The University of Southampton
University of Southampton Institutional Repository

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 stochastic Nash equilibrium problem in the wholesale electricity market.
stochastic nash equilibrium, exponential convergence, h-calmness, nash-c-stationary point
0926-6003
597–645
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 (2013) Stochastic nash equilibrium problems: sample average approximation and applications. Computational Optimization and Applications, 55 (3), 597–645. (doi:10.1007/s10589-013-9538-7).

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 stochastic Nash equilibrium problem in the wholesale electricity market.

Text
Nash-21-jan.pdf - Accepted Manuscript
Download (706kB)

More information

e-pub ahead of print date: 22 February 2013
Published date: July 2013
Keywords: stochastic nash equilibrium, exponential convergence, h-calmness, nash-c-stationary point
Organisations: Operational Research, Mathematics

Identifiers

Local EPrints ID: 145459
URI: http://eprints.soton.ac.uk/id/eprint/145459
ISSN: 0926-6003
PURE UUID: 87d288eb-ba09-4a96-9edd-6f43dd5124af
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

Catalogue record

Date deposited: 20 Apr 2010 14:13
Last modified: 09 Jan 2022 03:12

Export record

Altmetrics

Contributors

Author: Huifu Xu ORCID iD
Author: Dali Zhang

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×