Keep ‘hoping’ for rationality: a solution to the backward induction paradox
 Alexandru Baltag,
 Sonja Smets,
 Jonathan Alexander Zvesper
 … show all 3 hide
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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 Title
 Keep ‘hoping’ for rationality: a solution to the backward induction paradox
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Synthese
Volume 169, Issue 2 , pp 301333
 Cover Date
 20090701
 DOI
 10.1007/s112290099559z
 Print ISSN
 00397857
 Online ISSN
 15730964
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Backward induction
 Dynamic logic
 Epistemic logic
 Public announcements
 Rationality
 Industry Sectors
 Authors

 Alexandru Baltag ^{(1)}
 Sonja Smets ^{(2)} ^{(3)}
 Jonathan Alexander Zvesper ^{(4)}
 Author Affiliations

 1. Oxford University Computing Laboratory, University of Oxford, Oxford, OX1 3QD, UK
 2. University of Groningen, Groningen, The Netherlands
 3. Oxford University, Oxford, UK
 4. Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, Amsterdam, 1018 TV, Netherlands