Proper Scoring Rules with General Preferences: A Dual Characterization of Optimal Reports

Proper scoring rules incentivize truthful probability reports from risk-neutral individuals. For individuals with much more general preferences (including risk and ambiguity aversion) we use duality techniques to characterize the optimal report in any proper scoring rule. We apply this characterization to several well-known families of preferences, and provide several applications. In particular, for the case of CARA and CRRA preferences we (1) can back out true beliefs from optimal reports, (2) show that the quadratic scoring rule minimizes deviations from truthfulness, and (3) provide a two-step procedure in which the agent reports her risk aversion parameter and her belief truthfully.