Keep ‘hoping’ for rationality: a solution to the backward induction paradox

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.

Strategic Reasoning: Building Cognitive Models from Logical Formulas

“…Researchers such as Binmore have argued for the need for richer models of players, incorporating irrational as well as rational behavior (Binmore 1996). For more details on these issues see (Bicchieri 1988;Arló-Costa and Bicchieri 2007;Brandenburger 2007;Baltag et al 2009;Halpern and Pass 2009;Artemov 2009;Fitting 2011).…”

“…From the logical point of view, various characterisations of backward induction can be found in modal and temporal logic frameworks (Bonanno 2002;Harrenstein et al 2003;van der Hoek and Wooldridge 2003;Jamroga and van der Hoek 2004;Baltag et al 2009). There are also critical voices around backward induction arising from logical investigations of strategies.…”

The logic of empirical theories revisited

“…Indeed its internal styles of debate seem one of its main engines of progress. In logic, active contacts with game theory are developing these days (van Benthem 1999;de Bruin 2010;Baltag et al 2009;Dégrémont and Roy 2009). In the philosophy of science, interfaces have developed more in the area of evolutionary games (Skyrms 1990).…”

Reasoning About Strategies and Rational Play in Dynamic Games

“…Proposals in the literature include the notions of epistemic independence (Stalnaker (1998)), strong belief (Battigalli and Siniscalchi (2002)), stable belief (Baltag et al (2009)), substantive rationality (Aumann (1995), Halpern (2001)). For an overview of this literature the reader is referred to Brandenburger (2007) and Perea (2007).…”