Agreeing to Disagree with Conditional Probability Systems

In this note, we extend Aumann’s agreement theorem to a framework where beliefs are modelled by conditional probability systems à la Battigalli, P., and M. Siniscalchi. 1999. “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games.” Journal of Economic Theory 88: 188–230. We prove two independent generalizations of the agreement theorem, one where the agents share some common conditioning event, and one where they may not.

Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information

We analyze forward-induction reasoning in games with asymmetric information assuming some commonly understood restrictions on beliefs. Specifically, we assume that some given restrictions Δ on players’ initial or conditional first-order beliefs are transparent, that is, not only do the restrictions Δ hold but there is also common belief in Δ at every node. Most applied models of asymmetric information are covered as special cases whereby Δ pins down the probabilities initially assigned to states of nature. But the abstract analysis also allows for transparent restrictions on beliefs about behavior, e.g. independence restrictions or restrictions induced by the context behind the game. Our contribution is twofold. First, we use dynamic interactive epistemology to formalize assumptions that capture foward-induction reasoning given the transparency of Δ, and show that the behavioral implications of these assumptions are characterized by the Δ-rationalizability solution procedure of Battigalli (1999, 2003). Second, we study the differences and similarities between this solution concept and a simpler solution procedure put forward by Battigalli and Siniscalchi (2003). We show that the two procedures are equivalent if Δ is “closed under compositions” a property that holds in all the applications considered by Battigalli and Siniscalchi (2003). We also show that when Δ is not closed under compositions, the simpler solution procedure of Battigalli and Siniscalchi (2003) may fail to characterize the behavioral implications of forward-induction reasoning.

FORMAL EPISTEMOLOGY, CONTEXT AND CONTENT: INTRODUCTION TO SPECIAL ISSUE ON RECENT DEVELOPMENTS IN FORMAL EPISTEMOLOGY

This special issue presents a series of articles focusing on recent work in formal epistemology and formal philosophy. The articles in the latter category elaborate on the notion of context and content and their relationships. This work is not unrelated to recent developments in formal epistemology. Logical models of context, when connected with the representation of epistemic context, are clearly relevant for many issues considered by formal epistemologists. For example, the semantic framework Joe Halpern uses in his article for this issue has been applied elsewhere to solve problems in interactive epistemology (which can be seen as an active branch of contemporary formal epistemology).