I have a doubt regarding the well-known concepts of weak Pareto optimality and monotonicity.
Let $N$ be a finite set of players, let $A$ be a finite set of alternatives, let $\mathcal{P}$ be the set of all linear order profiles on $A$, and let $F:\mathcal{P}\to 2^A\backslash\{\emptyset\}$ be a social choice correspondence.
Let $F^*:\mathcal{P}\to 2^A\backslash\{\emptyset\}$ be the weak Pareto correspondence: namely, for all linear order profiles $P\in\mathcal{P}$, \begin{gather} F^*(P)=\{x\in A\mid(\nexists y\in A)[(\forall i\in N)(yP_ix)]\} \end{gather}
Given any player $i\in N$, any alternative $x\in A$ and any linear order profile $P\in\mathcal{P}$, let $L_i(x,P)=\{y\in A\mid xP_iy\}$ be player $i$'s lower contour set at $x$.
A social choice rule $F:\mathcal{P}^N\to 2^A\backslash\{\emptyset\}$ is monotonic if and only if for all alternatives $x\in A$ and all linear order profiles $P,P'\in\mathcal{P}$, the following is true: if $x\in F(P)$ and $L_i(x,P)\subseteq L_i(x,P')$ for all $i\in N$, then $x\in F(P')$.
We know by Maskin & Sjöström (Footnote 15, p. 248, 2002) that the weak Pareto correspondence is monotonic at the unrestricted domain of linear orders.
What I am wondering is whether all subcorrespondences of the weak Pareto correspondence is also monotonic at the unrestricted domain of linear orders.
