Interactive epistemology II: Probability

Abstract

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 and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The ®rst is the semantic, in which knowledge and beliefs are represented by a space W of states of the world, and for each player i, partitions Ii of W and probability distributions pi… ; o† on W for each state o of the world. The atom of Ii containing a given state o represents i 's knowledge at that state ± the set of those other states that i cannot distinguish from o; the probability distributions pi… ; o† represents i 's beliefs at the state o. The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows 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 W, the partitions Ii, and the probability distributions pi… ; o† can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach. * This paper is based on notes originally distributed in connection with a sequence of six lectures on Interactive Epistemology given at the Cowles Foundation for Research in Economics, Yale University, in the spring of 1989, and expanded and corrected several times since then. Previous versions were distributed in connection with the tutorial and workshop on ``Knowledge and Game Theory'' held at the Summer Institute in Game Theory, State University of New York at Stony Brook, July 20±24, 1992, and in connection with the Fifth Jerusalem Summer School in Economic Theory, on ``Rationality of Action and Belief in the Economy,'' June 13±23, 1994. The author is grateful to Prof. Dov Samet for important help in preparing the current version. Research support from the USNSF under grant SBR-9730205 and previous grants is gratefully acknowledged.

DOI: 10.1007/s001820050112

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@article{Aumann1999InteractiveEI, title={Interactive epistemology II: Probability}, author={Robert J. Aumann}, journal={Int. J. Game Theory}, year={1999}, volume={28}, pages={301-314} }