Economic Theory

, Volume 42, Issue 1, pp 55–96 | Cite as

Finding all Nash equilibria of a finite game using polynomial algebra

Open Access
Symposium

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.

Keywords

Nash equilibrium Normal form game Algebraic variety 

JEL Classification

C72 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.QB3 Institute, University of CaliforniaBerkeleyUSA

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