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Sion's mini-max theorem and Nash equilibrium in a five-players game with two groups which is zero-sum and symmetric in each group

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  • Atsuhiro Satoh
  • Yasuhito Tanaka

Abstract

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a five-players game with two groups which is zero-sum and symmetric in each group. We will show the following results. 1. The existence of Nash equilibrium which is symmetric in each group implies Sion's minimax theorem for a pair of playes in each group. 2. Sion's minimax theorem for a pair of playes in each group imply the existence of a Nash equilibrium which is symmetric in each group. Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.

Suggested Citation

  • Atsuhiro Satoh & Yasuhito Tanaka, 2018. "Sion's mini-max theorem and Nash equilibrium in a five-players game with two groups which is zero-sum and symmetric in each group," Papers 1809.02466, arXiv.org.
    • Handle: RePEc:arx:papers:1809.02466
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Atsuhiro Satoh & Yasuhito Tanaka, 2014. "Relative profit maximization in asymmetric oligopoly," Economics Bulletin, AccessEcon, vol. 34(3), pages 1653-1664.
    6. Satoh, Atsuhiro & Tanaka, Yasuhito, 2014. "Relative profit maximization and Bertrand equilibrium with convex cost functions," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 8, pages 1-15.
    7. Matsumura, Toshihiro & Matsushima, Noriaki & Cato, Susumu, 2013. "Competitiveness and R&D competition revisited," Economic Modelling, Elsevier, vol. 31(C), pages 541-547.
    8. Fernando Vega-Redondo, 1997. "The Evolution of Walrasian Behavior," Econometrica, Econometric Society, vol. 65(2), pages 375-384, March.
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    Cited by:

    1. Atsuhiro Satoh & Yasuhito Tanaka, 2018. "Nash equilibrium of partially asymmetric three-players zero-sum game with two strategic variables," Papers 1809.02465, arXiv.org.
    2. Satoh, Atsuhiro & Tanaka, Yasuhito, 2018. "Nash equilibrium in asymmetric multi-players zero-sum game with two strategic variables and only one alien," MPRA Paper 88978, University Library of Munich, Germany.

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