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Hard-to-Solve Bimatrix Games

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  • 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.

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File URL: http://hdl.handle.net/10.1111/j.1468-0262.2006.00667.x
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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 74 (2006)
Issue (Month): 2 (03)
Pages: 397-429

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Handle: RePEc:ecm:emetrp:v:74:y:2006:i:2:p:397-429

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Cited by:
  1. McLennan, Andrew & Tourky, Rabee, 2010. "Imitation games and computation," Games and Economic Behavior, Elsevier, vol. 70(1), pages 4-11, September.
  2. Tim Roughgarden, 2010. "Computing equilibria: a computational complexity perspective," Economic Theory, Springer, vol. 42(1), pages 193-236, January.
  3. Takuya Masuzawa, 2008. "Computing the cores of strategic games with punishment–dominance relations," International Journal of Game Theory, Springer, vol. 37(2), pages 185-201, June.
  4. Luciano De Castro, 2012. "Correlation of Types in Bayesian Games," Discussion Papers 1556, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  5. Srihari Govindan & Robert Wilson, 2010. "A decomposition algorithm for N-player games," Economic Theory, Springer, vol. 42(1), pages 97-117, January.
  6. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
  7. McLennan, Andrew & Tourky, Rabee, 2008. "Games in oriented matroids," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 807-821, July.
  8. Sobel, Joel, 2009. "ReGale: Some memorable results," Games and Economic Behavior, Elsevier, vol. 66(2), pages 632-642, July.
  9. 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.

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