Abstract
New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that \({(\vec{\sigma},\mu)}\) is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal \({\epsilon}\) and a completely mixed behavioral strategy profile σ′ (so that \({\sigma_i'}\) assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from \({\vec{\sigma}}\) such that at each information set I for player i, σ i is an \({\epsilon}\)-best response to \({\vec{\sigma}'_{-i}}\) conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ i must be a best response to \({\vec{\sigma}'_{-i}}\) conditional on having reached I. Yet another variant is used to characterize proper equilibrium.
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This work was supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, and by AFOSR under grant FA9550-05-1-0055.
An erratum to this article is available at http://dx.doi.org/10.1007/s00182-016-0537-7.
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Halpern, J.Y. A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium. Int J Game Theory 38, 37–49 (2009). https://doi.org/10.1007/s00182-008-0139-0
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DOI: https://doi.org/10.1007/s00182-008-0139-0