Value-based distance between the information structures.

Summary We dene the distance between two information structures as the largest possible dierence in the value across all zero-sum games. We provide a tractable characterization of the distance, as the minimal distance between 2 polytopes. We use it to show various results about the relation between games and single-agent problems, the value of additional information, informational substitutes, complements, etc. We show that approximate knowledge is similar to approximate common knowledge with respect to the value-based distance. Nevertheless, contrary to the weak topology, the value-based distance does not have a compact completion: there exists a sequence of information structures, where players acquire more and more information, and ε > 0 such that any two elements of the sequence have distance at least ε. This result answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper Repeated Games", ICM 1986.