Can you give an example of a social choice rule and a social welfare function that violates the conditions of Weak Pareto criterian and non dictatorship simultaneously?


Such an example does not exist.

Suppose non-dictatorship is violated. WLOG, let individual $1$ be the dictator among $n$ individuals. That is, the social choice function always selects an alternative that ranks highest in $1$'s preference, i.e. $$f(\succ_1,\dots,\succ_n)\in\arg\max\{\succ_1 | x\in X\}.$$

For weak Pareto efficiency to also be violated, we must have two alternatives $y,z\in X$ such that $y\succ_iz$ for every $i=1,\dots,n$, but the social choice function chooses $z$, i.e. $f(\succ_1,\dots,\succ_n)=z$. Yet this clearly contradicts the above condition of dictatorship, in that $z\notin\arg\max\{\succ_d | x\in X\}$.

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