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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

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, ...
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