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A generalization of von Neumann’s reduction from the assignment problem to zero-sum games

Author

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  • Adler, Ilan
  • Bullinger, Martin
  • Vazirani, Vijay V.

Abstract

The equivalence between von Neumann’s Minimax Theorem for zero-sum games and the LP Duality Theorem connects cornerstone problems of the two fields of game theory and optimization, respectively, and has been the subject of intense scrutiny for seven decades. Yet, as observed in this paper, the proof of the difficult direction of this equivalence is unsatisfactory: It does not assign distinct roles to the two players of the game, as is natural from the definition of a zero-sum game.

Suggested Citation

  • Adler, Ilan & Bullinger, Martin & Vazirani, Vijay V., 2026. "A generalization of von Neumann’s reduction from the assignment problem to zero-sum games," Games and Economic Behavior, Elsevier, vol. 157(C), pages 226-236.
    • Handle: RePEc:eee:gamebe:v:157:y:2026:i:c:p:226-236
    DOI: 10.1016/j.geb.2026.01.009
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    References listed on IDEAS

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    1. Ilan Adler, 2013. "The equivalence of linear programs and zero-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(1), pages 165-177, February.
    2. Benjamin Brooks & Philip J. Reny, 2023. "A canonical game—75 years in the making—showing the equivalence of matrix games and linear programming," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(2), pages 171-180, October.
    3. Galichon, A. & Jacquet, A., 2025. "The matching problem with linear transfers is equivalent to a hide-and-seek game," Games and Economic Behavior, Elsevier, vol. 152(C), pages 333-344.
    4. Bernhard von Stengel, 2024. "Zero-Sum Games and Linear Programming Duality," Mathematics of Operations Research, INFORMS, vol. 49(2), pages 1091-1108, May.
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