social choice theory Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...

, May's theorem states that

simple majority voting Majority rule is a principle that means the decision-making power belongs to the group that has the most members. In politics, majority rule requires the deciding vote to have majority, that is, more than half the votes. It is the binary deci ...

is the only anonymous,

neutral Neutral or neutrality may refer to: Mathematics and natural science Biology * Neutral organisms, in ecology, those that obey the unified neutral theory of biodiversity Chemistry and physics * Neutralization (chemistry), a chemical reaction in ...

, and positively responsive social choice function between two alternatives. Further, this procedure is resolute when there are an odd number of voters and ties (indecision) are not allowed.

Kenneth May Kenneth O. May (July 8, 1915, in Portland, Oregon – December 1977, in Toronto) 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 ...

first published this theorem in 1952. Various modifications have been suggested by others since the original publication. Mark Fey extended the proof to an infinite number of voters.

Robert Goodin Robert 'Bob' E. Goodin (born 30 November 1950) was Professor of Government at the University of Essex and is now Distinguished Professor of Philosophy and Social and Political theory at the Australian National University. Biography Goodin atten ...

and

Christian List Christian List (born 1973) is a German philosopher and political scientist who serves as professor of philosophy and decision theory at the Ludwig Maximilian University of Munich and co-director of the Munich Center for Mathematical Philosoph ...

showed that, among methods of aggregating first-preference votes over multiple alternatives, plurality rule uniquely satisfies May's conditions; under approval balloting, a similar statement can be made about approval voting.

Arrow's theorem Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syste ...

in particular does not apply to the case of two candidates, so this possibility result can be seen as a mirror analogue of that theorem. (Note that anonymity is a stronger form of 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 r ...

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

*Condition 1. The group decision function sends each set of preferences to a unique winner. (resolute, unrestricted domain) *Condition 2. The group decision function treats each voter identically. (anonymity) *Condition 3. The group decision function treats both outcomes the same, in that reversing each set of preferences reverses the group preference. (neutrality) *Condition 4. If the group decision was 0 or 1 and a voter raises a vote from −1 to 0 or 1, or from 0 to 1, the group decision is 1. (positive responsiveness) ''Theorem:'' A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

it is the simple majority method.

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