  We define the distance between two information structures as the largest
possible difference in value across all zero-sum games. We provide a tractable
characterization of distance and use it to discuss the relation between the
value of information in games versus single-agent problems, the value of
additional information, informational substitutes, complements, or joint
information. The convergence to a countable information structure under
value-based distance is equivalent to the weak convergence of belief
hierarchies, implying, among other things, that for zero-sum games, approximate
knowledge is equivalent to common knowledge. At the same time, the space of
information structures under the value-based distance is large: there exists a
sequence of information structures where players acquire increasingly more
information, and $\epsilon$ > 0 such that any two elements of the sequence have
distance of 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.
