Abstract
This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.
Similar content being viewed by others
References
Audet C., Belhaï za S., Hansen P.: Enumeration of all extreme equilibria in game theory: bimatrix and polymatrix games. J Optim Theory Appl 129, 349–372 (2006)
Audet, C., Belhaï za, S., Hansen, P.: A new sequence form approach for the enumeration of all extreme Nash equilibria for extensive form games. Int Game Theory Rev, to appear (2009)
Audet C., Hansen P., Jaumard B., Savard G.: Enumeration of all extreme equilibria of bimatrix games. SIAM J Sci Comput 23, 323–338 (2001)
Avis, D. lrs: a revised implementation of the reverse search vertex enumeration algorithm. In: Kalai, G., Ziegler, G. (eds.): Polytopes—Combinatorics and Computation, DMV Seminar Band 29, pp. 177–198. Basel: Birkhäuser (2000)
Avis, D.: User’s Guide for lrs. http://cgm.cs.mcgill.ca/~avis (2006)
Avis D., Fukuda K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput Geom 8, 295–313 (1992)
Avis D., Bremner D., Seidel R.: How good are convex hull algorithms?. Comput Geom 7, 265–301 (1997)
Azulay D.-O., Pique J.-F.: A revised simplex method with integer Q-matrices. ACM Trans Math Softw 27, 350–360 (2001)
Bron C., Kerbosch J.: Finding all cliques of an undirected graph. Comm ACM 16, 575–577 (1973)
Canty M.J.: Resolving Conflicts with Mathematica: Algorithms for Two-Person Games. Academic Press, Amsterdam (2003)
Chvátal V.: Linear Programming. Freeman, New York (1983)
Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge, Mass (2001)
Dickhaut J., Kaplan T.: A program for finding Nash equilibria. Mathematica J 1(4), 87–93 (1991)
Fukuda K., Prodon A. et al.: Double description method revisited. In: Deza, M. (eds) Combinatorics and Computer Science, Lecture Notes in Computer Science, vol 1120., pp. 91–121. Springer, Berlin (1996)
Gilboa I., Zemel E.: Nash and correlated equilibria: some complexity considerations. Games Econ Behav 1, 80–93 (1989)
Jansen M.J.M.: Maximal Nash subsets for bimatrix games. Naval Res Logist Quart 28, 147–152 (1981)
Kohlberg E., Mertens J.-F.: On the strategic stability of equilibria. Econometrica 54, 1003–1037 (1986)
Kuhn H.W.: An algorithm for equilibrium points in bimatrix games. Proc Nat Acad Sci USA 47, 1657–1662 (1961)
Lemke C.E., Howson J.T. Jr: Equilibrium points of bimatrix games. J Soc Ind Appl Math 12, 413–423 (1964)
Mangasarian O.L.: Equilibrium points in bimatrix games. J Soc Ind Appl Math 12, 778–780 (1964)
McKelvey, R.D., McLennan, A.M., Turocy, T.L.: Gambit: Software Tools for Game Theory, Version 0.2007.01.30 http://econweb.tamu.edu/gambit (2007)
Millham C.B.: On Nash subsets of bimatrix games. Naval Res Logist Quart 21, 307–317 (1974)
Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The double description method. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, pp. 51–73. Annals of Mathematics Studies 28, Princeton: Princeton University Press (1953)
Nash J.F.: Non-cooperative games. Ann Math 54, 286–295 (1951)
Rosenberg, G.D.: Enumeration of All Extreme Equilibria of Bimatrix Games with Integer Pivoting and Improved Degeneracy Check. CDAM Research Report LSE-CDAM-2005-18, London School of Economics (2005)
Savani, R.: Solve a bimatrix game. Interactive website. http://banach.lse.ac.uk/form.html (2005)
Shapley, L.S.: A note on the Lemke–Howson algorithm. Mathematical Programming Study 1: Pivoting and Extensions, pp. 175–189 (1974)
van den Elzen A.H., Talman A.J.J.: A procedure for finding Nash equilibria in bi-matrix games. Math Meth Oper Res 35, 27–43 (1991)
von Schemde A., von Stengel B.: Strategic characterization of the index of an equilibrium. In: Monien, B., Schroeder, U.-P. (eds) Symposium on Algorithmic Game Theory (SAGT) 2008, Lecture Notes in Computer Science, vol. 4997, pp. 242–254. Springer, Berlin (2008)
von Stengel B.: Efficient computation of behavior strategies. Games Econ Behav 14, 220–246 (1996)
von Stengel, B.: Improved equilibrium enumeration for bimatrix games. Extended Abstract, In: International Conference on Operations Research, ETH Zurich, 31 Aug–3 Sept. http://www.maths.lse.ac.uk/Personal/stengel/TEXTE/complement-enum.pdf (1998)
von Stengel B.: New maximal numbers of equilibria in bimatrix games. Discrete Comput Geom 21, 557–568 (1999)
von Stengel B.: Computing equilibria for two-person games. In: Aumann, R.J., Hart, S. (eds) Handbook of Game Theory, vol. 3., pp. 1723–1759. North-Holland, Amsterdam (2002)
Vorob’ev N.N.: Equilibrium points in bimatrix games. Theory Prob Appl 3, 297–309 (1958)
Winkels H.-M.: An algorithm to determine all equilibrium points of a bimatrix game. In: Moeschlin, O., Pallaschke, D. (eds) Game Theory and Related Topics., pp. 137–148. North-Holland, Amsterdam (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Rahul Savani: supported in part by EPSRC project EP/D067170/1.
We would like to thank Charles Audet for providing helpful detailed comments on an earlier draft of the paper.
Rights and permissions
About this article
Cite this article
Avis, D., Rosenberg, G.D., Savani, R. et al. Enumeration of Nash equilibria for two-player games. Econ Theory 42, 9–37 (2010). https://doi.org/10.1007/s00199-009-0449-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-009-0449-x