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The equivalence of linear programs and zero-sum games

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  • Ilan Adler

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. Copyright Springer-Verlag 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jogath:v:42:y:2013:i:1:p:165-177
    DOI: 10.1007/s00182-012-0328-8
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    References listed on IDEAS

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    1. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    2. Raghavan, T.E.S., 1994. "Zero-sum two-person games," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 20, pages 735-768, Elsevier.
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    Cited by:

    1. 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.
    2. 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.
    3. Daskalakis, Constantinos & Deckelbaum, Alan & Kim, Anthony, 2015. "Near-optimal no-regret algorithms for zero-sum games," Games and Economic Behavior, Elsevier, vol. 92(C), pages 327-348.
    4. Chen, Shun & Zhao, Xudong & Chen, Zhilong & Hou, Benwei & Wu, Yipeng, 2022. "A game-theoretic method to optimize allocation of defensive resource to protect urban water treatment plants against physical attacks," International Journal of Critical Infrastructure Protection, Elsevier, vol. 36(C).
    5. Amir Ali Ahmadi & Jeffrey Zhang, 2021. "Semidefinite Programming and Nash Equilibria in Bimatrix Games," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 607-628, May.
    6. Vikas Vikram Singh & Abdel Lisser, 2018. "Variational inequality formulation for the games with random payoffs," Journal of Global Optimization, Springer, vol. 72(4), pages 743-760, December.
    7. Tim Roughgarden, 2018. "Complexity Theory, Game Theory, and Economics: The Barbados Lectures," Papers 1801.00734, arXiv.org, revised Feb 2020.
    8. Hou, Benwei & Chen, Pengxu & Zhao, Xudong & Chen, Zhilong, 2025. "Optimal allocation of defensive resources in regional railway networks under intentional attacks," Reliability Engineering and System Safety, Elsevier, vol. 257(PB).
    9. Makoto Shimoji, 2022. "Bayesian persuasion in unlinked games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(3), pages 451-481, November.
    10. Ilan Adler & Martin Bullinger & Vijay V. Vazirani, 2024. "A Generalization of von Neumann's Reduction from the Assignment Problem to Zero-Sum Games," Papers 2410.10767, arXiv.org, revised Jan 2026.
    11. Valizadeh, Mehrdad & Gohari, Amin, 2019. "Playing games with bounded entropy," Games and Economic Behavior, Elsevier, vol. 115(C), pages 363-380.
    12. Han, Lin & Zhao, Xudong & Chen, Zhilong & Gong, Huadong & Hou, Benwei, 2021. "Assessing resilience of urban lifeline networks to intentional attacks," Reliability Engineering and System Safety, Elsevier, vol. 207(C).
    13. Ueda, Masahiko & Fujita, Ayaka, 2025. "Zero-determinant strategies in repeated continuously-relaxed games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 670(C).
    14. Constantin Ickstadt & Thorsten Theobald & Elias Tsigaridas, 2024. "Semidefinite games," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(3), pages 827-857, September.

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