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

Question

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 $<R_i>$.

$f(<R_i>)=R$

Suppose $i=1$ is the dictator. This means that $R=R_1$.

Now consider a new profile $<R^*>$ 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 $R_1^*=R^*$?

Answer 1

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, $f(\langle R^*_i \rangle_{i=1}^I)=R^*_1$.