Information hierachies

If experiment A is Blackwell more informative than experiment B, it is always possible that A and B are induced by signals A' and B' such that A' is a refinement of B', that is, A' entails observing B' plus some additional information. We first show that this result does not extend beyond pairs of experiments: There exist collections of experiments that cannot be induced by a collection of signals so that whenever two experiments are Blackwell ordered, the associated signals are refinement ordered. In other words, sometimes it is impossible for more informed agents to know everything that less informed agents know. More broadly, define an information hierarchy to be a partially ordered set that ranks experiments in terms of informativeness. Is it the case that for any choice of experiments indexed on the hierarchy such that higher experiments are Blackwell more informative, there are signals that induce these experiments with higher signals being refinements of lower signals? We show that the answers is affirmative if and only if the undirected graph of the information hierarchy is a forest.