Interactive Epistemology

Robert J. Aumann

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or "player". One is interested not only in each person's knowledge about substantive matters, but also in his knowledge about the others' knowledge. These notes examine two parallel approaches to the subject. The first is the semantic approach, in which knowledge is represented by a space % of states of the world, together with partitions %i of % for each player i; the atom of %i containing a given state % of the world represents the set of those states that i cannot distinguish from %. The second is the syntactic approach, in which knowledge is represented by abstract formulas constructed according to certain syntactic rules. These notes examine the relation between the two approaches, and show that they are in a sense equivalent. In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space % and the partitions %i can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach. Other topics that are taken up include various formalizations of "common knowledge", and the "Agreement Theorem" of J. Cave and M. Bacharach. The notes end with an application of these ideas to the context of probabilistic beliefs.

February, 1995
Published in: 
International Journal of Game Theory 28 (1999), 263-314