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The equivalence of linear programs and zero-sum games

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Abstract

In 1951, Dantzig showed the equivalence of linear programming problems and two-person zero-sum games. However, in the description of his reduction from linear programs to zero-sum games, he noted that there was one case in which the reduction does not work. This also led to incomplete proofs of the relationship between the Minimax Theorem of game theory and the Strong Duality Theorem of linear programming. In this note, we fill these gaps.

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References

  • Adler I, Beling PA (1997) Polynomial algorithms for linear programming over the algebraic numbers. Algorithmica 12(6): 436–457

    Article  Google Scholar 

  • Carathodory C (1911) ber den variabilittsbereich der Fourierschen konstanten von positiven harmonischen Funktionen. Rend Circ Mat Palermo 32: 193–217

    Article  Google Scholar 

  • Dantzig GB (1951) A proof of the equivalence of the programming problem and the game problem. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 330–335

    Google Scholar 

  • Dantzig GB (1963) Linear programming and extensions. Princeton University Press, Princeton

    Google Scholar 

  • Farkas J (1902) Uber die theorie der einfachen ungleichungen. J Reine Angew Math 124: 1–24

    Google Scholar 

  • Gale D, Kuhn HW, Tucker AW (1951) Linear programming and the theory of games. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 317–329

    Google Scholar 

  • Gordan P (1873) Uber die auflosung lincarer Glcichungcn mit reellen coefficienten. Math Ann 6: 23–28

    Article  Google Scholar 

  • Luce RD, Raiffa H (1957) Games and decisions: introduction and critical survey. Wiley, New York

    Google Scholar 

  • Megiddo N (1990) On solving the linear programming problem approximately. Contemp Math 114: 35–50

    Article  Google Scholar 

  • Raghavan TES (1994) Zero-sum two-person games. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 2. Elsevier, Amsterdam, pp 735–760

    Chapter  Google Scholar 

  • Tucker AW (1956) Dual systems of homogeneous linear relations. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Annals of Mathematics Studies, vol 38. Princeton University Press, Princeton, pp 3–18

    Google Scholar 

  • Ville J (1938) Sur a theorie generale des jeux ou intervient Vhabilite des joueurs. Trait Calc Probab Appl IV 2: 105–113

    Google Scholar 

  • von Neumann J (1928) Zur theorie der gesellschaftsspiele. Math Ann 100: 295–320

    Article  Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

Download references

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  1. Department of IEOR, University of California, Berkeley, CA, USA

    Ilan Adler

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  1. Ilan Adler
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Correspondence to Ilan Adler.

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Adler, I. The equivalence of linear programs and zero-sum games. Int J Game Theory 42, 165–177 (2013). https://doi.org/10.1007/s00182-012-0328-8

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  • DOI: https://doi.org/10.1007/s00182-012-0328-8

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