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social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
, May's theorem, also called the general possibility theorem, says that
majority vote A majority is more than half of a total; however, the term is commonly used with other meanings, as explained in the "#Related terms, Related terms" section below. It is a subset of a Set (mathematics), set consisting of more than half of the se ...
is the unique ranked social choice function between two candidates that satisfies the following criteria: *
Anonymity Anonymity describes situations where the acting person's identity is unknown. Anonymity may be created unintentionally through the loss of identifying information due to the passage of time or a destructive event, or intentionally if a person cho ...
– 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 Kenneth O. May (July8, 1915December 1977) was an American mathematician and historian of mathematics, who developed May's theorem. May was a prime mover behind the International Commission on the History of Mathematics, and was the first edit ...
in 1952. Various modifications have been suggested by others since the original publication. If
rated voting Rated, evaluative, graded, or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate, by giving each one a grade on a separate scale. The distribution of ratings for each candidate ...
is allowed, a wide variety of rules satisfy May's conditions, including
score voting Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...
or
highest median voting rules The highest median voting rules are a class of graded voting rules where the candidate with the highest median rating is elected. The various highest median rules differ in their treatment of ties, i.e., the method of ranking the candidates with ...
.
Arrow's theorem Arrow's impossibility theorem is a key result in social choice theory showing that no Ordinal utility, ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Kenneth Arrow, Arrow showed no ...
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'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 In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationality of preference aggregation rules (collective decision rules), such as voting rules. It is an indicator of the extent to which an aggregation ...
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 : F : \^N \to \ which aggregates individuals' preferences into a single preference. An -
tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
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

*
Social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
*
Arrow's impossibility theorem Arrow's impossibility theorem is a key result in social choice theory showing that no ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Arrow showed no such rule can satisfy the ind ...
*
Condorcet paradox In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory. The result implies that it is logically impossible for any voting system to guarante ...
* Gibbard–Satterthwaite theorem *
Gibbard's theorem In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properti ...


Notes

#May, Kenneth O. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", ''Econometrica'', Vol. 20, Issue 4, pp. 680–684. #Mark Fey,
May’s Theorem with an Infinite Population
, ''Social Choice and Welfare'', 2004, Vol. 23, issue 2, pages 275–293. #Goodin, Robert and Christian List (2006). "A conditional defense of plurality rule: generalizing May's theorem in a restricted informational environment," ''American Journal of Political Science'', Vol. 50, issue 4, pages 940-949.


References

*Alan D. Taylor (2005). ''Social Choice and the Mathematics of Manipulation'', 1st edition, Cambridge University Press. {{isbn, 0-521-00883-2. Chapter 1.
Logrolling, May’s theorem and Bureaucracy
Social choice theory 1952 in science Theorems in discrete mathematics Voting theory