This question is closely related to another question I asked recently but is more to the point than that other question.

Let $mathcal P$ be the set of all probability measures on the Borel algebra of $(0,1)$. A measurable scoring rule $S: mathcal P times (0,1) to (-infty, 0)$ is called *strictly proper* if

$$int S(P,x)P(dx) > int S(Q, x)P(dx)$$

holds for all $P neq Q$ in $mathcal P$.

Can anyone provide a concrete example of a strictly proper scoring rule?