<p>I am reading <a href="https://link.springer.com/article/10.1007/s00355-003-0291-1" rel="nofollow noreferrer">Eliaz (2004</a>) for a general treatment of social-choice impossibilities. I am confused by the fact that it seems that we should get completeness even in situations where we should not have it.</p>
<p>Some terminology: let <span class="math-container">$V$</span> be a set of <span class="math-container">$n \geq 2$</span> voters and <span class="math-container">$X$</span> a set of 3 or more alternatives. Let <span class="math-container">$P$</span> be the set of all linear orderings on <span class="math-container">$X$</span>, and let <span class="math-container">$\succ$</span> be a typical element of <span class="math-container">$P$</span>. Let <span class="math-container">$R$</span> be the set of all binary relations on <span class="math-container">$X$</span>. A <strong>social aggregator</strong> is a function <span class="math-container">$F: P \to R$</span>. Properties of social aggregators may include:</p>
<ul>
<li><strong>C</strong>ompleteness: for all <span class="math-container">$(a,b) \in X^2, a R b \lor b R a$</span></li>
<li><strong>B</strong>est <strong>A</strong>lternative: there is some <span class="math-container">$a \in X: a R b $</span> for all <span class="math-container">$b$</span></li>
<li><strong>Par</strong>eto: if <span class="math-container">$a \succ_i b$</span> for all <span class="math-container">$i \in V$</span>, then <span class="math-container">$b \lnot R a$</span></li>
<li><strong>P</strong>reference <strong>R</strong>eversal: for all <span class="math-container">$\succ, \succ' \in P$</span>, if <span class="math-container">$a F(\succ) b$</span> and <span class="math-container">$a \lnot F(\succ') b$</span>, then for some <span class="math-container">$i \in V$</span>, <span class="math-container">$a \succ b$</span> and <span class="math-container">$b \succ' a$</span>.</li>
</ul>
<p>I now want to show the following:</p>
<p><strong>Claim.</strong> Suppose <span class="math-container">$F$</span> satisfies <strong>BA</strong>, <strong>Par</strong>, <strong>PR</strong>. Then <span class="math-container">$F$</span> satisfies <strong>C</strong>.</p>
<p><em>Proof</em>. Consider some profile <span class="math-container">$\succ$</span> where for all <span class="math-container">$i \in N$</span>, <span class="math-container">$a \succ c$</span> and <span class="math-container">$b \succ c$</span> for all <span class="math-container">$c \in X \setminus \{a,b\}$</span> for some <span class="math-container">$a \neq b$</span> (i.e. <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are the two top alternatives for all voters). By Pareto, <span class="math-container">$c \lnot F(\succ) a,b$</span>. By best alternative, then <span class="math-container">$a F(\succ) b$</span> or <span class="math-container">$b F(\succ) a$</span>. WLOG, suppose <span class="math-container">$a F(\succ) b$</span>. Then consider any profile <span class="math-container">$\succ'$</span> such that for all <span class="math-container">$i$</span>, <span class="math-container">$a \succ_i b \iff a \succ'_i b$</span>. By preference reversal, <span class="math-container">$a F(\succ') b$</span>. Because we have not placed any restriction on <span class="math-container">$\{i \mid a \succ_i b\}$</span>, and <span class="math-container">$(a,b)$</span> was chosen arbitrarily, we conclude that <span class="math-container">$F$</span> satisfies completeness. QED.</p>
<p>Is there a mistake? If not, doesn't Eliaz's results (including those connecting PR to monotonicity) imply that a monotonic social choice function (the subject of Gibbard-Statterthwaite's theorem) must be complete?</p>

CC BY-SA 4.0
2022-06-02T19:40:09.077Z
Your proof looks correct to me. But your definitions of BA and PR differ from the ones used by Eliaz.
Michael Greinecker
CC BY-SA 4.0
2022-06-02T19:44:22.137Z
I see how the difference in my definition of PR might change the result, but the difference in the definition of BA should be immaterial, no?
raving-bandit
CC BY-SA 4.0
2022-06-02T19:46:50.863Z
I think so too, apart from the resulting relation possibly not being reflexive.
Michael Greinecker
CC BY-SA 4.0
2022-06-02T20:34:48.017Z
Thank you! I am looking at Proposition 1 in Eliaz. Let PR' be preference reversal as defined by him (weaker than my definition). I'm having trouble seeing if my definition of PR also fits this result, since for $F : P \to R_0 =  \{(a,b) \mid b \in X \} \subset R$, it seems that PR  and PR' are equivalent: if $a R_0 b$, then $b \lnot R_0 a$ (assuming $a \neq b$), and thus PR and PR' collapse, no? So don't we still get completeness of $R_0$? (Assume reflexivity for now).
raving-bandit
CC BY-SA 4.0
2022-06-02T20:56:17.303Z
Your definition of preference reversal differs from the one of ELiaz in more than one way. The one of Eliaz says that if $a$ is socially strictly preferred to $b$ in one profile and $b$ socially weakly preferred to $a$ in the second profile, then someone must prefer $a$ to $b$ in the former profile who prefers $b$ to $a$ in the latter profile. Your definition says that if $a$ is socially weakly preferred to $b$ in one profile but not in another profile, then someone must prefer $a$ to $b$ in the former profile who prefers $b$ to $a$ in the latter profile. So the definition differs in two ways
Michael Greinecker
CC BY-SA 4.0
2022-06-02T20:57:45.650Z
Now, it is true that every weak preference between two different alternatives is a strict preference in $R_0$. But $a$ not being preferred to $b$ in $R_0$ does not mean that $b$ is preferred to $a$ in $R_0$.
Michael Greinecker
CC BY-SA 4.0
2022-06-02T21:22:14.207Z
I see the difference better now, thanks. So my definition is not necessarily stronger, right? (Although it yields completeness, which rules out social choice functions, and that's a bit too strong). I'm going to think more about under what conditions PR and PR' are equivalent. You are always extremely helpful!
raving-bandit
<microeconomics><theory><choice-theory><social-choice>
What social aggregators are complete?
raving-bandit
# What social aggregators are complete?

