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Nash Equilibria via Polynomial Equations

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LATIN 2004: Theoretical Informatics (LATIN 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2976))

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Abstract

We consider the problem of computing a Nash equilibrium in multiple-player games. It is known that there exist games, in which all the equilibria have irrational entries in their probability distributions [19]. This suggests that either we should look for symbolic representations of equilibria or we should focus on computing approximate equilibria. We show that every finite game has an equilibrium such that all the entries in the probability distributions are algebraic numbers and hence can be finitely represented. We also propose an algorithm which computes an approximate equilibrium in the following sense: the strategies output by the algorithm are close with respect to l  ∞ -norm to those of an exact Nash equilibrium and also the players have only a negligible incentive to deviate to another strategy. The running time of the algorithm is exponential in the number of strategies and polynomial in the digits of accuracy. We obtain similar results for approximating market equilibria in the neoclassical exchange model under certain assumptions.

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Author information

Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Richard J. Lipton & Evangelos Markakis

  2. Telcordia Research, Morristown, NJ, 07960, USA

    Richard J. Lipton

Authors
  1. Richard J. Lipton
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  2. Evangelos Markakis
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Rutgers University, NJ 08854, Piscataway

    Martín Farach-Colton

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© 2004 Springer-Verlag Berlin Heidelberg

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Lipton, R.J., Markakis, E. (2004). Nash Equilibria via Polynomial Equations. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_45

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  • DOI: https://doi.org/10.1007/978-3-540-24698-5_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21258-4

  • Online ISBN: 978-3-540-24698-5

  • eBook Packages: Springer Book Archive

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