May's theorem

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 {{math|{A, B, 0}}}, where represents neither. Let 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 -tuple {{math|(R, …, R) ∈ {A, B, 0}}} 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: • Anonymity: The social choice function treats all voters the same, i.e. permuting the order of the voters does not change the result. • Neutrality: The social choice function treats all outcomes the same, i.e. permuting the order of the outcomes does not change the result. • 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 . ==See also==