Interactive epistemology in simple dynamic games with a continuum of strategies

Abstract

We extend the epistemic analysis of dynamic games of Battigalli and Siniscalchi (1999, 2002, 2007 from …nite dynamic games to all simple games, that is, …nite and in…nite-horizon games with …nite action sets at non-terminal stages and compact action sets at terminal stages. We prove a generalization of Lubin's (1974) extension result to deal with conditional probability systems and strong belief. With this, we can provide a short proof of the following result: in every simple dynamic game, strong rationalizability characterizes the behavioral implications of rationality and common strong belief in rationality.

Epistemic game theory without types structures: An application to psychological games

“…To the best of our knowledge, this is the …rst paper on epistemic game theory that does not assume transparency of coherence and hence does not rely on standard epistemic structures. As we explained above, our epistemic justi…cation of solution concepts adapts techniques borrowed from Battigalli & Tebaldi (2018). Our analysis of hierarchies of conditional beliefs builds on Battigalli & Siniscalchi (1999), our analysis of common strong belief in rationality (forward-induction reasoning) builds on Battigalli & Siniscalchi (2002) and Battigalli et al (2013).…”

“…For each i 2 I, can be extended to a corresponding precedence relation on H i in an obvious way. 6 The set of terminal histories or paths is…”

“…We considered …nite games forms, but our analysis extends to all multistage games where players' feasible action sets are …nite at all histories of height 2 or more and their psychological utility functions do not depend on terminal beliefs, as in this case we can still adapt the techniques of Battigalli & Tebaldi (2018). This covers, for example, all compact-continuous games with simultaneous moves where utility depends only on initial beliefs.…”

“…Application to psychological games Besides its conceptual appeal, our approach has also a technical advantage: it allows us to extend to so called "psychological games"the techniques used by Battigalli & Tebaldi (2018) in the epistemic analysis of a class of games with compact strategy spaces. In a psychological game utilities of outcomes depend on beliefs.…”

“…The technical problem is to show that a k + 1-order belief that justi…es a player's behavior as rationalizable can be extended to an in…nite hierarchy of beliefs that makes such behavior consistent with rationality and common strong belief in rationality. Battigalli & Tebaldi (2018) rely on the possibility to factorize the uncertainty space of any player i as follows i = Y i M i , where Y i is a set of utility-relevant states and M i is a space of opponents'beliefs (probability measures). In a psychological game, Y i should be the set of possible behaviors and opponents'beliefs up to order k, and M i should be the set of opponents'beliefs of order k + 1 or higher.…”