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Hard-to-Solve Bimatrix Games

Author

Listed:
  • Rahul Savani
  • Bernhard Stengel

Abstract

The Lemke-Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space. The construction is extended to nonsquare games where, in addition to exponentially long Lemke-Howson computations, finding an equilibrium by support enumeration takes on average exponential time. Copyright The Econometric Society 2006.

Suggested Citation

  • Rahul Savani & Bernhard Stengel, 2006. "Hard-to-Solve Bimatrix Games," Econometrica, Econometric Society, vol. 74(2), pages 397-429, March.
  • Handle: RePEc:ecm:emetrp:v:74:y:2006:i:2:p:397-429
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    File URL: http://hdl.handle.net/10.1111/j.1468-0262.2006.00667.x
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    Citations

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    Cited by:

    1. Luciano De Castro, 2012. "Correlation of Types in Bayesian Games," Discussion Papers 1556, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Takuya Masuzawa, 2008. "Computing the cores of strategic games with punishment–dominance relations," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 185-201, June.
    3. Rahul Savani & Bernhard Stengel, 2015. "Game Theory Explorer: software for the applied game theorist," Computational Management Science, Springer, vol. 12(1), pages 5-33, January.
    4. McLennan, Andrew & Tourky, Rabee, 2010. "Imitation games and computation," Games and Economic Behavior, Elsevier, vol. 70(1), pages 4-11, September.
    5. Senthil K. Veeraraghavan & Laurens G. Debo, 2011. "Herding in Queues with Waiting Costs: Rationality and Regret," Manufacturing & Service Operations Management, INFORMS, vol. 13(3), pages 329-346, July.
    6. Zschocke, Mark S. & Mantin, Benny & Jewkes, Elizabeth M., 2013. "Mature or emerging markets: Competitive duopoly investment decisions," European Journal of Operational Research, Elsevier, vol. 228(3), pages 612-622.
    7. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    8. Tim Roughgarden, 2018. "Complexity Theory, Game Theory, and Economics," Papers 1801.00734, arXiv.org.
    9. Rahul Savani & Bernhard von Stengel, 2016. "Unit vector games," International Journal of Economic Theory, The International Society for Economic Theory, vol. 12(1), pages 7-27, March.
    10. Srihari Govindan & Robert Wilson, 2010. "A decomposition algorithm for N-player games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 97-117, January.
    11. Sobel, Joel, 2009. "ReGale: Some memorable results," Games and Economic Behavior, Elsevier, vol. 66(2), pages 632-642, July.
    12. McLennan, Andrew & Tourky, Rabee, 2008. "Games in oriented matroids," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 807-821, July.
    13. Tim Roughgarden, 2010. "Computing equilibria: a computational complexity perspective," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 193-236, January.
    14. repec:eee:reensy:v:93:y:2008:i:11:p:1740-1750 is not listed on IDEAS
    15. Morales, Dolores Romero & Vermeulen, Dries, 2009. "Existence of equilibria in a decentralized two-level supply chain," European Journal of Operational Research, Elsevier, vol. 197(2), pages 642-658, September.

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