Comparison of Experiments through Summary Statistics

Blackwell's Theorem establishes an equivalence of different partial orders to represent informativeness. A source is said to be Bayesian Better if any Bayesian decision maker would prefer to receive information from it. This notion leaves preferences unrestricted. I consider a partial order that only requires any Bayesian decision maker who's preferences are measurable with respect to the expected state to prefer a source. This ranking can equivalently be described by a mean-preserving spread order on the distribution of posterior means. In this note I provide a necessary and sufficient condition for when the notion of Bayesian Better under mean measurable preferences is equivalent to the more fundamental notion of a garbling also considered by Blackwell: The less informative signalling structure needs to be monotone non-overlapping. Monotone non-overlapping signalling structures split the state space into convex sets that only overlap at their extremes and send a signal realisation for each such set. Since information design has largely focused on agents with mean-measurable preferences this result is useful for studying constrained information design problems. In settings where agents are partially informed about the fundamental state, reducing the information design problem by optimizing over the posterior mean distribution is only a valid simplification of the problem when the agents' partial information is generated by a monotone non-overlapping signalling structure.