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John_Maxwell_IV comments on Bead Jar Guesses - Less Wrong
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In my opinion, a more interesting question is what game Omega can devise in which revealing your probability estimate is part of the winning strategy. If he asks you to name even odds and bet with him, for example, you could name ridiculous odds if you wanted more money. The only thing I can think of is for Omega to pour out part of the jar and reward you depending on your deviation from the correct percentage.
You could use a proper scoring rule.
A more comprehensive description:
T. Gneiting & A. E. Raftery (2007). `Strictly Proper Scoring Rules, Prediction, and Estimation'. Journal of the American Statistical Association 102(477):359-378. [pdf]