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Finding all Nash equilibria of a finite game using polynomial algebra

Abstract

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article, we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Correspondence to Ruchira S. Datta.

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Our earlier paper (Datta 2003c) contains much of the material which is surveyed more expansively here. We would like to express our gratitude to the following for generously taking the time to personally discuss with us the use of their software packages: Andrew McLennan and Ted Turocy (Gambit McKelvey et al. 2006), Gert-Martin Greuel (Singular Greuel et al. 2001), and Jan Verschelde (PHC Verschelde 1999). We would also like to thank Gabriela Jeronimo for sending us a preprint of her paper with Daniel Perrucci and Juan Sabia, and Andrew McLennan for suggesting she do so. We would like to thank Richard Fateman and Bernd Sturmfels for supervising the research leading up to that paper, during which the author was partially supported by NSF grant DMS 0138323. We would also like to acknowledge our debt to Bernd Sturmfels, especially for teaching us about the application of polynomial algebra to Nash equilibria, in the lectures leading to Sturmfels (2002).

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Datta, R.S. Finding all Nash equilibria of a finite game using polynomial algebra. Econ Theory 42, 55–96 (2010). https://doi.org/10.1007/s00199-009-0447-z

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Keywords

  • Nash equilibrium
  • Normal form game
  • Algebraic variety

JEL Classification

  • C72