic nash equilibrium, exponential convergence, h-calmness, nash-c-stationary point
Organisations: Operational Research

Identifiers

Local EPrints ID: 182197
URI: http://eprints.soton.ac.uk/id/eprint/182197
ISSN: 0233-1934
PURE UUID: 502bd062-413f-4948-981d-f5bda2209cea
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

Catalogue record

Date deposited: 27 Apr 2011 13:26
Last modified: 11 Dec 2021 03:51

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Contributors

Author: Huifu Xu ORCID iD
Author: Dali Zhang

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