Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders?

Question

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.

Answer 1

The question in the title seems to differ from the question in the body: all/any. At least the question in the title does have a negative answer:

Let $A=\{a,b,c\}$. Consider the sub-correspondence of the weak Pareto correspondence in which $c$ is removed unless $c$ is the only weak Pareto optimum or $b$ is a weak Pareto optimum. There are two agents. The value under the profile consisting of $a\succ b\succ c$ and $c\succ b\succ a$ is $\{a,b,c\}$. However, under the profile consisting of $a\succ c\succ b$ and $c\succ b\succ a$, the value is $\{a\}$. So this sub-correspondence is not monotonic.