Finding the Dictator in Arrow's Impossibility Theorem Arrow's Impossibility Theorem states that if we have at least three different social states and a finite number of individuals (voters), any social welfare function that satisfies the conditions of universal domain ( _U_ ), independence of irrelevant alternatives ( _IIA_ ), and the Pareto principle ( _PP_ ) must have a dictator. Given a social welfare function that respects _U, IIA_ , and _PP_ , and a social profile (a set of individual preferences), how would one find the dictator? What is the most efficient algorithm to do so? Has anyone looked into the computational complexity of this?

The social welfare function ranks the $n$ available alternatives as $x_1 \succ x_2 \succ x_3 \succ \ldots \succ x_n$.

Every agent whose ranking over the available alternatives matches the social ranking is a "dictator".

Note that "dictator" in Arrow's theorem does not necessarily mean that that person decides the social ranking. As far as the individual and the society have the same ranking, that person is indistinguishable from a dictator.