A probability forecaster is asked to give a density $p$ of a random variable $\omega$. In return he gets a reward (or score) depending on $p$ and on a subsequently observed value of $\omega$. A scoring rule is called proper if the expected score is maximized when the true density is chosen. The present paper uses convex analysis to generalize McCarthy's characterization of proper scoring rules.
"Proper Scores for Probability Forecasters." Ann. Math. Statist. 42 (6) 1916 - 1921, December, 1971. https://doi.org/10.1214/aoms/1177693057