<p>I am currently searching for collections of axioms that imply 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>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><strong>I am looking for collections of axioms that, once combined, imply monotonicity. Do you know any?</strong></p>

CC BY-SA 4.0
2024-07-08T08:03:26.107Z
Being *decisive* (single-valued) and *strategy-proof* works.
Michael Greinecker
CC BY-SA 4.0
2024-07-08T08:06:54.750Z
Thank you for your comment. I suspect this comes from the fact that decisivenes and strategy-proofness, together with the unrestricted preference domain; imply dictatorship, which is obviously monotonic. Any other one that comes to mind, involving perhaps some form of IIA?
EoDmnFOr3q
CC BY-SA 4.0
2024-07-08T08:12:18.330Z
To get to dictatorship, you need to also assume that there are at least three alternatives and that every alternative can be elected. It is generally true that monotonicity and strategy-proofness are equivalent on a full domain for social choice-functions.
Michael Greinecker
CC BY-SA 4.0
2024-07-08T08:43:42.420Z
Thank you again for your comment—I didn’t know the equivalence holds so generally. Given the relationship between IIA and strategy-proofness found by Satterthwaite (1975), I’m kinda surprised there’s no known collection of axioms containing some form of IIA that together imply monotonicity.
EoDmnFOr3q
<microeconomics><theory><social-choice>
Collections of axioms that imply monotonicity?
EoDmnFOr3q
# Collections of axioms that imply monotonicity?

(Asked by: username_0 on 2024-07-08)

I am currently searching for collections of axioms that imply 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*.

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 $x\in F(P')$.

**I am looking for collections of axioms that, once combined, imply monotonicity. Do you know any?**

**Tags:** microeconomics, theory, social-choice
> **Comments:**
> **username_1**: Being *decisive* (single-valued) and *strategy-proof* works.  
> **username_0**: Thank you for your comment. I suspect this comes from the fact that decisivenes and strategy-proofness, together with the unrestricted preference domain; imply dictatorship, which is obviously monotonic. Any other one that comes to mind, involving perhaps some form of IIA?  
> **username_1**: To get to dictatorship, you need to also assume that there are at least three alternatives and that every alternative can be elected. It is generally true that monotonicity and strategy-proofness are equivalent on a full domain for social choice-functions.  
> **username_0**: Thank you again for your comment—I didn’t know the equivalence holds so generally. Given the relationship between IIA and strategy-proofness found by Satterthwaite (1975), I’m kinda surprised there’s no known collection of axioms containing some form of IIA that together imply monotonicity.  

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## 0 answers

