Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
For simplicity, we assume in this paper that \(\Omega \) is finite, and all subsets of \(\Omega \) are measurable.
While this arguably is a reasonable assumption for strategic-form games, when we move to extensive-form games, agents will be able to learn more in the course of a game.
We consistently use S, possibly subscripted, to denote a pure strategy, while \(\sigma \), possibly subscripted or with a prime, denotes a mixed strategy.
Aumann and Brandenburger (1995) show that common knowledge of rationality is not required for \(\sigma \) to be a Nash equilibrium. This is not a contradiction to Theorem 2.1, which simply says that \(\sigma \) is a Nash equilibrium iff there exists a model M describing the beliefs of the players where rationality is common knowledge. There may be other models where the players play \(\sigma \) and rationality is not common knowledge.
The construction of \(\mathbb {R}(\varepsilon )\) apparently goes back to Robinson (1973).
There is a natural extension of \(\mathbb {R}(\varepsilon )\) called \(\mathbb {R}^*(\varepsilon )\) that is normal. As shown by Halpern (2009, (2013), Theorems 3.3 and 3.5 could be strengthened to use \(\mathbb {R}^*(\varepsilon )\) rather than an existentially quantified normal non-Archimedean field.
As is standard, we assume that the same set of actions is available to i at all histories in I.
The characterization of perfect equilibrium given in (Halpern 2009) involved best responses. In (Halpern 2013), it was pointed out that this was incorrect; \(\sigma _i\) needed to be a local best response to get a characterization of perfect equilibrium, but taking it to be a best response gave a characterization of quasi-perfect equilibrium.
There is no notion of multiplication in SCLPs, so this statement is not quite accurate. Nevertheless, consequences of the chain rule, such as that \(\mu (A \mid B) = \mu (A' \mid B)\) implies \(\mu (A \mid C) = \mu (A' \mid C)\) do not hold for SCLPs.
Perea (2012) also provides epistemic characterizations of iterated admissibility and EFR that do not require complete type structures.
References
Aghassi M, Bertsimas D (2006) Robust game theory. Math Program Ser B 107(1–2):231–273
Arieli I, Aumann RJ (2015) The logic of backward induction. J Econ Theory 159:443–464
Asheim GB, Perea A (2005) Sequential and quasi-perfect rationalizability in extensive games. Games Econ Behav 53:15–42
Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18
Aumann RJ, Brandenburger A (1995) Epistemic conditions for Nash equilibrium. Econometrica 63(5):1161–1180
Battigalli P, Siniscalchi M (2002) Strong beliefs and forward-induction reasoning. J Econ Theory 106:356–391
Blume L, Brandenburger A, Dekel E (1991a) Lexicographic probabilities and choice under uncertainty. Econometrica 59(1):61–79
Blume L, Brandenburger A, Dekel E (1991b) Lexicographic probabilities and equilibrium refinements. Econometrica 59(1):81–98
Brandenburger A, Dekel E (1987) Rationalizability and correlated equilibria. Econometrica 55:1391–1402
Brandenburger A, Friedenberg A, Keisler J (2008) Admissibility in games. Econometrica 76(2):307–352
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 619–702
Enderton HB (1972) A Mathematical Introduction to Logic. Academic Press, New York
Fagin R, Halpern JY, Moses Y, Vardi MY (1995) Reasoning About Knowledge. MIT Press, Cambridge Mass. A slightly revised paperback version was published in 2003
Fagin R, Halpern JY, Moses Y, Vardi MY (1997) Knowledge-based programs. Distrib Comput 10(4):199–225
Halpern JY (2009) A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium. Int J Game Theory 38(1):37–50
Halpern JY (2010) Lexicographic probability, conditional probability, and nonstandard probability. Games Econ Behav 68(1):155–179
Halpern JY (2013) A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium: Erratum (to appear in IJGT)
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 1300–1307
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 146–155
Halpern JY, Pass R (2012) Iterated regret minimization: a new solution concept. Games Econ Behav 74(1):194–207
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 268–277
Kreps DM, Wilson RB (1982) Sequential equilibria. Econometrica 50:863–894
Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge (Mass)
Pearce DG (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4):1029–1050
Perea A (2012) Epistemic game theory. Cambridge University Press, Cambridge
Robinson A (1973) Function theory on some nonarchimedean fields. Am Math Mon Pap Found Math 80:S87–S109
Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55
van Damme E (1984) A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games. Int J Game Theory 13:1–13
Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Y. Halpern was supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR under grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1-0055.
Y. Moses was the Israel Pollak academic chair at the Technion; work supported in part by Israel Science Foundation under grant 1520/11.
About this article
Cite this article
Halpern, J.Y., Moses, Y. Characterizing solution concepts in terms of common knowledge of rationality. Int J Game Theory 46, 457–473 (2017). https://doi.org/10.1007/s00182-016-0535-9
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-016-0535-9