May's theorem
In social choice theory, May's theorem states that simple majority voting is the only anonymous, neutral, and positively responsive social choice function between two alternatives. Further, this procedure is resolute when there are an odd number of voters and ties (indecision) are not allowed. Kenneth May first published this theorem in 1952. Various modifications have been suggested by others since the original publication. Mark Fey extended the proof to an infinite number of voters. Robert Goodin and Christian List showed that, among methods of aggregating first-preference votes over multiple alternatives, plurality rule uniquely satisfies May 's conditions; under approval balloting, a similar statement can be made about approval voting. Arrow's theorem in particular does not apply to the case of two candidates, so this possibility result can be seen as a mirror analogue of that theorem. (Note that anonymity is a stronger form of non-dictatorship.) Another way of explaining the fact that simple majority voting can successfully deal with at most two alternatives is to cite Nakamura's theorem. The theorem states that the number of alternatives that a rule can deal with successfully is less than the Nakamura number of the rule . The Nakamura number of simple majority voting is 3, except in the case of four voters. Supermajority rules may have greater Nakamura numbers. Formal statement. "Theorem:" A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.