Social choice theory

Question

Is it possible for a social choice rule to simultaneously violate the condition of non dictatorship and weak pareto property? Non dictatorship is when there is no individual such that if he prefers x over y, then so does the society irrespective of what the society feels. And Pareto property is that if all individuals prefer x over y, then so should the society. It is about how society preserves total agreement. 

I don't think it's possible because of the following reason. Now, if we have dictatorship and rest all also agree with him, then society is essentially preserving that opinion (so Pareto property satisfied). On the other hand, if Pareto property is not satisfied, ie total agreement of x preferred over y by each individual doesn't not lead to society choosing x over y, this means there is no individual such that if he prefers x over y, then society prefers x over y. So non dictatorship is satisfied. Please see if this line of thought is correct.

Answer 1

You are correct that it is not possible to violate Weak Pareto and Nondictatorship at the same time. But your explanation (second paragraph) is a bit muddled. Here is how I would put it.

To prove, "it is not possible to violate Weak Pareto and Nondictatorship at the same time", it suffices to prove: "Any Dictatorship satisfies Weak Pareto." To prove this, suppose F is a social choice rule in which voter i is a dictator. So for any alternatives a and b, if i prefers a over b, then so does the output of rule F. To see that F satisfies weak Pareto, let a and b be social alternatives, and suppose that everyone prefers a over b. Then in particular, i does. Thus, the output of F prefers a over b (because i is a dictator in F). Therefore, F satisfies Weak Pareto.