International Journal of Game Theory

, Volume 46, Issue 2, pp 457–473 | Cite as

Characterizing solution concepts in terms of common knowledge of rationality

  • Joseph Y. Halpern
  • Yoram Moses
Original Paper


Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known. Analogous characterizations of sequential equilibrium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using earlier results of Halpern characterizing these solution concepts using non-Archimedean fields.


Characterizing solution concepts Common knowledge of rationality Sequential equilibrium Perfect equilibrium Quasi-perfect equilibrium 



We would like to thank the anonymous reviewers for their detailed reading of the paper and useful comments that helped improve the paper. In particular, we thank an anonymous reviewer for encouraging us to compare our results to those of Asheim and Perea (2005).


  1. Aghassi M, Bertsimas D (2006) Robust game theory. Math Program Ser B 107(1–2):231–273CrossRefGoogle Scholar
  2. Arieli I, Aumann RJ (2015) The logic of backward induction. J Econ Theory 159:443–464CrossRefGoogle Scholar
  3. Asheim GB, Perea A (2005) Sequential and quasi-perfect rationalizability in extensive games. Games Econ Behav 53:15–42CrossRefGoogle Scholar
  4. Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18CrossRefGoogle Scholar
  5. Aumann RJ, Brandenburger A (1995) Epistemic conditions for Nash equilibrium. Econometrica 63(5):1161–1180CrossRefGoogle Scholar
  6. Battigalli P, Siniscalchi M (2002) Strong beliefs and forward-induction reasoning. J Econ Theory 106:356–391CrossRefGoogle Scholar
  7. Blume L, Brandenburger A, Dekel E (1991a) Lexicographic probabilities and choice under uncertainty. Econometrica 59(1):61–79CrossRefGoogle Scholar
  8. Blume L, Brandenburger A, Dekel E (1991b) Lexicographic probabilities and equilibrium refinements. Econometrica 59(1):81–98CrossRefGoogle Scholar
  9. Brandenburger A, Dekel E (1987) Rationalizability and correlated equilibria. Econometrica 55:1391–1402CrossRefGoogle Scholar
  10. Brandenburger A, Friedenberg A, Keisler J (2008) Admissibility in games. Econometrica 76(2):307–352CrossRefGoogle Scholar
  11. Dekel E, Siniscalchi M (2015) Epistemic game theory. In: Young HP, Zamir S (eds) Handbook of Game Theory with Economic Applications, vol 4. North-Holland, pp 619–702Google Scholar
  12. Enderton HB (1972) A Mathematical Introduction to Logic. Academic Press, New YorkGoogle Scholar
  13. Fagin R, Halpern JY, Moses Y, Vardi MY (1995) Reasoning About Knowledge. MIT Press, Cambridge Mass. A slightly revised paperback version was published in 2003Google Scholar
  14. Fagin R, Halpern JY, Moses Y, Vardi MY (1997) Knowledge-based programs. Distrib Comput 10(4):199–225CrossRefGoogle Scholar
  15. Halpern JY (2009) A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium. Int J Game Theory 38(1):37–50CrossRefGoogle Scholar
  16. Halpern JY (2010) Lexicographic probability, conditional probability, and nonstandard probability. Games Econ Behav 68(1):155–179CrossRefGoogle Scholar
  17. Halpern JY (2013) A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium: Erratum (to appear in IJGT)Google Scholar
  18. Halpern JY, Moses Y (2007) Characterizing solution concepts in games using knowledge-based programs. In: Proc. Twentieth International Joint Conference on Artificial Intelligence (IJCAI ’07), pp 1300–1307Google Scholar
  19. Halpern JY, Pass R (2009) A logical characterization of iterated admissibility and extensive-form rationalizability (Unpublished manuscript). A preliminary version, with the title A logical characterization of iterated admissibility, appears in Proc. Twelfth Conference on Theoretical Aspects of Rationality and Knowledge (TARK), 2009, pp 146–155Google Scholar
  20. Halpern JY, Pass R (2012) Iterated regret minimization: a new solution concept. Games Econ Behav 74(1):194–207CrossRefGoogle Scholar
  21. Hyafil N, Boutilier C (2004) Regret minimizing equilibria and mechanisms for games with strict type uncertainty. In: Proc. Twentieth Conference on Uncertainty in Artificial Intelligence (UAI 2004), pp 268–277Google Scholar
  22. Kreps DM, Wilson RB (1982) Sequential equilibria. Econometrica 50:863–894CrossRefGoogle Scholar
  23. Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge (Mass)Google Scholar
  24. Pearce DG (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4):1029–1050CrossRefGoogle Scholar
  25. Perea A (2012) Epistemic game theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  26. Robinson A (1973) Function theory on some nonarchimedean fields. Am Math Mon Pap Found Math 80:S87–S109CrossRefGoogle Scholar
  27. Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55CrossRefGoogle Scholar
  28. van Damme E (1984) A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games. Int J Game Theory 13:1–13CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA
  2. 2.Department of Electrical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations