May's theorem


In social choice theory, May's theorem, also called the general possibility theorem, says that simple [majority voting|majority vote] is the unique ranked social choice function between two candidates that satisfies the following criteria:
  • Anonymity: the decision rule treats each voter identically. Who casts a vote makes no difference; the voter's identity need not be disclosed.
  • Neutrality: the decision rule treats each alternative or candidate equally.
  • Decisiveness: if the vote is tied, adding a single voter will break the tie.Positive response: If a voter changes a preference, MR never switches the outcome against that voter. If the outcome the voter now prefers would have won, it still does so.
  • Ordinality: the decision rule relies only on which of two outcomes a voter prefers, not how much.
  • * This can be replaced by strategyproofness, i.e. every person's dominant strategy is to honestly disclose their preferences.
The theorem was first published by Kenneth May in 1952.
Various modifications have been suggested by others since the original publication. If rated voting is allowed, a wide variety of rules satisfy May's conditions, including score voting or highest median voting rules.
Arrow's theorem does not apply to the case of two candidates, so this possibility result can be seen as the mirror analogue of that theorem. Note that anonymity is a stronger requirement than Arrow's 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

Let and be two possible choices, often called alternatives or candidates. A preference is then simply a choice of whether,, or neither is preferred. Denote the set of preferences by, where represents neither.
Let be a positive integer. In this context, a ordinal (ranked) social choice function is a function
which aggregates individuals' preferences into a single preference. An -tuple of voters' preferences is called a preference profile.
Define a social choice function called simple majority voting as follows:
  • If the number of preferences for is greater than the number of preferences for, simple majority voting returns,
  • If the number of preferences for is less than the number of preferences for, simple majority voting returns,
  • If the number of preferences for is equal to the number of preferences for, simple majority voting returns.
May's theorem states that simple majority voting is the unique social welfare function satisfying all three of the following conditions:
  1. Anonymity: The social choice function treats all voters the same, i.e. permuting the order of the voters does not change the result.
  2. Neutrality: The social choice function treats all outcomes the same, i.e. permuting the order of the outcomes does not change the result.
  3. Positive responsiveness: If the social choice was indifferent between and, but a voter who previously preferred changes their preference to, then the social choice becomes.