<p>Yes, this is correct.  If <span class="math-container">$i=1$</span> is a dictator for the social choice rule <span class="math-container">$f$</span>, then this means that for <em>any</em> profile <span class="math-container">$\langle R_i \rangle_{i=1}^I$</span>, we have <span class="math-container">$f(\langle R_i \rangle_{i=1}^I)=R_1$</span>.  Hence, in your example, <span class="math-container">$f(\langle R^*_i \rangle_{i=1}^I)=R^*_1$</span>.</p>

CC BY-SA 4.0
2023-11-26T07:09:01.353Z
2023-11-26T07:09:01.353Z
Marcus Pivato
<p>Suppose <span class="math-container">$R$</span> is a social ordering, <span class="math-container">$f$</span> is the social choice function, and <span class="math-container">$R_i$</span> is an individual preference. A profile of individual preference is <span class="math-container">$&lt;R_i&gt;$</span>.</p>
<p><span class="math-container">$f(&lt;R_i&gt;)=R$</span></p>
<p>Suppose  <span class="math-container">$i=1$</span> is the dictator. This means that <span class="math-container">$R=R_1$</span>.</p>
<p>Now consider a new profile <span class="math-container">$&lt;R^*&gt;$</span> where only the taste of the dictator is updated: <span class="math-container">$R^*_i=R_i$</span> for all <span class="math-container">$i\neq 1$</span>; and <span class="math-container">$R^*_1\neq R_1$</span>.</p>
<p>In this case, do we still have <span class="math-container">$R_1^*=R^*$</span>?</p>

<microeconomics><theory><decision-theory><social-welfare>
Is my understanding of Arrow's dictatorship correct? The dictator is free to update her preference and the social choice will always follow her taste
dodo
# Is my understanding of Arrow's dictatorship correct? The dictator is free to update her preference and the social choice will always follow her taste

(Asked by: username_0 on 2023-11-25)

Suppose $R$ is a social ordering, $f$ is the social choice function, and $R\_i$ is an individual preference. A profile of individual preference is $$.

$f()=R$

Suppose $i=1$ is the dictator. This means that $R=R\_1$.

Now consider a new profile $$ where only the taste of the dictator is updated: $R^\*\_i=R\_i$ for all $i\neq 1$; and $R^\*\_1\neq R\_1$.

In this case, do we still have $R\_1^\*=R^\*$?

**Tags:** microeconomics, theory, decision-theory, social-welfare

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## 1 answers

### Answer by username_1 on 2023-11-26. Score: 3 | Accepted answer

Yes, this is correct. If $i=1$ is a dictator for the social choice rule $f$, then this means that for *any* profile $\langle R\_i \rangle\_{i=1}^I$, we have $f(\langle R\_i \rangle\_{i=1}^I)=R\_1$. Hence, in your example, $f(\langle R^\*\_i \rangle\_{i=1}^I)=R^\*\_1$.


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