Non-compactness of information structures.

Summary We say that two type spaces over a fixed space of uncertainty are δ-away if there exists a zero-sum payoff function (uniformly bounded by 1) such that the values of the zero-sum game on the two type spaces are δ-away from each other. We show that the induced topology is not pre-compact: there exists δ > 0 and a set of infinitely many type spaces such that any two of them are δ-away from each other. Thus, it is impossible to approximate the entire universe of type spaces with finite sets. Moreover, this construction shows that there exists type spaces having the same joint distribution of beliefs of arbitrarily high-order that are δ-away from each other.