Arrow’s impossibility theorem states that there is no procedure for aggregating individual preference orderings into a collective preference ordering that satisfies certain apparently desirable axioms. This has often been taken to mean that there is no such thing as a good voting system. Actual voting systems, however, do not typically attempt to generate a (complete) social ordering. The goal, rather, is to simply select one candidate who will govern — and the task of the ranking the others, while perhaps interesting, is not of immediate practical importance. This raises a simple question: does Arrow’s result extend to preference aggregation procedures that choose a single candidate based on a profile of individual preferences?
To make this more precise, let me (informally) restate Arrow’s axioms as adapted to the problem at hand:
Unrestricted domain. The procedure chooses a best candidate for every possible list of individual orderings.
Weak Pareto. If all voters prefer candidate A to candidate B, then candidate B is not elected.
Non-dictatorship. There is no voter who alone determines which candidate is elected (regardless of the other voters’ preference orderings).
Independence of Irrelevant Alternatives. If candidate A is elected when opposed to a set of rival candidates, then A will still be elected when opposed by some subset of this set of rival candidates.
(Notice that the transitivity axiom cannot, it would seem, be adapted to this context.)
Question: does there exist a voting scheme that satisfies all of the axioms listed above?