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A scoring rule in the reward convention gives $S(P,x)$ for announcing the probability distribution $P$ and then observing $x$. Under genuine belief $Q$, the expected score isA proper scoring rule satisfies $s(Q,Q)\ge s(P,Q)$ for every admissible $P,Q$. A strictly proper scoring rule additionally requires equality only when the distributions coincide:The expected scores must be well defined. A loss convention reverses the inequalities, but the question uses rewards.