<p>Here is a partial answer that assumes <span class="math-container">$a\geq 4$</span>. The main point is that the existence of a Condorcet winner doesn't do that much in this setting.</p>
<p>Let <span class="math-container">$x$</span> be any alternative, and let <span class="math-container">$\mathcal{C}_x$</span> be the set of linear orders that rank <span class="math-container">$x$</span> top.  Then, any profile of linear orders over the remaining <span class="math-container">$a-1$</span> alternatives corresponds to a unique profile in <span class="math-container">$\mathcal{C}_x^N$</span>. Moreover, a Condorcet winner always exists, namely <span class="math-container">$x$</span>. Every social welfare function defined on profiles of linear orders over the remaining <span class="math-container">$a-1$</span> alternatives that satisfy the conditions must clearly extend to a unique social welfare function on <span class="math-container">$\mathcal{C}_x^N$</span>. So a dictator over these <span class="math-container">$a-1$</span> alternatives exists by Arrow's theorem. Since this works for all <span class="math-container">$x$</span>, a plain dictator must exist too.</p>

CC BY-SA 4.0
2024-11-12T23:19:07.223Z
2024-11-12T23:19:07.223Z
Michael Greinecker
<p>I am trying to show that independence of irrelevant alternatives and Pareto optimality imply monotonicity in the restricted strict preference domain with Condorcet winners. I am looking for any of these things:</p>
<ol>
<li>a counter-example that proves my conjecture wrong,</li>
<li>a proof strategy to show my conjecture,</li>
<li>a helpful reference.</li>
</ol>
<p>Let <span class="math-container">$N=\{1,\dots,n\}$</span> be a <em>finite set of voters</em>; let <span class="math-container">$A=\{1,\dots,a\}$</span> be a <em>finite set of alternatives</em>, where <span class="math-container">$a\geq 3$</span>; let <span class="math-container">$\mathcal{P}$</span> be the <em>set of all linear orders on <span class="math-container">$A$</span></em>; let <span class="math-container">$\mathcal{C}^N\subsetneq\mathcal{P}^N$</span> be the <em>set of all linear order profiles with Condorcet winners</em>, where an alternative <span class="math-container">$x\in A$</span> is a <em>Condorcet winner</em> at profile <span class="math-container">$\mathbf{P}\in\mathcal{P}^N$</span> if and only if <span class="math-container">$\{i\in N\mid xP_iy\}&gt;n/2$</span> for all alternatives <span class="math-container">$y\in A\backslash\{x\}$</span>; and let <span class="math-container">$f:\mathcal{C}^N\to\mathcal{P}$</span> be a <em>social welfare function</em>.</p>
<ul>
<li><p>A social welfare function <span class="math-container">$f:\mathcal{C}^N\to\mathcal{P}$</span> is <strong>independent of irrelevant alternatives</strong> if and only if given any two alternatives <span class="math-container">$x,y\in A$</span> and any two profiles <span class="math-container">$\mathbf{P},\mathbf{P}'\in\mathcal{C}^N$</span> such that for all voters <span class="math-container">$i\in N$</span>, we have <span class="math-container">$xP_iy$</span> if and only if <span class="math-container">$xP_i'y$</span>; then we also have <span class="math-container">$xf(\mathbf{P})y$</span> if and only if <span class="math-container">$xf(\mathbf{P}')y$</span>.</p>
</li>
<li><p>A social welfare function <span class="math-container">$f:\mathcal{C}^N\to\mathcal{P}$</span> is <strong>Pareto optimal</strong> if and only if for all profiles <span class="math-container">$\mathbf{P}\in\mathcal{C}^N$</span> and all alternatives <span class="math-container">$x,y\in A$</span>, we have that <span class="math-container">$xP_iy$</span> for all voters <span class="math-container">$i\in N$</span> implies that <span class="math-container">$xf(\mathbf{P})y$</span>.</p>
</li>
</ul>
<p>Given any profile <span class="math-container">$\mathbf{P}\in\mathcal{C}^N$</span> and any alternative <span class="math-container">$x\in A$</span>, let <span class="math-container">$L_i(x,\mathbf{P})=\{y\in A\mid xP_iy\}$</span> be <em>voter <span class="math-container">$i$</span>'s lower contour set at <span class="math-container">$x$</span></em>, and let <span class="math-container">$\mathcal{C}^N(\mathbf{P},x)=\{\mathbf{P}'\in\mathcal{C}^N\mid(\forall i\in N)[L_i(x,\mathbf{P})\subseteq L_i(x,\mathbf{P}')]\}$</span> be the <em>set of preference profiles in which alternative <span class="math-container">$x$</span> has not fallen in any voter's preference</em>.</p>
<ul>
<li>A social welfare function <span class="math-container">$f:\mathcal{C}^N\to\mathcal{P}$</span> is <strong>monotonic</strong> if and only if for all alternatives <span class="math-container">$x\in A$</span>, all profiles <span class="math-container">$\mathbf{P}\in\mathcal{C}^N$</span> and all profiles <span class="math-container">$\mathbf{P}'\in\mathcal{C}^N(\mathbf{P},x)$</span>; we have that for all alternatives <span class="math-container">$y\in A\backslash\{x\}$</span>, <span class="math-container">$xf(\mathbf{P})y$</span> implies <span class="math-container">$xf(\mathbf{P}')y$</span>.</li>
</ul>
<p>By <a href="https://www.journals.uchicago.edu/doi/abs/10.1086/256963" rel="nofollow noreferrer">Arrow (1950)</a>, we know that under the unrestricted strict preference domain, a social welfare function satisfying Pareto optimality and independence of irrelevant alternatives is dictatorial, and thus monotonic. My feeling by reading contemporary versions of <a href="https://www.journals.uchicago.edu/doi/abs/10.1086/256963" rel="nofollow noreferrer">Arrow’s (1950)</a> proof is that the unrestricted domain is necessary to attain the dictatorial result, but not monotonicity. However, I cannot prove it nor find a counter-example.</p>
<p><strong>Can you provide any help?</strong></p>

