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Equivalent Results in Minimax Theory

Author

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

Abstract

In this paper we review known minimax results with applications in game theory and show that these results are easy consequences of the first minimax result for a two person zero sum game with finite strategy sets published by von Neumann in 1928: Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky-Fan, König, Neumann and Gwinner-Oettli. Actually it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show these implications the authors only use simple properties of compact sets and the well-known Weierstrass Lebesgue lemma.

Suggested Citation

  • Frenk, J.B.G. & Kassay, G. & Kolumban, J., 2002. "Equivalent Results in Minimax Theory," ERIM Report Series Research in Management ERS-2002-08-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:158
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    File URL: https://repub.eur.nl/pub/158/erimrs20020124104200.pdf
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    References listed on IDEAS

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    1. T. Illés & G. Kassay, 1999. "Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 243-257, May.
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    Cited by:

    1. Frenk, J.B.G. & Kas, P. & Kassay, G., 2004. "On linear programming duality and necessary and sufficient conditions in minimax theory," Econometric Institute Research Papers EI 2004-14, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Friedman, Craig & Sandow, Sven, 2006. "Utility-based performance measures for regression models," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 541-560, February.

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    More about this item

    Keywords

    convex analysis; finite dimensional separation of convex sets; game theory; generalized convexity; minimax theory;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • 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

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