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Rational Generating Functions and Integer Programming Games

Author

Listed:
  • Matthias Köppe

    (Department of Mathematics, University of California, Davis, Davis, California 95616)

  • Christopher Thomas Ryan

    (Booth School of Business, University of Chicago, Chicago, Illinois 60637)

  • Maurice Queyranne

    (Sauder School of Business, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada)

Abstract

We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games, with differences of piecewise-linear convex functions as payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy and pure threat point, when the dimension and number of “convex” linear pieces in the payoff functions are fixed. Sequential games where a leader is followed by competing followers (a Stackelberg--Nash setting) are also considered.

Suggested Citation

  • Matthias Köppe & Christopher Thomas Ryan & Maurice Queyranne, 2011. "Rational Generating Functions and Integer Programming Games," Operations Research, INFORMS, vol. 59(6), pages 1445-1460, December.
    • Handle: RePEc:inm:oropre:v:59:y:2011:i:6:p:1445-1460
    DOI: 10.1287/opre.1110.0964
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    References listed on IDEAS

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    7. M. Köppe & M. Queyranne & C. T. Ryan, 2010. "Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 137-150, July.
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    Cited by:

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    6. Carvalho, Margarida & Lodi, Andrea & Pedroso, João.P., 2022. "Computing equilibria for integer programming games," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1057-1070.
    7. Crönert, Tobias & Martin, Layla & Minner, Stefan & Tang, Christopher S., 2024. "Inverse optimization of integer programming games for parameter estimation arising from competitive retail location selection," European Journal of Operational Research, Elsevier, vol. 312(3), pages 938-953.
    8. Stefan Schwarze & Oliver Stein, 2023. "A branch-and-prune algorithm for discrete Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 491-519, November.

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