Learning Efficiency of Multi-Agent Information Structures

We study settings in which, prior to playing an incomplete information game, players observe many draws of private signals about the state from some information structure. Signals are i.i.d. across draws, but may display arbitrary correlation across players. For each information structure, we define a simple learning efficiency index, which only considers the statistical distance between the worst-informed player's marginal signal distributions in different states. We show, first, that this index characterizes the speed of common learning (Cripps, Ely, Mailath, and Samuelson, 2008): In particular, the speed at which players achieve approximate common knowledge of the state coincides with the slowest player's speed of individual learning, and does not depend on the correlation across players' signals. Second, we build on this characterization to provide a ranking over information structures: We show that, with sufficiently many signal draws, information structures with a higher learning efficiency index lead to better equilibrium outcomes, robustly for a rich class of games and objective functions that are "aligned at certainty." We discuss implications of our results for constrained information design in games and for the question when information structures are complements vs. substitutes.