Consider a finite set $S$ and let $\Delta S$ be the set of all probability distributions on S. Consider a partition $\Pi$ of $S$, which is a collection of mutually disjoint subsets $E$ of $S$, whose union is $S$.
Let ${\cal F}$ be the collection of all functions $f: {\mathbb R}^S \rightarrow {\mathbb R}_{++}$ and let $u: {\cal F} \rightarrow {\mathbb R}$ be an increasing function.
Does there exist function $u$, partition $\Pi$ and a set $M \subseteq \Delta S$, both with at least two elements, such that for all $p,p' \in M$, for all $E \in \Pi$ and all $f \in {\cal F}$, we have $p(E),p'(E) > 0$ and
$\frac{\underset{s \in E}{\sum} \frac{p(s)}{p(E)}u(f(s))}{\underset{s \in S}{\sum} p(s) u(f(s))} = \frac{\underset{s \in E}{\sum} \frac{p'(s)}{p'(E)}u(f(s))}{\underset{s \in S}{\sum} p'(s) u(f(s))}$?