  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.