(Asked by: username_0 on 2022-06-02)

I am reading [Eliaz (2004](https://link.springer.com/article/10.1007/s00355-003-0291-1)) for a general treatment of social-choice impossibilities. I am confused by the fact that it seems that we should get completeness even in situations where we should not have it.

Some terminology: let $V$ be a set of $n \geq 2$ voters and $X$ a set of 3 or more alternatives. Let $P$ be the set of all linear orderings on $X$, and let $\succ$ be a typical element of $P$. Let $R$ be the set of all binary relations on $X$. A **social aggregator** is a function $F: P \to R$. Properties of social aggregators may include:

* **C**ompleteness: for all $(a,b) \in X^2, a R b \lor b R a$
* **B**est **A**lternative: there is some $a \in X: a R b $ for all $b$
* **Par**eto: if $a \succ\_i b$ for all $i \in V$, then $b \lnot R a$
* **P**reference **R**eversal: for all $\succ, \succ' \in P$, if $a F(\succ) b$ and $a \lnot F(\succ') b$, then for some $i \in V$, $a \succ b$ and $b \succ' a$.

I now want to show the following:

**Claim.** Suppose $F$ satisfies **BA**, **Par**, **PR**. Then $F$ satisfies **C**.

*Proof*. Consider some profile $\succ$ where for all $i \in N$, $a \succ c$ and $b \succ c$ for all $c \in X \setminus \{a,b\}$ for some $a \neq b$ (i.e. $a$ and $b$ are the two top alternatives for all voters). By Pareto, $c \lnot F(\succ) a,b$. By best alternative, then $a F(\succ) b$ or $b F(\succ) a$. WLOG, suppose $a F(\succ) b$. Then consider any profile $\succ'$ such that for all $i$, $a \succ\_i b \iff a \succ'\_i b$. By preference reversal, $a F(\succ') b$. Because we have not placed any restriction on $\{i \mid a \succ\_i b\}$, and $(a,b)$ was chosen arbitrarily, we conclude that $F$ satisfies completeness. QED.

Is there a mistake? If not, doesn't Eliaz's results (including those connecting PR to monotonicity) imply that a monotonic social choice function (the subject of Gibbard-Statterthwaite's theorem) must be complete?

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

> **Comments:**
> **username_1**: Your proof looks correct to me. But your definitions of BA and PR differ from the ones used by Eliaz.  
> **username_0**: I see how the difference in my definition of PR might change the result, but the difference in the definition of BA should be immaterial, no?  
> **username_1**: I think so too, apart from the resulting relation possibly not being reflexive.  
> **username_0**: Thank you! I am looking at Proposition 1 in Eliaz. Let PR' be preference reversal as defined by him (weaker than my definition). I'm having trouble seeing if my definition of PR also fits this result, since for $F : P \to R_0 =  \{(a,b) \mid b \in X \} \subset R$, it seems that PR  and PR' are equivalent: if $a R_0 b$, then $b \lnot R_0 a$ (assuming $a \neq b$), and thus PR and PR' collapse, no? So don't we still get completeness of $R_0$? (Assume reflexivity for now).  
> **username_1**: Your definition of preference reversal differs from the one of ELiaz in more than one way. The one of Eliaz says that if $a$ is socially strictly preferred to $b$ in one profile and $b$ socially weakly preferred to $a$ in the second profile, then someone must prefer $a$ to $b$ in the former profile who prefers $b$ to $a$ in the latter profile. Your definition says that if $a$ is socially weakly preferred to $b$ in one profile but not in another profile, then someone must prefer $a$ to $b$ in the former profile who prefers $b$ to $a$ in the latter profile. So the definition differs in two ways  
> **username_1**: Now, it is true that every weak preference between two different alternatives is a strict preference in $R_0$. But $a$ not being preferred to $b$ in $R_0$ does not mean that $b$ is preferred to $a$ in $R_0$.  
> **username_0**: I see the difference better now, thanks. So my definition is not necessarily stronger, right? (Although it yields completeness, which rules out social choice functions, and that's a bit too strong). I'm going to think more about under what conditions PR and PR' are equivalent. You are always extremely helpful!  

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## 0 answers

