Information Hierarchies

If Anne knows more than Bob about the state of the world, she may or may not know what Bob thinks, but it is always possible that she does. In other words, if the distribution of Anne's belief about the state is a mean-preserving spread of the distribution of Bob's belief, we can construct signals for Anne and Bob that induce these distributions of beliefs and provide Anne with full information about Bob's belief. We establish that with more agents, the analogous result does not hold. It might be that Anne knows more than Bob and Charles, who in turn both know more than David, yet what they know about the state precludes the possibility that Anne knows what Bob and Charles think and that everyone knows what David thinks. More generally, we define an information hierarchy as a partially ordered set and ask whether higher elements being Blackwell more informed always makes the hierarchy compatible with higher elements having more information (under various notions of that term) than lower elements. We show that the answer is affirmative if and only if the graph of the hierarchy is a forest. We discuss applications of this result to rationalizing a decision maker's reaction to unknown sources of information and to information design in organizations.