Let $S$ be a scoring rule for probability functions. Define 


$EXP\_{S}(Q|P) = \sum \limits\_{w} P(w)S(Q, w)$.


Say that $S$ is striclty proper if and only if $P$ always minimises $EXP\_{S}(Q|P)$ as a function of $Q$. Define


$D\_{S}(P, Q) = EXP\_{S}(Q|P) - EXP\_{S}(P|P)$.


If $S$ is the logarithmic scoring rule defined by $S(P, w) = -ln(P(w))$, then $D\_{S}(P, Q)$ is just the Kullback-Leibler divergence between $P$ and $Q$, or equivalently, the inverse Kullback-Leibler divergence between $Q$ and $P$. Note that the inverse Kullback-Leibler divergence is an $f$-divergence. 


My question is this: *is there any other strictly proper scoring rule $S$ such that $D\_{S}(P, Q)$ is equal to $F(Q, P)$ for some $f$-divergence $F$?*


I think that $D\_{S}(P, Q)$ is always a Bregman divergence, and Amari proved that the only $f$-divergence that is also a Bregman divergence is the Kullback-Leibler divergence (on the space of probability functions). Is this enough to imply that there are no other strictly proper scoring rules with this property?



Strictly Proper Scoring Rules and f-Divergences
In a word, yes, KL is the only one. You're correct that $S$ is strictly proper if and only if $D\_S$ is a Bregman divergence of some strictly convex function[1] (note you should swap the terms in your definition of $D\_S$). You're also apparently right (going from the abstract) that the only f-divergence on the simplex that is a Bregman divergence is KL-divergence[2], so your conclusion follows.


One more direct way to see this is that $D\_S$ is of the form
 $$ D\_S(Q;P) = \sum\_w P(w) \left[ S(P,w) - S(Q,w) \right] $$
while an $f$-divergence is of the form
\begin{align}
 D\_f(Q;P) = \sum\_w P(w) \left[ f\left(\frac{Q(w)}{P(w)}\right) \right]
\end{align}
Non-rigorously, for these to be equal, at the very least we would need $S(P,w)$ to be only a function of $P(w)$ and not the rest of $P$, and we already know the log scoring rule is the only one that satisfies this; magically it also converts the difference in score into the log of the ratio, as is needed.


[1] e.g. Gneiting and Raftery 2007 <https://www.stat.washington.edu/raftery/Research/PDF/Gneiting2007jasa.pdf>


[2] <https://ieeexplore.ieee.org/document/5290302>


