Lattice structure of knowledge and agreeing to disagree

Abstract

This paper is a contribution to the study of the underlying mathematical structure of common-knowledge, which gives the well-known result of Aumann about the impossibility of ‘agreeing to disagree’. We present the Bayesian subjective probability model with player's belief: i.e. a triple (ℒ %plane1D;4AF;, μ), in which i is a player. ℒ is a lattice in the field of sets of a state space Ω, %plane1D;4AF;, is a correspondence assigning to each state ω a filter %plane1D;4AF;( ω) in ℒ, and μ is a common-prior. For this model, we impose none of the important restrictions on the information structure in the Aumann-Bacharach model: axiom of knowledge K 1. axiom of transparency K 2 and axiom of wisdom K 3. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ℒ is an Artinian lattice.