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Economic Theory

, Volume 42, Issue 1, pp 119–156 | Cite as

Homotopy methods to compute equilibria in game theory

  • P. Jean-Jacques HeringsEmail author
  • Ronald Peeters
Open Access
Symposium
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Abstract

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.

Keywords

Homotopy Equilibrium computation Non-cooperative games Nash equilibrium 

JEL Classification

C62 C63 C72 C73 
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Notes

Open Access

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.

References

  1. Allgower E.L., Georg K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990)Google Scholar
  2. Avis, D., Rosenberg, G., Savani, R., von Stengel, B.: Enumeration of Nash equilibria for two-player games. Econ Theory (2009, this issue)Google Scholar
  3. Balthasar, A.: Equilibrium tracing in strategic-form games. Econ Theory (2009, this issue)Google Scholar
  4. Browder F.E.: On continuity of fixed points under deformations of continuous mappings. Summa Brasiliensis Math 4, 183–191 (1960)Google Scholar
  5. Datta, R.S.: Finding all Nash equilibria of a finite game using polynomial algebra. Econ Theory (2009, this issue)Google Scholar
  6. Eaves B.C.: Polymatrix games with joint constraints. SIAM J Appl Math 24, 418–423 (1973)CrossRefGoogle Scholar
  7. Eaves B.C., Schmedders K.: General equilibrium models and homotopy methods. J Econ Dyn Control 23, 1249–1279 (1999)CrossRefGoogle Scholar
  8. van den Elzen A.H., Talman A.J.J.: A procedure for finding Nash equilibria in bi-matrix games. ZOR Methods Models Oper Res 35, 27–43 (1991)CrossRefGoogle Scholar
  9. van den Elzen A.H., Talman A.J.J.: An algorithmic approach toward the tracing procedure for bi-matrix games. Games Econ Behav 28, 130–145 (1999)CrossRefGoogle Scholar
  10. Filar J.A., Raghavan T.E.S.: A matrix game solution of a single controller stochastic game. Math Oper Res 9, 356–362 (1984)CrossRefGoogle Scholar
  11. Garcia C.B., Zangwill W.I.: Pathways to Solutions, Fixed Points, and Equilibria. Prentice Hall, Englewood Cliffs (1981)Google Scholar
  12. Garcia C.B., Lemke C.E., Lüthi H.J.: Simplicial approximation of an equilibrium point of noncooperative n-person games. In: Hu, T.C., Robinson, S.M. (eds) Mathematical Programming, pp. 227–260. Academic Press, New York (1973)Google Scholar
  13. Geanakoplos J.D.: Nash and Walras equilibrium via Brouwer. Econ Theory 21, 585–603 (2003)CrossRefGoogle Scholar
  14. Gilboa I., Zemel E.: Nash and correlated equilibria: Some complexity considerations. Games Econ Behav 1, 80–93 (1989)CrossRefGoogle Scholar
  15. Govindan S., Wilson R.: A global Newton method to compute Nash equilibria. J Econ Theory 110, 65–86 (2003)Google Scholar
  16. Govindan S., Wilson R.: Computing Nash equilibria by iterated polymatrix approximation. J Econ Dyn Control 28, 1229–1241 (2004)CrossRefGoogle Scholar
  17. Harsanyi J.C.: The tracing procedure: a Bayesian approach to defining a solution for n-person noncooperative games. Int J Game Theory 4, 61–94 (1975)CrossRefGoogle Scholar
  18. Harsanyi J.C., Selten R.: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge (1988)Google Scholar
  19. Herings P.J.J.: A globally and universally stable price adjustment process. J Math Econ 27, 163–193 (1997)CrossRefGoogle Scholar
  20. Herings P.J.J.: Two simple proofs of the feasibility of the linear tracing procedure. Econ Theory 15, 485–490 (2000)CrossRefGoogle Scholar
  21. Herings P.J.J.: Universally converging adjustment processes—a unifying approach. J Math Econ 38, 341–370 (2002)CrossRefGoogle Scholar
  22. Herings P.J.J., van den Elzen A.H.: Computation of the Nash equilibrium selected by the tracing procedure in n-person games. Games Econ Behav 38, 89–117 (2002)CrossRefGoogle Scholar
  23. Herings P.J.J., Peeters R.: A differentiable homotopy to compute Nash equilibria of n-person games. Econ Theory 18, 159–185 (2001)CrossRefGoogle Scholar
  24. Herings P.J.J., Peeters R.: Stationary equilibria in stochastic games: structure, selection and computation. J Econ Theory 118, 32–60 (2004)CrossRefGoogle Scholar
  25. 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
  26. Howson J.T. Jr: Equilibria of polymatrix games. Manag Sci 21, 313–315 (1972)CrossRefGoogle Scholar
  27. Howson J.T. Jr, Rosenthal R.W.: Bayesian equilibria of finite two-person games with incomplete information. Manag Sci 21, 313–315 (1974)CrossRefGoogle Scholar
  28. Jongen, H.Th., Jonker, P., Twilt, F.: Nonlinear optimization in \({\mathbb {R}^n}\) , I. Morse Theory, Chebyshev Approximation. Methoden und Verfahren der Mathematischen Physik, 29. Frankfurt: Peter Lang (1983)Google Scholar
  29. Judd K.L.: Computational economics and economic theory: substitutes or complements?. J Econ Dyn Control 21, 907–942 (1997)CrossRefGoogle Scholar
  30. Kohlberg E., Mertens J.F.: On the strategic stability of equilibria. Econometrica 54, 1003–1037 (1986)CrossRefGoogle Scholar
