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.
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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.
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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).
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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.
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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
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DOI: https://doi.org/10.1007/s00182-016-0535-9