Editing May's theorem

{{Electoral systems sidebar}}{{short description|Social choice theorem on superiority of majority voting}}
In [[social choice theory]], '''May's theorem''', also called the '''general possibility theorem''',<ref name="May">{{Cite journal |last=May |first=Kenneth O. |date=1952 |title=A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision |url=https://www.jstor.org/stable/1907651 |journal=Econometrica |volume=20 |issue=4 |pages=680–684 |doi=10.2307/1907651 |jstor=1907651 |issn=0012-9682|url-access=subscription }}</ref> says that [[simple majority voting|majority vote]] is the unique [[Ranked voting|ranked social choice function]] between two candidates that satisfies the following criteria: 

* [[Anonymity (social choice)|Anonymity]] – each voter is treated identically,
* [[Neutrality (social choice)|Neutrality]] – each candidate is treated identically,
* [[Monotonicity criterion|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.{{ref|May}}

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.{{Cn|date=April 2024}}

==Formal statement==
Let {{math|''A''}} and {{math|''B''}} be two possible choices, often called alternatives or candidates. A [[preference (economics)|''preference'']] is then simply a choice of whether {{math|''A''}}, {{math|''B''}}, or neither is preferred.<ref name="May" /> Denote the set of preferences by {{math|{''A'', ''B'', 0}}}, where {{math|0}} represents neither.

Let {{math|''N''}} be a positive integer. In this context, a [[Ranked voting|''ordinal (ranked)'']] ''social choice function'' is a function

: <math>F : \{A,B,0\}^N \to \{A,B,0\}</math>

which aggregates individuals' preferences into a single preference.<ref name="May" /> An {{math|''N''}}-[[tuple]] {{math|(''R''{{sub|1}}, …, ''R''{{sub|''N''}}) ∈ {''A'', ''B'', 0}{{sup|''N''}}}} of voters' preferences is called a ''preference profile''.

Define a social choice function called ''simple majority voting'' as follows:<ref name="May" />
* If the number of preferences for {{math|''A''}} is greater than the number of preferences for {{math|''B''}}, simple majority voting returns {{math|''A''}},
* If the number of preferences for {{math|''A''}} is less than the number of preferences for {{math|''B''}}, simple majority voting returns {{math|''B''}},
* If the number of preferences for {{math|''A''}} is equal to the number of preferences for {{math|''B''}}, simple majority voting returns {{math|0}}.

May's theorem states that simple majority voting is the unique social welfare function satisfying all three of the following conditions:<ref name="May" />

# [[Anonymity (social choice)|'''Anonymity''']]: The social choice function treats all voters the same, i.e. permuting the order of the voters does not change the result.
# [[Anonymity (social choice)|'''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 {{math|''A''}} and {{math|''B''}}, but a voter who previously preferred {{math|''B''}} changes their preference to {{math|''A''}}, then the social choice becomes {{math|''A''}}.

==See also==
* [[Social choice theory]]
* [[Arrow's impossibility theorem]]
* [[Condorcet paradox]]
* [[Gibbard–Satterthwaite theorem]]
* [[Gibbard's theorem]]

==Notes==
#{{note|May}}May, Kenneth O. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", ''Econometrica'', Vol. 20, Issue 4, pp.&nbsp;680–684. {{JSTOR|1907651}}
#{{note|Fey}}Mark Fey, "[http://rangevoting.org/FeyMay.pdf May’s Theorem with an Infinite Population]", ''Social Choice and Welfare'', 2004, Vol. 23, issue 2, pages 275–293.
#{{note|List}}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==
{{reflist}}
*Alan D. Taylor (2005). ''Social Choice and the Mathematics of Manipulation'', 1st edition, Cambridge University Press. {{isbn|0-521-00883-2}}. Chapter 1.
*[https://web.archive.org/web/20110719120639/http://mit.econ.au.dk/vip_htm/povergaard/pbohome/Courses_Pre_2007/Applied_Econ_Fall_2003/e03/materialer/Logrolling.pdf Logrolling, May’s theorem and Bureaucracy]

[[Category:Social choice theory]]
[[Category:1952 in science]]
[[Category:Theorems in discrete mathematics]]
[[Category:Voting theory]]