= Solution
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 is
$$
s(P,Q)=\mathbb E_{X\sim Q}[S(P,X)]
=\int S(P,x)\,Q(dx).
$$
A <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:
$$
\boxed{s(Q,Q)>s(P,Q)\quad\text{for all }P\ne Q.}
$$
The expected scores must be well defined. A loss convention reverses the inequalities, but the question uses rewards.
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