IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v55y2013i3p597-645.html
   My bibliography  Save this article

Stochastic Nash equilibrium problems: sample average approximation and applications

Author

Listed:
  • Huifu Xu
  • Dali Zhang

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. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Huifu Xu & Dali Zhang, 2013. "Stochastic Nash equilibrium problems: sample average approximation and applications," Computational Optimization and Applications, Springer, vol. 55(3), pages 597-645, July.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:597-645
    DOI: 10.1007/s10589-013-9538-7
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-013-9538-7
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-013-9538-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hanif D. Sherali & Allen L. Soyster & Frederic H. Murphy, 1983. "Stackelberg-Nash-Cournot Equilibria: Characterizations and Computations," Operations Research, INFORMS, vol. 31(2), pages 253-276, April.
    2. Joshua S. Gans & Danny Price & Kim Woods, 1998. "Contracts and Electricity Pool Prices," Australian Journal of Management, Australian School of Business, vol. 23(1), pages 83-96, June.
    3. WEI, Jing-Yuan & SMEERS, Yves, 1999. "Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices," LIDAM Reprints CORE 1454, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Papageorgiou, Nikolaos S., 1985. "On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 17(2), pages 185-206, October.
    5. Aumann, Robert J, 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality," Econometrica, Econometric Society, vol. 55(1), pages 1-18, January.
    6. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    7. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    8. Huifu Xu & Fanwen Meng, 2007. "Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 648-668, August.
    9. Jian Yao & Ilan Adler & Shmuel S. Oren, 2008. "Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network," Operations Research, INFORMS, vol. 56(1), pages 34-47, February.
    10. Watling, David, 2006. "User equilibrium traffic network assignment with stochastic travel times and late arrival penalty," European Journal of Operational Research, Elsevier, vol. 175(3), pages 1539-1556, December.
    11. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.
    12. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
    13. Yao, Jian & Oren, Shmuel S. & Adler, Ilan, 2007. "Two-settlement electricity markets with price caps and Cournot generation firms," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1279-1296, September.
    14. Xinmin Hu & Daniel Ralph, 2007. "Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices," Operations Research, INFORMS, vol. 55(5), pages 809-827, October.
    15. Jeff Linderoth & Alexander Shapiro & Stephen Wright, 2006. "The empirical behavior of sampling methods for stochastic programming," Annals of Operations Research, Springer, vol. 142(1), pages 215-241, February.
    16. John C. Harsanyi, 1967. "Games with Incomplete Information Played by "Bayesian" Players, I-III Part I. The Basic Model," Management Science, INFORMS, vol. 14(3), pages 159-182, November.
    17. L. Dai & C. H. Chen & J. R. Birge, 2000. "Convergence Properties of Two-Stage Stochastic Programming," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 489-509, September.
    18. D. Ralph & H. Xu, 2005. "Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 673-699, March.
    19. Dali Zhang & Huifu Xu & Yue Wu, 2010. "A two stage stochastic equilibrium model for electricity markets with two way contracts," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 1-45, February.
    20. Klemperer, Paul D & Meyer, Margaret A, 1989. "Supply Function Equilibria in Oligopoly under Uncertainty," Econometrica, Econometric Society, vol. 57(6), pages 1243-1277, November.
    21. Huifu Xu, 2010. "Sample Average Approximation Methods For A Class Of Stochastic Variational Inequality Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 27(01), pages 103-119.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. David Pozo & Enzo Sauma & Javier Contreras, 2017. "Basic theoretical foundations and insights on bilevel models and their applications to power systems," Annals of Operations Research, Springer, vol. 254(1), pages 303-334, July.
    2. Wang, Zheng & Li, Chaojie & Zhou, Xiaojun & Xie, Renyou & Li, Xiangyu & Dong, Zhaoyang, 2023. "Stochastic bidding for VPPs enabled ancillary services: A case study," Applied Energy, Elsevier, vol. 352(C).
