I wrote a rough draft of a paper proving geometrically that any strictly proper scoring rule continuous on the probabilities has every score of a non-probability dominated by a score of a probability, without assuming additivity of score. My proof is very much geometric.
Notes: Richard Pettigrew first announced this result in a forthcoming paper, but his proof is flawed. Then Michael Nielsen found a proof in the special case of bounded scoring rules. Finally, Nielsen and I approximately simultaneously (within hours of each other) found quite different proofs without the assumption of boundedness (though there could still be problems in one or the other proof). Research continues regarding how far the condition of continuity can be weakened.
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