# Finding all Nash equilibria of a finite game using polynomial algebra

- 792 Downloads
- 21 Citations

## 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### References

- Basu S., Pollack R., Roy M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)Google Scholar
- Bernstein D.N.: The number of roots of a system of equations. Funct Anal Appl
**9**, 183–185 (1975)CrossRefGoogle Scholar - Bochnak J., Coste M., Roy M.-F.: Real Algebraic Geometry. Springer, Heidelberg (1998)Google Scholar
- Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Springer, Heidelberg (1997)Google Scholar
- Cox D., Little J., O’Shea D.: Using Algebraic Geometry. Springer, Heidelberg (1998)Google Scholar
- Datta, R.S.: Algebraic Methods In Game Theory. Ph.D. thesis, University of California at Berkeley (2003a)Google Scholar
- Datta R.S.: Universality of Nash equilibria. Math Oper Res
**28**, 424–432 (2003b)CrossRefGoogle Scholar - Datta, R.S.: Using computer algebra to find Nash equilibria. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 74–79 (electronic). ACM, New York (2003c)Google Scholar
- Dickenstein A., Emiris I.Z. (2005) Solving Polynomial Equations. Springer, HeidelbergGoogle Scholar
- Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Basel (1994)CrossRefGoogle Scholar
- Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 2.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern. http://www.singular.uni-kl.de (2001)
- Harsanyi J.: Oddness of the number of equilibrium points: a new proof. Int J Game Theory
**2**, 235–250 (1973)CrossRefGoogle Scholar - Herings P.J.-J., Peeters R.: A globally convergent algorithm to compute all Nash equilibria for
*n*-person games. Ann Oper Res**137**, 349–368 (2005)CrossRefGoogle Scholar - Herings, P.J.-J., Peeters, R.: Homotopy methods to compute equilibria in game theory. Econ Theory (2009) (this issue)Google Scholar
- Huber B., Sturmfels B.: A polyhedral method for solving sparse polynomial systems. Math Comput
**64**, 1541–1555 (1995)CrossRefGoogle Scholar - Kouchnirenko A.G.: Newton polytopes and the Bezout theorem. Funct Anal Appl
**10**, 233–235 (1976)CrossRefGoogle Scholar - Lazard D., Rouillier F.: Solving parametric polynomial systems. J Symb Comput
**42**, 636–667 (2007)CrossRefGoogle Scholar - McKelvey R., McLennan A.: The maximal number of regular totally mixed Nash equilibria. J Econ Theory
**72**, 411–425 (1997)CrossRefGoogle Scholar - McKelvey, R.D., McLennan, A.M., Turocy, T.L.: Gambit: Software tools for game theory, version 0.2006.01.20 (2006). Available at http://econweb.tamu.edu/gambit/
- McLennan A.M.: The expected number of real roots of a multihomogeneous system of polynomial equations. Am J Math
**124**, 49–73 (2002)CrossRefGoogle Scholar - Montes A.: A new algorithm for discussing Groebner bases with parameters. J Symb Comput
**33**, 183–208 (2002)CrossRefGoogle Scholar - Osborne M.J., Rubinstein A.: A Course in Game Theory. MIT Press, Cambridge (1994)Google Scholar
- Porter R., Nudelman E., Shoham Y.: Simple search methods for finding a Nash equilibrium. Games Econ Behav
**63**, 642–662 (2008)CrossRefGoogle Scholar - Sommese A.J., Wampler C.W.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005)Google Scholar
- Sturmfels B.: Solving Systems of Polynomial Equations. American Mathematical Society, Providence (2002)Google Scholar
- Torregrosa J.R., Jordán C., el Ghamry R.: The nonsingular matrix completion problem. Int J Contemp Math Sci
**2**, 349–355 (2007)Google Scholar - Verschelde J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans Math Softw
**25**, 251–276 (1999)CrossRefGoogle Scholar