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For a binary outcome, rewarding the reported probability of the realized outcome gives expected score $1-q+(2q-1)p$ under genuine probability $q$. The optimal report is an endpoint whenever $q\ne 1/2$, so this is not a proper scoring rule.

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.

A scoring rule assigns a reward or loss to an announced probability distribution and realized outcome. In the reward convention, a proper scoring rule makes truthful reporting maximize expected reward; the loss convention reverses the comparison.

A proper scoring rule is strictly proper when truthful reporting is the unique optimizing distribution. For densities, equality is understood almost everywhere relative to the reference measure.