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A canonical game—75 years in the making—showing the equivalence of matrix games and linear programming

Author

Listed:
  • Benjamin Brooks

    (University of Chicago)

  • Philip J. Reny

    (University of Chicago)

Abstract

According to Dantzig (Econometrica, 17, p.200, 1949), von Neumann was the first to observe that for any finite two-person zero-sum game, there is a feasible linear programming (LP) problem whose saddle points yield equilibria of the game, thus providing an immediate proof of the minimax theorem from the strong duality theorem. We provide an analogous construction going in the other direction. For any LP problem, we define a game and, with a brief and elementary proof, show that every equilibrium either yields a saddle point of the LP problem or certifies that one of the primal or dual programs is infeasible and the other is infeasible or unbounded. We thus obtain an immediate proof of the strong duality theorem from the minimax theorem. Taken together, von Neumann’s and our results provide a succinct and elementary demonstration that matrix games and linear programming are “equivalent” in a classical sense.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:etbull:v:11:y:2023:i:2:d:10.1007_s40505-023-00252-8
    DOI: 10.1007/s40505-023-00252-8
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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.
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    More about this item

    Keywords

    Matrix games; Linear programming; Equivalence;
    All these keywords.

    JEL classification:

    • D00 - Microeconomics - - General - - - General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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