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Rational Generating Functions and Integer Programming Games

Author

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  • 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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    1. Dominique Lepelley & Ahmed Louichi & Hatem Smaoui, 2008. "On Ehrhart polynomials and probability calculations in voting theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(3), pages 363-383, April.
    2. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    3. Alexander I. Barvinok, 1994. "A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 769-779, November.
    4. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    5. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    6. Juliane Dunkel & Andreas S. Schulz, 2008. "On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 851-868, November.
    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.
    8. Jesús A. De Loera & Raymond Hemmecke & Matthias Köppe & Robert Weismantel, 2006. "Integer Polynomial Optimization in Fixed Dimension," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 147-153, February.
    9. James T. Moore & Jonathan F. Bard, 1990. "The Mixed Integer Linear Bilevel Programming Problem," Operations Research, INFORMS, vol. 38(5), pages 911-921, October.
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    Cited by:

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