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
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Zhang, Dali
e6ceaf3b-e99f-45f9-b302-2159f9315810
22 May 2009
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.
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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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
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Date deposited: 27 Apr 2011 13:26
Last modified: 15 Mar 2024 03:15
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Author:
Huifu Xu
Author:
Dali Zhang
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