Minimax theorem and Nash equilibrium of symmetric three-players zero-sum game with two strategic variables

Hattori, Masahiko and Satoh, Atsuhiro and Tanaka, Yasuhito (2018): Minimax theorem and Nash equilibrium of symmetric three-players zero-sum game with two strategic variables.

Abstract

We consider a symmetric three-players zero-sum game with two strategic variables. Three players are Players A, B and C. Two strategic variables are ti and si, i = A;B;C. They are related by invertible functions. Using the minimax theorem by Sion (1958) and the fixed point theorem by Glicksberg (1952) we will show that Nash equilibria in the following four states are equivalent. 1. All players, Players A, B and C choose ti; i = A;B;C, (as their strategic variables). 2. Two players choose ti's, and one player chooses si. 3. One player chooses ti, and two players choose si's. 4. All players, Players A, B and C choose si; i = A;B;C.

• Item Type: MPRA Paper
• Language: English
• Keywords: symmetric three-person zero-sum game, Nash equilibrium, two strategic variables
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Glicksberg, I.L. (1952) "A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points." Proceedings of the American Mathematical Society, 3, pp.170-174.

Kindler, J. (2005), "A simple proof of Sion's minimax theorem," American Mathematical Monthly, 112, pp. 356-358.

Komiya, H. (1988), "Elementary proof for Sion's minimax theorem," Kodai Mathematical Journal, 11, pp. 5-7.

Matsumura, T., N. Matsushima and S. Cato (2013) "Competitiveness and R&D competition revisited," Economic Modelling, 31, pp. 541-547.

Satoh, A. and Y. Tanaka (2013) "Relative profit maximization and Bertrand equilibrium with quadratic cost functions," Economics and Business Letters, 2, pp. 134-139, 2013.

Satoh, A. and Y. Tanaka (2014a) "Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly," Economics Bulletin, 34, pp. 819-827, 2014.

Satoh, A. and Y. Tanaka (2014b), "Relative profit maximization in asymmetric oligopoly," Economics Bulletin, 34, pp. 1653-1664.

Satoh, A. and Y. Tanaka (2017), "Two person zero-sum game with two sets of strategic variables," MPRA Paper 73272, University Library of Munich, Germany.

Sion, M. (1958), "On general minimax theorems," Pacific Journal of Mathematics, 8, pp. 171-176.

Tanaka, Y. (2013a) "Equivalence of Cournot and Bertrand equilibria in differentiated duopoly under relative profit maximization with linear demand," Economics Bulletin, 33, pp. 1479-1486.

Tanaka, Y. (2013b) "Irrelevance of the choice of strategic variables in duopoly under relative profit maximization," Economics and Business Letters, 2, pp. 75-83, 2013.

Vega-Redondo, F. (1997) "The evolution of Walrasian behavior,", Econometrica, 65, pp. 375-384.