Misplaced Pages

A joint Politics and Economics series
Social choice and electoral systems
Social choice Mechanism design Comparative politics Comparison List (By country)
Single-winner methodsSingle vote - plurality methods First preference plurality (FPP) Two-round (US: Jungle primary) Partisan primary Instant-runoff UK: Alternative vote (AV) US: Ranked-choice (RCV) Condorcet methods Condorcet-IRV Round-robin voting Minimax Schulze Ranked pairs Maximal lottery Positional voting Plurality (el. IRV) Borda count (el. Baldwin) Antiplurality (el. Coombs) Cardinal voting Score voting Approval voting Majority judgment STAR voting
Proportional representationParty-list Apportionment Highest averages Largest remainders National remnant Biproportional List type Closed list Open list Panachage List-free PR Localized list Quota-remainder methods Hare STV Schulze STV CPO-STV Quota Borda Approval-based committees Thiele's method Phragmen's method Expanding approvals rule Method of equal shares Fractional social choice Direct representation Interactive representation Liquid democracy Fractional approval voting Maximal lottery Random ballot Semi-proportional representation Cumulative SNTV Limited voting
Mixed systemsBy results of combination Mixed-member majoritarian Mixed-member proportional By mechanism of combination Non-compensatory Parallel (superposition) Coexistence Conditional Fusion (majority bonus) Compensatory Seat linkage system UK: 'AMS' NZ: 'MMP' Vote linkage system Negative vote transfer Mixed ballot Supermixed systems Dual-member proportional Rural–urban proportional Majority jackpot By ballot type Single vote Double simultaneous vote Dual-vote
Paradoxes and pathologiesSpoiler effects Spoiler effect Cloning paradox Frustrated majorities paradox Center squeeze Pathological response Perverse response Best-is-worst paradox No-show paradox Multiple districts paradox Strategic voting Lesser evil voting Exaggeration Truncation Turkey-raising Paradoxes of majority rule Tyranny of the majority Discursive dilemma Conflicting majorities paradox
Social and collective choiceImpossibility theorems Arrow's theorem Majority impossibility Moulin's impossibility theorem McKelvey–Schofield chaos theorem Gibbard's theorem Positive results Median voter theorem Condorcet's jury theorem May's theorem Condorcet dominance theorems Harsanyi's utilitarian theorem
Politics portal Economics portal Mathematics portal
v t e

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'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 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

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

which aggregates individuals' preferences into a single preference. An N-tuple (R₁, …, R_N) ∈ {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 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 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 A and B, but a voter who previously preferred B changes their preference to A, then the social choice is still A.

See also

• Social choice theory
• Arrow's impossibility theorem
• Condorcet paradox
• Gibbard–Satterthwaite theorem
• Gibbard's theorem

Notes

1. May, Kenneth O. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", Econometrica, Vol. 20, Issue 4, pp. 680–684. JSTOR 1907651
2. Mark Fey, "May’s Theorem with an Infinite Population", Social Choice and Welfare, 2004, Vol. 23, issue 2, pages 275–293.
3. 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. doi:10.1111/j.1540-5907.2006.00225.x

References

1. ^ May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN 0012-9682. JSTOR 1907651.

• 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