Publication Date: August 2021
Revision Date: December 2021
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.
Keywords: Common learning, Speed of learning, Higher-order beliefs, Comparison of information structures
JEL Classification Codes: D80, D83, C70CFDP 2299CFDP 2299R2