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 Groebner 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.Comment: Invited contribution to Journal of Economic Theory; includes color
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