The Extreme Points of Mean-Preserving Contractions

We study multidimensional mean-preserving contractions (MPC) and their ex treme points. Proposition 1 focuses on extreme MPCs of a measure µ: each finitely supported extreme MPC ν induces a partition of X, the domain of µ, into convex sets such that the support of ν on each element of the partition is an affinely indepen dent set, and such that the restriction of ν on each element of the partition is itself an MPC of the restriction of the prior µ on that element. We apply our results to several questions concerning moment persuasion and categorization. In particular, we prove that categories and prototypes (key elements typically assumed to be exogenous in the classical literature on categorization) can endogenously emerge as outcomes of optimization when the decision maker faces limits on the amount of information that she can either acquire or process.