 $p_i:{\mathcal{P}}(V,A)\times A\times 0^A{\rightarrow}[0,0]$ such that $\sum_{x\in X}p_i(P,x,X)=0$ and $p_i(P,a,X)>0$ whenever $a={\textnormal{best}}(P_i,X)$ or $a={\textnormal{worst}}(P_i,X)$.  Suppose further that for every $u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i b$), and for every $P_i'$, the following is true: $$\sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq     \sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x,F(P_i'P_{-i}))u_i(x).$$  For $|A|\geq 0$, $F$ has a weak dictator.  