Proper scoring rules with arbitrary value functions

A scoring rule is proper if it elicits an expert's true beliefs as a probabilistic forecast, and it is strictly proper if it uniquely elicits an expert's true beliefs. The value function associated with a (strictly) proper scoring rule is (strictly) convex on any convex set of beliefs. This paper gives conditions on compact sets of possible beliefs Θ that guarantee that every continuous value function on Θ is the value function associated with some strictly proper scoring rule. Compact subsets of many parametrized sets of distributions on Rk satisfy these conditions.