    3. Vikas Vikram Singh & Abdel Lisser, 2018. "A Characterization of Nash Equilibrium for the Games with Random Payoffs," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 998-1013, September.
    4. Liu, Yongchao & Xu, Huifu & Yang, Shu-Jung Sunny & Zhang, Jin, 2018. "Distributionally robust equilibrium for continuous games: Nash and Stackelberg models," European Journal of Operational Research, Elsevier, vol. 265(2), pages 631-643.
    5. Vikas Vikram Singh & Abdel Lisser, 2018. "Variational inequality formulation for the games with random payoffs," Journal of Global Optimization, Springer, vol. 72(4), pages 743-760, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dali Zhang & Huifu Xu & Yue Wu, 2010. "A two stage stochastic equilibrium model for electricity markets with two way contracts," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 1-45, February.
    2. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.
    3. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    4. Desmond Cai & Anish Agarwal & Adam Wierman, 2020. "On the Inefficiency of Forward Markets in Leader–Follower Competition," Operations Research, INFORMS, vol. 68(1), pages 35-52, January.
    5. Tsimopoulos, Evangelos G. & Georgiadis, Michael C., 2021. "Nash equilibria in electricity pool markets with large-scale wind power integration," Energy, Elsevier, vol. 228(C).
    6. Pär Holmberg & Andy Philpott, 2014. "Supply function equilibria in transportation networks," Cambridge Working Papers in Economics 1421, Faculty of Economics, University of Cambridge.
    7. Jörg Fliege & Huifu Xu, 2011. "Stochastic Multiobjective Optimization: Sample Average Approximation and Applications," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 135-162, October.
    8. Dongyan Chen & Chunying Tian & Zhaobo Chen & Ding Zhang, 2022. "Competition among supply chains: the choice of financing strategy," Operational Research, Springer, vol. 22(2), pages 977-1000, April.
    9. Holmberg, Pär & Philpott, Andrew, 2012. "Supply Function Equilibria in Networks with Transport Constraints," Working Paper Series 945, Research Institute of Industrial Economics, revised 10 Aug 2015.
    10. Uday V. Shanbhag & Gerd Infanger & Peter W. Glynn, 2011. "A Complementarity Framework for Forward Contracting Under Uncertainty," Operations Research, INFORMS, vol. 59(4), pages 810-834, August.
    11. David Pozo & Enzo Sauma & Javier Contreras, 2017. "Basic theoretical foundations and insights on bilevel models and their applications to power systems," Annals of Operations Research, Springer, vol. 254(1), pages 303-334, July.
    12. Andreas Ehrenmann & Karsten Neuhoff, 2009. "A Comparison of Electricity Market Designs in Networks," Operations Research, INFORMS, vol. 57(2), pages 274-286, April.
    13. Javad Khazaei & Golbon Zakeri & Shmuel S. Oren, 2017. "Single and Multisettlement Approaches to Market Clearing Under Demand Uncertainty," Operations Research, INFORMS, vol. 65(5), pages 1147-1164, October.
    14. Julien, Ludovic A., 2017. "On noncooperative oligopoly equilibrium in the multiple leader–follower game," European Journal of Operational Research, Elsevier, vol. 256(2), pages 650-662.
    15. Olufolajimi Oke & Daniel Huppmann & Max Marshall & Ricky Poulton & Sauleh Siddiqui, 2019. "Multimodal Transportation Flows in Energy Networks with an Application to Crude Oil Markets," Networks and Spatial Economics, Springer, vol. 19(2), pages 521-555, June.
    16. Elisabetta Allevi & Didier Aussel & Rossana Riccardi, 2018. "On an equilibrium problem with complementarity constraints formulation of pay-as-clear electricity market with demand elasticity," Journal of Global Optimization, Springer, vol. 70(2), pages 329-346, February.
    17. Ming Hu & Masao Fukushima, 2011. "Variational Inequality Formulation of a Class of Multi-Leader-Follower Games," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 455-473, December.
    18. Gürkan, G. & Ozdemir, O. & Smeers, Y., 2013. "Strategic Generation Capacity Choice under Demand Uncertainty : Analysis of Nash Equilibria in Electricity Markets," Discussion Paper 2013-044, Tilburg University, Center for Economic Research.
    19. Hailin Sun & Huifu Xu & Yong Wang, 2014. "Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 257-284, April.
    20. David P. Brown & David E. M. Sappington, 2022. "Vertical integration and capacity investment in the electricity sector," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 31(1), pages 193-226, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:597-645. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.