Necessary and Sufficient Conditions for Domination Results for Proper Scoring Rules

TL;DR

This paper establishes the optimal necessary and sufficient conditions under which proper scoring rules guarantee that probabilistically inconsistent forecasts are dominated by consistent ones, extending previous results.

Contribution

It introduces a weaker, optimal condition than continuity that ensures domination results for proper scoring rules.

Findings

Weaker condition than continuity suffices for domination results.

The condition is proven to be optimal.

Extension of domination results to non-additive scoring rules.

Abstract

Scoring rules measure the deviation between a probabilistic forecast 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. Probabilistic forecasts need not satisfy the axioms of the probability calculus, but Predd, et al. (2009) have shown that given a finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal.