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On the consistency among prior, posteriors, and information sets

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Abstract

This paper studies implications of the consistency conditions among prior, posteriors, and information sets on introspective properties of qualitative belief induced from information sets. The main result reformulates the consistency conditions as: (i) the information sets, without any assumption, almost surely form a partition; and (ii) the posterior at a state is equal to the Bayes conditional probability given the corresponding information set. The main implication of this result is to provide a tractable epistemic model which dispenses with the technical assumptions inherent in the standard epistemic model such as the countable number of information sets. Applications are agreement theorem, no-trade theorem, and the epistemic characterization of correlated equilibria. Implications are as follows. First, since qualitative belief reduces to fully introspective knowledge in the standard environment, a care must be taken when one studies non-veridical belief or non-introspective knowledge. Second, an information partition compatible with the consistency conditions is uniquely determined by the posteriors. Third, qualitative and probability-one beliefs satisfy truth axiom almost surely. The paper also sheds light on how the additivity of the posteriors yields negative introspective properties of beliefs.

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  1. Department of Decision Sciences and IGIER, Bocconi University, 20136, Milan, Italy

    Satoshi Fukuda

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I would like to thank Atsushi Kajii, Nobuo Koida, Massimo Marinacci, Sujoy Mukerji, Eric Pacuit, Burkhard Schipper, Takashi Ui, and the seminar participants at TARK 2019 and SWET/DTW 2019 for their insightful comments and discussions. I would also like to thank two anonymous reviewers and the handling editor for their valuable comments.

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Fukuda, S. On the consistency among prior, posteriors, and information sets. Econ Theory (2024). https://doi.org/10.1007/s00199-024-01558-9

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