  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 $\epsilon$ > 0 such that
any two elements of the sequence have distance at least $\epsilon$. 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.