<microeconomics><reference-request><theory><social-welfare><social-choice>
Do Independence of Irrelevant Alternatives and Pareto optimality imply monotonicity whenever there are Condorcet winners?
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# Do Independence of Irrelevant Alternatives and Pareto optimality imply monotonicity whenever there are Condorcet winners?

(Asked by: username_0 on 2024-11-12)

I am trying to show that independence of irrelevant alternatives and Pareto optimality imply monotonicity in the restricted strict preference domain with Condorcet winners. I am looking for any of these things:

1. a counter-example that proves my conjecture wrong,
2. a proof strategy to show my conjecture,
3. a helpful reference.

Let $N=\{1,\dots,n\}$ be a *finite set of voters*; let $A=\{1,\dots,a\}$ be a *finite set of alternatives*, where $a\geq 3$; let $\mathcal{P}$ be the *set of all linear orders on $A$*; let $\mathcal{C}^N\subsetneq\mathcal{P}^N$ be the *set of all linear order profiles with Condorcet winners*, where an alternative $x\in A$ is a *Condorcet winner* at profile $\mathbf{P}\in\mathcal{P}^N$ if and only if $\{i\in N\mid xP\_iy\}>n/2$ for all alternatives $y\in A\backslash\{x\}$; and let $f:\mathcal{C}^N\to\mathcal{P}$ be a *social welfare function*.

* A social welfare function $f:\mathcal{C}^N\to\mathcal{P}$ is **independent of irrelevant alternatives** if and only if given any two alternatives $x,y\in A$ and any two profiles $\mathbf{P},\mathbf{P}'\in\mathcal{C}^N$ such that for all voters $i\in N$, we have $xP\_iy$ if and only if $xP\_i'y$; then we also have $xf(\mathbf{P})y$ if and only if $xf(\mathbf{P}')y$.
* A social welfare function $f:\mathcal{C}^N\to\mathcal{P}$ is **Pareto optimal** if and only if for all profiles $\mathbf{P}\in\mathcal{C}^N$ and all alternatives $x,y\in A$, we have that $xP\_iy$ for all voters $i\in N$ implies that $xf(\mathbf{P})y$.

Given any profile $\mathbf{P}\in\mathcal{C}^N$ and any alternative $x\in A$, let $L\_i(x,\mathbf{P})=\{y\in A\mid xP\_iy\}$ be *voter $i$'s lower contour set at $x$*, and let $\mathcal{C}^N(\mathbf{P},x)=\{\mathbf{P}'\in\mathcal{C}^N\mid(\forall i\in N)[L\_i(x,\mathbf{P})\subseteq L\_i(x,\mathbf{P}')]\}$ be the *set of preference profiles in which alternative $x$ has not fallen in any voter's preference*.

* A social welfare function $f:\mathcal{C}^N\to\mathcal{P}$ is **monotonic** if and only if for all alternatives $x\in A$, all profiles $\mathbf{P}\in\mathcal{C}^N$ and all profiles $\mathbf{P}'\in\mathcal{C}^N(\mathbf{P},x)$; we have that for all alternatives $y\in A\backslash\{x\}$, $xf(\mathbf{P})y$ implies $xf(\mathbf{P}')y$.

By [Arrow (1950)](https://www.journals.uchicago.edu/doi/abs/10.1086/256963), we know that under the unrestricted strict preference domain, a social welfare function satisfying Pareto optimality and independence of irrelevant alternatives is dictatorial, and thus monotonic. My feeling by reading contemporary versions of [Arrow’s (1950)](https://www.journals.uchicago.edu/doi/abs/10.1086/256963) proof is that the unrestricted domain is necessary to attain the dictatorial result, but not monotonicity. However, I cannot prove it nor find a counter-example.

**Can you provide any help?**

**Tags:** microeconomics, reference-request, theory, social-welfare, social-choice

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## 1 answers

### Answer by username_1 on 2024-11-12. Score: 1 | Accepted answer

Here is a partial answer that assumes $a\geq 4$. The main point is that the existence of a Condorcet winner doesn't do that much in this setting.

Let $x$ be any alternative, and let $\mathcal{C}\_x$ be the set of linear orders that rank $x$ top. Then, any profile of linear orders over the remaining $a-1$ alternatives corresponds to a unique profile in $\mathcal{C}\_x^N$. Moreover, a Condorcet winner always exists, namely $x$. Every social welfare function defined on profiles of linear orders over the remaining $a-1$ alternatives that satisfy the conditions must clearly extend to a unique social welfare function on $\mathcal{C}\_x^N$. So a dictator over these $a-1$ alternatives exists by Arrow's theorem. Since this works for all $x$, a plain dictator must exist too.


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