The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Purification

  • Stephen Morris
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_2679

Abstract

Many complete information games have equilibria only in mixed strategies, where players are required to randomize over pure strategies among which they are indifferent according to a fixed probability distribution. Harsanyi showed that, if such a game were perturbed – with each player observing an idiosyncratic independent payoff shock with continuous support – then the perturbed games will have pure strategy equilibria which will converge to the mixed strategy equilibrium as the perturbations become small. We review the strong conditions on perturbations under which Harsanyi’s result holds and discuss weaker conditions on perturbations which ensure the existence of pure strategy equilibria without approximating all mixed equilibria of the unperturbed game.

Keywords

Approachability Complete information games Correlation Extensive form games Finite dynamic games Harsanyi, J. C. Incomplete information games Mixed strategy equilibria Nash equilibrium Normal form games Overlapping generations model Pure strategy equilibria Purification Regular Nash equilibrium Subgame perfection 

JEL Classifications

C7 
This is a preview of subscription content, log in to check access.

Bibliography

  1. Aumann, R. 1974. Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1: 67–96.CrossRefGoogle Scholar
  2. Aumann, R., Y. Katznelson, R. Radner, R. Rosenthal, and B. Weiss. 1983. Approximate purification of mixed strategies. Mathematics of Operations Research 8: 327–341.CrossRefGoogle Scholar
  3. Bhaskar, V. 1998. Informational constraints and overlapping generations model: Folk and anti-folk theorems. Review of Economic Studies 65: 135–149.CrossRefGoogle Scholar
  4. Bhaskar, V. 2000. The robustness of repeated game equilibria to incomplete payoff information. Mimeo: University College London.Google Scholar
  5. Bhaskar, V., G. Mailath, and S. Morris. 2006. Purification in the infinitely-repeated Prisoners’ Dilemma. Discussion paper no. 1571, Cowles Foundation.Google Scholar
  6. Carlsson, H., and E. van Damme. 1993. Global games and equilibrium selection. Econometrica 61: 989–1018.CrossRefGoogle Scholar
  7. Govindan, S., P. Reny, and A. Robson. 2003. A short proof of Harsanyi’s purification th. Games and Economic Behavior 45: 369–374.CrossRefGoogle Scholar
  8. Harsanyi, J. 1973. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Games Theory 2: 1–23.CrossRefGoogle Scholar
  9. Milgrom, P., and R. Weber. 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10: 619–632.CrossRefGoogle Scholar
  10. Radner, R., and R. Rosenthal. 1982. Private information and pure-strategy equilibria. Mathematics of Operations Research 7: 401–409.CrossRefGoogle Scholar
  11. Reny, P., and A. Robson. 2004. Reinterpreting mixed strategy equilibria: A unification of the classical and Bayesian views. Games and Economic Behavior 48: 355–384.CrossRefGoogle Scholar
  12. Schmeidler, D. 1973. Equilibrium points of non-atomic games. Journal of Statistical Physics 7: 295–301.CrossRefGoogle Scholar
  13. van Damme, E. 1991. The stability and perfection of nash equilibria. New York: Springer-Verlag.CrossRefGoogle Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Stephen Morris
    • 1
  1. 1.