IDEAS home Printed from https://ideas.repec.org/p/ems/eureri/7809.html
   My bibliography  Save this paper

On Noncooperative Games, Minimax Theorems and Equilibrium Problems

Author

Listed:
  • Frenk, J.B.G.
  • Kassay, G.

Abstract

In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions.

Suggested Citation

  • Frenk, J.B.G. & Kassay, G., 2006. "On Noncooperative Games, Minimax Theorems and Equilibrium Problems," ERIM Report Series Research in Management ERS-2006-022-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    • Handle: RePEc:ems:eureri:7809
    as

    Download full text from publisher

    File URL: https://repub.eur.nl/pub/7809/ERS-2006-022-LIS.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Frenk, J. B. G. & Kassay, G. & Kolumban, J., 2004. "On equivalent results in minimax theory," European Journal of Operational Research, Elsevier, vol. 157(1), pages 46-58, August.
    2. Yang, Z.F., 1996. "Simplicial fixed point algorithms and applications," Other publications TiSEM 60fbb5f7-785c-4c91-8b84-5, Tilburg University, School of Economics and Management.
    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. Frenk, J.B.G. & Kassay, G., 2006. "On noncooperative games, minimax theorems and equilibrium problems," Econometric Institute Research Papers EI 2006-21, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Frenk, J.B.G. & Kassay, G., 2005. "Lagrangian duality and cone convexlike functions," ERIM Report Series Research in Management ERS-2005-019-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    3. Birbil, S.I. & Bouza, G. & Frenk, J.B.G. & Still, G.J., 2003. "Equilibrium Constrained Optimization Problems," Econometric Institute Research Papers ERS-2003-085-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Gaury, E.G.A., 2000. "Designing pull production control systems : Customization and robustness," Other publications TiSEM c9b49495-f647-4583-b2ef-0, Tilburg University, School of Economics and Management.
    5. Gong, X., 2001. "Empirical studies on the labor market and on consumer demand," Other publications TiSEM eed29455-f1bf-4cc3-aff5-c, Tilburg University, School of Economics and Management.
    6. Gerard van der Laan & A.F. Tieman, 1996. "Evolutionary Game Theory and the Modelling of Economic Behavior," Tinbergen Institute Discussion Papers 96-172/8, Tinbergen Institute.
    7. Stremersch, S., 2001. "Essays on marketing strategy in technology-intensive markets," Other publications TiSEM 51d17923-2aae-485b-a59b-5, Tilburg University, School of Economics and Management.
    8. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," ERIM Report Series Research in Management ERS-2004-074-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    9. Birbil, S.I. & Bouza, G. & Frenk, J.B.G. & Still, G.J., 2003. "Equilibrium Constrained Optimization Problems," ERIM Report Series Research in Management ERS-2003-085-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    10. Mingchao Xia & Qingying Lai & Yajiao Zhong & Canbing Li & Hsiao-Dong Chiang, 2016. "Aggregator-Based Interactive Charging Management System for Electric Vehicle Charging," Energies, MDPI, vol. 9(3), pages 1-14, March.
    11. Girndt, T., 2000. "Cultural diversity and work-group performance : Detecting the rules," Other publications TiSEM 79060da1-f8e7-45e5-bb6b-8, Tilburg University, School of Economics and Management.
    12. Fernando Luque-Vásquez & J. Adolfo Minjárez-Sosa & Max E. Mitre-Báez, 2016. "A Note on König and Close Convexity in Minimax Theorems," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 65-71, July.
    13. Onderstal, A.M., 2002. "Papers in auction theory," Other publications TiSEM 7d84e23f-2bb6-4e3c-a689-5, Tilburg University, School of Economics and Management.
    14. van Lomwel, A.G.C., 2000. "Essays on labour economics," Other publications TiSEM 957b3422-61d0-461d-9d22-e, Tilburg University, School of Economics and Management.
    15. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," Econometric Institute Research Papers ERS-2004-074-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    16. Possajennikov, A., 2000. "Learning and evolution in games and oligopoly models," Other publications TiSEM be1a3e81-e186-46b5-9101-3, Tilburg University, School of Economics and Management.
    17. van den Broek, W.A., 2001. "Uncertainty in differential games," Other publications TiSEM 195bcb68-8943-49c1-8acb-0, Tilburg University, School of Economics and Management.
    18. Tomás Prieto-Rumeau & José Lorenzo, 2015. "Approximation of zero-sum continuous-time Markov games under the discounted payoff criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 799-836, October.
    19. H. Boualam & A. Roubi, 2019. "Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs," Journal of Global Optimization, Springer, vol. 74(2), pages 255-284, June.
    20. Camlibel, M.K., 2001. "Complementarity methods in the analysis of piecewise linear dynamical systems," Other publications TiSEM c3e08484-56d2-4f1a-9db0-8, Tilburg University, School of Economics and Management.

    More about this item

    Keywords

    Equilibrium Problems; KKM Lemma; Minimax Theorems; Nash Equilibrium Point; Non-Cooperative Game Theory;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • M - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics
    • M11 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Administration - - - Production Management
    • R4 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - Transportation Economics

    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:ems:eureri:7809. 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: RePub (email available below). General contact details of provider: https://edirc.repec.org/data/erimanl.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.