Domination for Finite Proper Scoring Rules

Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast $p$ by the lights of $p$ is strictly better than the mathematical expectation of any other forecast $q$ by the lights of $p$. It has recently been shown that any strictly proper scoring rule that is continuous on the probabilities has the property that the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. I shall show that in the case of a finite score, we only need continuity on the regular probabilities---those that assign assign a non-zero value to every point