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

CC BY-SA 4.0
2024-07-07T07:55:51.563Z
Thank you very much for your answer. I have edited the body of the question to ensure it matches the question in the title.
EoDmnFOr3q
CC BY-SA 4.0
2024-07-06T23:01:15.097Z
2024-07-06T23:01:15.097Z
Michael Greinecker
<p>I have a doubt regarding the well-known concepts of weak Pareto optimality and monotonicity.</p>
<p>Let <span class="math-container">$N$</span> be a <em>finite set of players</em>, let <span class="math-container">$A$</span> be a <em>finite set of alternatives</em>, let <span class="math-container">$\mathcal{P}$</span> be the <em>set of all linear order profiles</em> on <span class="math-container">$A$</span>, and let <span class="math-container">$F:\mathcal{P}\to 2^A\backslash\{\emptyset\}$</span> be a <em>social choice correspondence</em>.</p>
<p>Let <span class="math-container">$F^*:\mathcal{P}\to 2^A\backslash\{\emptyset\}$</span> be the <em>weak Pareto correspondence</em>: namely, for all linear order profiles <span class="math-container">$P\in\mathcal{P}$</span>,
<span class="math-container">\begin{gather}
F^*(P)=\{x\in A\mid(\nexists y\in A)[(\forall i\in N)(yP_ix)]\}
\end{gather}</span></p>
<p>Given any player <span class="math-container">$i\in N$</span>, any alternative <span class="math-container">$x\in A$</span> and any linear order profile <span class="math-container">$P\in\mathcal{P}$</span>, let <span class="math-container">$L_i(x,P)=\{y\in A\mid xP_iy\}$</span> be <em>player <span class="math-container">$i$</span>'s lower contour set at <span class="math-container">$x$</span></em>.</p>
<p>A social choice rule <span class="math-container">$F:\mathcal{P}^N\to 2^A\backslash\{\emptyset\}$</span> is <em>monotonic</em> if and only if for all alternatives <span class="math-container">$x\in A$</span> and all linear order profiles <span class="math-container">$P,P'\in\mathcal{P}$</span>, the following is true: if <span class="math-container">$x\in F(P)$</span> and <span class="math-container">$L_i(x,P)\subseteq L_i(x,P')$</span> for all <span class="math-container">$i\in N$</span>, then <span class="math-container">$x\in F(P')$</span>.</p>
<p>We know by <a href="https://scholar.harvard.edu/files/maskin/files/implementation_theory_published_version.pdf" rel="nofollow noreferrer">Maskin &amp; Sjöström (Footnote 15, p. 248, 2002)</a> that the weak Pareto correspondence is monotonic at the unrestricted domain of linear orders.</p>
<p><strong>What I am wondering is whether <em>all</em> subcorrespondences of the weak Pareto correspondence is also monotonic at the unrestricted domain of linear orders.</strong></p>

<microeconomics><theory><social-choice>
Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders?
EoDmnFOr3q
# Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders?

(Asked by: username_0 on 2024-07-06)

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)](https://scholar.harvard.edu/files/maskin/files/implementation_theory_published_version.pdf) 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.**

**Tags:** microeconomics, theory, social-choice

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## 1 answers

### Answer by username_1 on 2024-07-06. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you very much for your answer. I have edited the body of the question to ensure it matches the question in the title.  

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