<p>Let <span class="math-container">$S$</span> be a set of <em>social states</em> with at least three elements. Let <span class="math-container">$\mathcal{R}$</span> be the set of complete and transitive relations on <span class="math-container">$S$</span>. A function <span class="math-container">$f:\mathcal{R}^n\to\mathcal{R}$</span> satisfies <em>independence of irrelevant alternatives</em> if the restriction of <span class="math-container">$f(R)$</span> to <span class="math-container">$A\subseteq S$</span> depends only on the corresponding restriction of <span class="math-container">$\mathcal{R}^n$</span> for every subset <span class="math-container">$A\subseteq S$</span>. For <span class="math-container">$R\in\mathcal{R}^n$</span>, and <span class="math-container">$i=1,\ldots,n$</span>, let <span class="math-container">$R_i$</span> be the <span class="math-container">$i$</span>th entry of <span class="math-container">$R$</span>.</p>
<p>Wilson's (first) theorem in the paper says that if <span class="math-container">$f$</span> satisfies independence of irrelevant alternatives, and if for any <span class="math-container">$x,y\in S$</span> there exists some <span class="math-container">$R\in\mathcal{R}^n$</span> such that <span class="math-container">$x f(R) y$</span>, then <span class="math-container">$f$</span> is either constant with universal indifference as its value, or there is some <span class="math-container">$i$</span> such that <span class="math-container">$x f(R)y$</span> holds exactly when <span class="math-container">$xR_i y$</span> holds (<span class="math-container">$i$</span> is a dictator), or there is some <span class="math-container">$i$</span> such that <span class="math-container">$xf(R)y$</span> holds exactly when <span class="math-container">$yR_i x$</span> holds (<span class="math-container">$i$</span> is an inverse dictator).</p>
<p>In Arrow's theorem, the case of constant <span class="math-container">$f$</span> or the existence of an inverse dictator is excluded. But the Pareto condition used by Arrow is also stronger than the condition that for any <span class="math-container">$x,y\in $</span> there exists some <span class="math-container">$R\in\mathcal{R}^n$</span> such that <span class="math-container">$x f(R) y$</span>.</p>

CC BY-SA 4.0
2024-10-10T04:22:11.803Z
"In Arrow's theorem, the case of constant for the existence of an inverse dictator is excluded." I think they are exactly excluded by Pareto, so pareto is not redundent in Arrow's theorem?
High GPA
CC BY-SA 4.0
2024-10-10T08:55:34.727Z
Exactly, Pareto excludes this two cases and is not redundant in Arrow's theorem.
Michael Greinecker
CC BY-SA 4.0
2024-10-09T14:46:46.433Z
2024-10-10T08:09:46.727Z
2024-10-10T08:09:46.727Z
Michael Greinecker
<p>I recently read a note on &quot;Arrow's generalized impossibility theorem&quot;, claiming that we don't need the pareto principle.</p>
<p><a href="https://www.sciencedirect.com/science/article/pii/0022053172900518" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/pii/0022053172900518</a></p>
<p>I wonder if this note means that the Arrow's theorem we learnt in class does not need the pareto condition.</p>
<p>The paper is difficult to read and I am not sure it is exactly the same set-up as the classic impossibility theorem.</p>

<microeconomics><macroeconomics><game-theory><social-welfare><social-choice>
What is the "Arrow's generalized impossibility theorem" noted by Robert Wilson?
High GPA
# What is the "Arrow's generalized impossibility theorem" noted by Robert Wilson?

(Asked by: username_0 on 2024-10-09)

I recently read a note on "Arrow's generalized impossibility theorem", claiming that we don't need the pareto principle.

<https://www.sciencedirect.com/science/article/pii/0022053172900518>

I wonder if this note means that the Arrow's theorem we learnt in class does not need the pareto condition.

The paper is difficult to read and I am not sure it is exactly the same set-up as the classic impossibility theorem.

**Tags:** microeconomics, macroeconomics, game-theory, social-welfare, social-choice

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## 1 answers

### Answer by username_1 on 2024-10-09. Score: 2 | Accepted answer

Let $S$ be a set of *social states* with at least three elements. Let $\mathcal{R}$ be the set of complete and transitive relations on $S$. A function $f:\mathcal{R}^n\to\mathcal{R}$ satisfies *independence of irrelevant alternatives* if the restriction of $f(R)$ to $A\subseteq S$ depends only on the corresponding restriction of $\mathcal{R}^n$ for every subset $A\subseteq S$. For $R\in\mathcal{R}^n$, and $i=1,\ldots,n$, let $R\_i$ be the $i$th entry of $R$.

Wilson's (first) theorem in the paper says that if $f$ satisfies independence of irrelevant alternatives, and if for any $x,y\in S$ there exists some $R\in\mathcal{R}^n$ such that $x f(R) y$, then $f$ is either constant with universal indifference as its value, or there is some $i$ such that $x f(R)y$ holds exactly when $xR\_i y$ holds ($i$ is a dictator), or there is some $i$ such that $xf(R)y$ holds exactly when $yR\_i x$ holds ($i$ is an inverse dictator).

In Arrow's theorem, the case of constant $f$ or the existence of an inverse dictator is excluded. But the Pareto condition used by Arrow is also stronger than the condition that for any $x,y\in $ there exists some $R\in\mathcal{R}^n$ such that $x f(R) y$.

> **Comments:**
> **username_0**: "In Arrow's theorem, the case of constant for the existence of an inverse dictator is excluded." I think they are exactly excluded by Pareto, so pareto is not redundent in Arrow's theorem?  
> **username_1**: Exactly, Pareto excludes this two cases and is not redundant in Arrow's theorem.  

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