# Example which violates weak pareto and non dictatorship

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?

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 $$z\notin\arg\max\{\succ_d | x\in X\}$$.