  31. Koller D., Megiddo N., von Stengel B.: Efficient computation of equilibria for extensive two-person games. Games Econ Behav 14, 247–259 (1996)CrossRefGoogle Scholar
  32. Kostreva M.M., Kinard L.A.: A differential homotopy approach for solving polynomial optimization problems and noncooperative games. Comput Math Appl 21, 135–143 (1991)CrossRefGoogle Scholar
  33. van der Laan G., Talman A.J.J., Heijden L.: Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling. Math Oper Res 12, 377–397 (1987)CrossRefGoogle Scholar
  34. Lemke C.E.: Bimatrix equilibrium points and mathematical programming. Manag Sci 11, 681–689 (1965)CrossRefGoogle Scholar
  35. Lemke C.E., Howson J.T. Jr: Equilibrium points of bimatrix games. SIAM J Appl Math 12, 413–423 (1964)CrossRefGoogle Scholar
  36. Mas-Colell A.: A note on a theorem of F. Browder. Math Program 6, 229–233 (1974)CrossRefGoogle Scholar
  37. Mas-Colell A.: The Theory of General Economic Equilibrium, a Differentiable Approach. Cambridge University Press, Cambridge (1985)Google Scholar
  38. McKelvey R.D., McLennan A.: Computation of equilibria in finite games. In: Amman, H.M., Kendrick, D.A., Rust, J. (eds) Handbook of Computational Economics, vol. I, pp. 87–142. Elsevier/North-Holland, Amsterdam (1996)Google Scholar
  39. McKelvey R.D., Palfrey T.R.: Quantal response equilibria for normal form games. Games Econ Behav 10, 6–38 (1995)CrossRefGoogle Scholar
  40. McKelvey R.D., Palfrey T.R.: Quantal response equilibria for extensive form games. Exp Econ 1, 9–41 (1998)Google Scholar
  41. McLennan A.: The expected number of Nash equilibria of a normal form game. Econometrica 73, 141–174 (2005)CrossRefGoogle Scholar
  42. Nowak A.S., Raghavan T.E.S.: A finite step algorithm via a bimatrix game to a single controller nonzero-sum stochastic game. Math Program 59, 249–259 (1993)CrossRefGoogle Scholar
  43. Raghavan T.E.S., Syed Z.: Computing stationary Nsh equilibria of undiscounted single-controller stochastic games. Math Oper Res 22, 384–400 (2002)CrossRefGoogle Scholar
  44. Rosenmüller J.: On a generalization of the Lemke–Howson algorithm to noncooperative n-person games. SIAM J Appl Math 21, 73–79 (1971)CrossRefGoogle Scholar
  45. Rosenthal R.W.: A bounded-rationality approach to the study of noncooperative games. Int J Game Theory 18, 273–292 (1989)CrossRefGoogle Scholar
  46. Schanuel S.H., Simon L.K., Zame W.R.: The algebraic geometry of games and the tracing procedure. In: Selten, R. (eds) Game Equilibrium Models II: Methods, Morals and Markets, pp. 9–43. Springer, Berlin (1991)Google Scholar
  47. Shapley L.S.: A note on the Lemke–Howson algorithm. Mathematical programming study. Pivoting Ext 1, 175–189 (1974)Google Scholar
  48. von Stengel B.: Efficient computation of behavior strategies. Games Econ Behav 14, 220–246 (1996)CrossRefGoogle Scholar
  49. von Stengel B.: Computing equilibria for two-person games. In: Aumann, R.J., Hart, S. (eds) Handbook of Game Theory, chap. 45, vol. 3, pp. 1723–1759. North-Holland, Amsterdam (2002)Google Scholar
  50. von Stengel B., van den Elzen A.H., Talman A.J.J.: Computing normal form perfect equilibria for extensive two-person games. Econometrica 70, 693–715 (2002)CrossRefGoogle Scholar
  51. Sturmfels, B.: Solving Systems of Polynomial Equations. American Mathematical Society, CBMS Regional Conferences Series, No. 97. Rhode Island, Providence (2002)Google Scholar
  52. Todd M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems, vol. 124. Springer, Berlin (1976)Google Scholar
  53. Turocy T.L.: A dynamic homotopy interpretation of the logistical quantal response equilibrium correspondence. Games Econ Behav 51, 243–263 (2005)CrossRefGoogle Scholar
  54. Voorneveld M.: Probabilistic choice in games: properties of Rosenthal’s t-solutions. Int J Game Theory 34, 105–122 (2006)CrossRefGoogle Scholar
  55. Watson L.T., Billups S.C., Morgan A.P.: HOMPACK: a suite of codes for globally convergent homotopy algorithms. ACM Trans Math Softw 13, 281–310 (1987)CrossRefGoogle Scholar
  56. Wilson R.: Computing equilibria of n-person games. SIAM J Appl Math 21, 80–87 (1971)CrossRefGoogle Scholar
  57. Wilson R.: Computing equilibria of two-person games from the extensive form. Manag Sci 18, 448–460 (1972)CrossRefGoogle Scholar
  58. Wilson R.: Computing simply stable equilibria. Econometrica 60, 1039–1070 (1992)CrossRefGoogle Scholar
  59. Yamamoto Y.: A path-following procedure to find a proper equilibrium of finite games. Int J Game Theory 22, 249–259 (1993)CrossRefGoogle Scholar
  60. Yanovskaya E.B.: Equilibrium points in polymatrix games. Litovskii Matematicheskii Sbornik 8, 381–384 (1986) (in Russian) (see also Math Rev 39, # 3831)Google Scholar

Copyright information

© The Author(s) 2009

Open AccessThis 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.

Authors and Affiliations

  1. 1.Department of EconomicsMaastricht UniversityMaastrichtThe Netherlands

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