Economic Theory

, Volume 42, Issue 1, pp 39–54 | Cite as

Equilibrium tracing in strategic-form games

Symposium

Abstract

We analyze the relationships of the van den Elzen–Talman algorithm, the Lemke–Howson algorithm and the global Newton method for equilibrium computation by Govindan and Wilson. For two-player games, all three can be implemented as complementary pivoting algorithms. The algorithms by Lemke and Howson and by van den Elzen and Talman start at a pair of strategies: the first method at a pure strategy and its best reply, the latter anywhere in the strategy space. However, we show that even with the same starting point they may find different equilibria. Our second result is that the van den Elzen–Talman algorithm is a special case of the global Newton method, which was known only for the Lemke–Howson algorithm. More generally, the global Newton method implements the linear tracing procedure for any number of players. All three algorithms find generically only equilibria of positive index. Even though the van den Elzen–Talman algorithm is extremely flexible in the choice of starting point, we show that there are generic coordination games where the completely mixed equilibrium, which has positive index, is generically not found by the algorithm.

Keywords

Bimatrix game Equilibrium computation Homotopy methods Index 

JEL Classification

C72 

References

  1. Eaves B.C.: The linear complementarity problem. Manage Sci 17, 612–634 (1971)CrossRefGoogle Scholar
  2. Govindan S., Wilson R.: A global Newton method to compute Nash equilibria. J Econ Theory 10, 65–86 (2003a)Google Scholar
  3. Govindan, S., Wilson, R.: Supplement to: a global Newton method to compute Nash equilibria. Accessed online at www.nyu.edu/jet/supplementary.html (2003b)
  4. 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
  5. Harsanyi J.C., Selten R.: A General Theory of Equilibrium Selection in Games. MIT press, Cambridge (1988)Google Scholar
  6. Herings, P.J.-J., Peeters, R.: Homotopy methods to compute equilibria in game theory. Econ Theory (2009, this issue)Google Scholar
  7. Hofbauer, J.: Some thoughts on sustainable/learnable equilibria. Paper presented at the 15th Italian Meeting on Game Theory and Applications, Urbino, Italy, July 9–12, 2003. Accessed online at http://www.econ.uniurb.it/imgta/PlenaryLecture/Hofbauer.pdf (2003)
  8. Kohlberg E., Mertens J.-F.: On the strategic stability of equilibria. Econometrica 54, 1003–1037 (1986)CrossRefGoogle Scholar
  9. Lemke C.E.: Bimatrix equilibrium points and mathematical programming. Manage Sci 11, 681–689 (1965)CrossRefGoogle Scholar
  10. Lemke C.E., Howson J.T. Jr: Equilibrium points of bimatrix games. J Soc Indus Appl Math 12, 413–423 (1964)CrossRefGoogle Scholar
  11. Myerson R.B. et al.: Sustainable equilibria in culturally familiar games. In: Albers, W. (eds) Understanding Strategic Interaction: Essays in Honor of Reinhard Selten, pp. 111–121. Springer, Heidelberg (1997)Google Scholar
  12. Ritzberger K.: Foundations of Non-Cooperative Game Theory. Oxford University Press, Oxford (2002)Google Scholar
  13. Shapley, L.S.: A note on the Lemke–Howson algorithm. Mathematical Programming Study 1: Pivoting and Extensions, pp. 175–189 (1974)Google Scholar
  14. Smale S.: A convergent process of price adjustment and global Newton methods. J Math Econ 3, 107–120 (1976)CrossRefGoogle Scholar
  15. 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
  16. 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
  17. von Stengel, B.: Computing equilibria for two-person games, Chap. 45. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory, vol. 3, pp. 1723--1759. North-Holland, Amsterdam (2002)Google Scholar
  18. 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

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsLondon School of EconomicsLondonUK

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