Abstract
We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.
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Acknowledgements
The authors would like to give special thanks to Johan van Benthem for his insights and the stimulating discussions that lead to this paper. The last two authors would also like to thank Larry Moss for his help, including for organising the Bloomington workshop at which a draft of this paper was first presented. We are grateful to the two anonymous referees for their comments. The research of the first author was partially supported by the Netherlands Organisation for Scientific Research (NWO), grant number B 62-635, which is herewith gratefully acknowledged. The second author acknowledges the support by the Flemish Fund for Scientific Research in the early stage of this paper and support by the University of Groningen via a Rosalind Franklin research position in the last stage of this paper. The last author was supported by a Marie Curie Early Stage Research fellowship in the project GLoRiClass (MEST-CT-2005-020841).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baltag, A., Smets, S. & Zvesper, J.A. Keep ‘hoping’ for rationality: a solution to the backward induction paradox. Synthese 169, 301–333 (2009). https://doi.org/10.1007/s11229-009-9559-z
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DOI: https://doi.org/10.1007/s11229-009-9559-z