May's theorem

In social choice theory, May's theorem, also called the general possibility theorem, says that majority vote is the unique ranked social choice function between two candidates that satisfies the following criteria:


 * Anonymity – each voter is treated identically,
 * Neutrality – each candidate is treated identically,
 * Positive responsiveness – a voter changing their mind to support a candidate cannot cause that candidate to lose, had the candidate not also lost without that voters' support.

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 (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 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 $A$ and $B$ be two possible choices, often called alternatives or candidates. A preference is then simply a choice of whether $A$, $B$, or neither is preferred. Denote the set of preferences by ${A, B, 0}$, where $0$ represents neither.

Let $N$ be a positive integer. In this context, a ordinal (ranked) social choice function is a function


 * $$F : \{A,B,0\}^N \to \{A,B,0\}$$

which aggregates individuals' preferences into a single preference. An $N$-tuple $(R1, …, RN) ∈ {A, B, 0}N$ 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 $A$ is greater than the number of preferences for $B$, simple majority voting returns $A$,
 * If the number of preferences for $A$ is less than the number of preferences for $B$, simple majority voting returns $B$,
 * If the number of preferences for $A$ is equal to than the number of preferences for $B$, simple majority voting returns $0$.

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 each voter identically, in that permuting the order of the voters does not change the social choice,
 * 2) (Neutrality) The social choice function treats all outcomes the same, in that reversing each set of preferences reverses the group preference,
 * 3) (Positive responsiveness) If the social choice was to prefer $A$ and a voter who previously preferred $B$ changes their preference to $A$, then the social choice is still $A$.