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Insocial choice theory,May's theorem, also called thegeneral possibility theorem,[1] says thatmajority vote is the uniqueranked social choice function between two candidates that satisfies the following criteria:
The theorem was first published byKenneth May in 1952.[1]
Various modifications have been suggested by others since the original publication. Ifrated voting is allowed, a wide variety of rules satisfy May's conditions, includingscore voting orhighest median voting rules.
Arrow's theorem does not apply to the case of two candidates (when there are trivially no "independent alternatives"), so this possibility result can be seen as the mirror analogue of that theorem. Note that anonymity is a stronger requirement than Arrow'snon-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 theNakamura 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.[citation needed]
LetA andB be two possible choices, often called alternatives or candidates. Apreference is then simply a choice of whetherA,B, or neither is preferred.[1] Denote the set of preferences by{A,B, 0}, where0 represents neither.
LetN be a positive integer. In this context, aordinal (ranked)social choice function is a function
which aggregates individuals' preferences into a single preference.[1] AnN-tuple(R1, …,RN) ∈ {A,B, 0}N of voters' preferences is called apreference profile.
Define a social choice function calledsimple majority voting as follows:[1]
May's theorem states that simple majority voting is the unique social welfare function satisfying all three of the following conditions:[1]
