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The equivalence of mini-max theorem and existence of Nash equilibrium in asymmetric three-players zero-sum game with two groups

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  • Satoh, Atsuhiro
  • Tanaka, Yasuhito

Abstract

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric three-players zero-sum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then, 1. The existence of a Nash equilibrium, which is symmetric in Group A, implies Sion's minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A. 2. Sion's minimax theorem for pairs of a player in Group A and Player C with symmetry in Group A implies the existence of a Nash equilibrium which is symmetric in Group A. Thus, they are equivalent.

Suggested Citation

  • Satoh, Atsuhiro & Tanaka, Yasuhito, 2018. "The equivalence of mini-max theorem and existence of Nash equilibrium in asymmetric three-players zero-sum game with two groups," MPRA Paper 87249, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:87249
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    References listed on IDEAS

    as
    1. Hattori, Masahiko & Satoh, Atsuhiro & Tanaka, Yasuhito, 2018. "Minimax theorem and Nash equilibrium of symmetric three-players zero-sum game with two strategic variables," MPRA Paper 85503, University Library of Munich, Germany.
    2. Matsumura, Toshihiro & Matsushima, Noriaki & Cato, Susumu, 2013. "Competitiveness and R&D competition revisited," Economic Modelling, Elsevier, vol. 31(C), pages 541-547.
    3. Yasuhito Tanaka, 2013. "Irrelevance of the choice of strategic variables in duopoly under relative profit maximization," Economics and Business Letters, Oviedo University Press, vol. 2(2), pages 75-83.
    4. Atsuhiro Satoh & Yasuhito Tanaka, 2020. "Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game," International Journal of Mathematics in Operational Research, Inderscience Enterprises Ltd, vol. 16(2), pages 279-289.
    5. Yasuhito Tanaka, 2013. "Equivalance of Cournot and Bertrand equilibria in differentiated duopoly under relative profit maximization with linear demand," Economics Bulletin, AccessEcon, vol. 33(2), pages 1479-1486.
    6. Atsuhiro Satoh & Yasuhito Tanaka, 2014. "Relative profit maximization in asymmetric oligopoly," Economics Bulletin, AccessEcon, vol. 34(3), pages 1653-1664.
    7. Satoh, Atsuhiro & Tanaka, Yasuhito, 2014. "Relative profit maximization in asymmetric oligopoly: Cournot and Bertrand equilibria," MPRA Paper 55883, University Library of Munich, Germany.
    8. Atsuhiro Satoh & Yasuhito Tanaka, 2014. "Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly," Economics Bulletin, AccessEcon, vol. 34(2), pages 819-827.
    9. Atsuhiro Satoh & Yasuhito Tanaka, 2013. "Relative profit maximization and Bertrand equilibrium with quadratic cost functions," Economics and Business Letters, Oviedo University Press, vol. 2(3), pages 134-139.
    10. Fernando Vega-Redondo, 1997. "The Evolution of Walrasian Behavior," Econometrica, Econometric Society, vol. 65(2), pages 375-384, March.
    11. Satoh, Atsuhiro & Tanaka, Yasuhito, 2014. "Relative profit maximization and equivalence of Cournot and Bertrand equilibria in an asymmetric differentiated duopoly," MPRA Paper 55895, University Library of Munich, Germany.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    three-players zero-sum game; two groups; Nash equilibrium; Sion's minimax theorem;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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