IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/344.html
   My bibliography  Save this paper

Perfection and stability of stationary points with applications to noncooperative games

Author

Listed:
  • van der Laan, Gerard

    (Center for Mathematical Economics, Bielefeld University)

  • Talman, Dolf

    (Center for Mathematical Economics, Bielefeld University)

  • Yang, Zaifu

    (Center for Mathematical Economics, Bielefeld University)

Abstract

This discussion paper resulted in a publication in the 'SIAM Journal on Optimization', 2006, 16, 854-870. It is well known that an upper semi-continuous compact- and convex-valued mapping fi from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image fi(x) has a nonempty intersection with the normal cone at x. In many circumstances there may be more than one stationary point. In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept. In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stabIe with respect to this sequenceof sets and the mapping which defines the perturbed solution. It is shown that stable stationary points exist for a large class of perturbations. A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X. It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory. We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept. Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game. In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium. Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics. We show that the refined solution always exists, contrary to many weIl known refinement concepts in the field that may fail
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2017. "Perfection and stability of stationary points with applications to noncooperative games," Center for Mathematical Economics Working Papers 344, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:344
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/2909893/2909995
    File Function: First Version, 2002
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
    2. Dai, Y. & van der Laan, G. & Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron," Other publications TiSEM 82992276-1868-4b56-a937-0, Tilburg University, School of Economics and Management.
    3. Yamamoto, Yoshitsugu, 1993. "A Path-Following Procedure to Find a Proper Equilibrium of Finite Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(3), pages 249-259.
    4. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    5. A. J. J. Talman & Y. Yamamoto, 1989. "A Simplicial Algorithm for Stationary Point Problems on Polytopes," Mathematics of Operations Research, INFORMS, vol. 14(3), pages 383-399, August.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    8. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    9. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    Full references (including those not matched with items on IDEAS)

    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. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    2. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    3. Talman, A.J.J. & Yamamoto, M., 2001. "Contiuum of Zero Points of a Mapping on a Compact Convex Set," Other publications TiSEM 57411440-5b14-448e-8c27-3, Tilburg University, School of Economics and Management.
    4. Zhiqiang Zheng & Balaji Padmanabhan & Steven O. Kimbrough, 2003. "On the Existence and Significance of Data Preprocessing Biases in Web-Usage Mining," INFORMS Journal on Computing, INFORMS, vol. 15(2), pages 148-170, May.
    5. Carlos R. Handy & Daniel Vrinceanu & Carl B. Marth & Harold A. Brooks, 2015. "Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure," Mathematics, MDPI, vol. 3(4), pages 1-24, November.
    6. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    7. Bode, Sven & Michaelowa, Axel, 2003. "Avoiding perverse effects of baseline and investment additionality determination in the case of renewable energy projects," Energy Policy, Elsevier, vol. 31(6), pages 505-517, May.
    8. Ala, Guido & Fasshauer, Gregory E. & Francomano, Elisa & Ganci, Salvatore & McCourt, Michael J., 2017. "An augmented MFS approach for brain activity reconstruction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 3-15.
    9. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    10. Haider A. Khan, 2004. "General Conclusions: From Crisis to a Global Political Economy of Freedom," Palgrave Macmillan Books, in: Global Markets and Financial Crises in Asia, chapter 9, pages 193-211, Palgrave Macmillan.
    11. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    12. Broekhuis, Manda & Vos, Janita F.J., 2003. "Improving organizational sustainability using a quality perspective," Research Report 03A43, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    13. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    14. Edcarlos D. Silva & J. C. Albuquerque & T. R. Cavalcante, 2021. "Fourth-order nonlocal type elliptic problems with indefinite nonlinearities," Partial Differential Equations and Applications, Springer, vol. 2(2), pages 1-22, April.
    15. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    16. Montijano, J.I. & Rández, L. & Van Daele, M. & Calvo, M., 2020. "On the numerical stability of the exponentially fitted methods for first order IVPs," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    17. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    18. Jensen, Nathan M. & Li, Quan & Rahman, Aminur, 2007. "Heard melodies are sweet, but those unheard are sweeter : understanding corruption using cross-national firm-level surveys," Policy Research Working Paper Series 4413, The World Bank.
    19. Oscar J. Cacho & Robyn L. Hean & Russell M. Wise, 2003. "Carbon‐accounting methods and reforestation incentives," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 47(2), pages 153-179, June.
    20. Walter M. Cadette, 1999. "Financing Long-Term Care: Options for Policy," Economics Working Paper Archive wp_283, Levy Economics Institute.

    More about this item

    Keywords

    Kritischer Punkt ; Stabilität; Nichtkooperatives Spiel;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    Statistics

    Access and download statistics

    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:bie:wpaper:344. 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: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    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.