Wikipedia - User contributions [en]

Social choice theory

2023-04-01T21:26:35Z

92.237.98.153: /* Overlap with public choice theory */

{{Short description|Academic discipline}}
{{Citation style|date=April 2021}}
{{Economics sidebar}}
{{Politics sidebar}}

'''Social choice theory''' or '''social choice''' is a [[Theory|theoretical]] framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.<ref name="Sen2008">[[Amartya Sen]] (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition, [http://www.dictionaryofeconomics.com/article?id=pde2008_S000164&q=Social%20choice&topicid=&result_number=2 Abstract & TOC.]</ref> Whereas choice theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with how to translate the preferences of individuals into the preferences of a group. A non-theoretical example of a collective decision is enacting a law or set of laws under a [[constitution]]. Another example is voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences.<ref name=":1">{{Citation|last1=Zwicker|first1=William S.|title=Introduction to the Theory of Voting|date=2016|url=https://www.cambridge.org/core/product/identifier/CBO9781107446984A019/type/book_part|work=Handbook of Computational Social Choice|pages=23–56|editor-last=Brandt|editor-first=Felix|place=Cambridge|publisher=Cambridge University Press|doi=10.1017/cbo9781107446984.003|isbn=978-1-107-44698-4|access-date=2021-12-24|last2=Moulin|first2=Herve|editor2-last=Conitzer|editor2-first=Vincent|editor3-last=Endriss|editor3-first=Ulle|editor4-last=Lang|editor4-first=Jerome}}</ref>

Social choice blends elements of [[welfare economics]] and [[public choice]] theory. It is [[Methodological individualism|methodologically individualistic]], in that it aggregates preferences and behaviors of individual members of society. Using elements of [[formal logic]] for generality, analysis proceeds from a set of seemingly reasonable [[axiom]]s of social choice to form a ''[[social welfare function]]'' (or [[Social welfare function#Arrow social welfare function (constitution)|''constitution'']]).<ref>For example, in Kenneth J. Arrow (1951). ''Social Choice and Individual Values'', New York: Wiley, ch. [http://cowles.econ.yale.edu/P/cm/m12-2/m12-2-02.pdf II], section 2, A Notation for Preferences and Choice, and ch. [http://cowles.econ.yale.edu/P/cm/m12-2/m12-2-03.pdf III], "The Social Welfare Function".</ref> Results uncovered the logical incompatibility of various axioms, as in [[Arrow's theorem]], revealing an [[aggregation problem]] and suggesting reformulation or theoretical [[triage]] in dropping some axiom(s).<ref name="Sen2008"/>

== Overlap with public choice theory ==
{{See also|Public choice}}
"Public choice" and "social choice" are heavily overlapping fields of endeavor.

Social choice and [[public choice]] theory may overlap but are disjoint if narrowly construed. The [[JEL classification codes|Journal of Economic Literature classification codes]] place Social Choice under [[Microeconomics]] at [[JEL classification codes#Microeconomics JEL: D Subcategories|JEL D71]] (with Clubs, Committees, and Associations) whereas most Public Choice subcategories are in JEL D72 (Economic Models of Political Processes: [[rent-seeking|Rent-Seeking]], Elections, Legislatures, and Voting Behavior).

Social choice theory (and public choice theory) dates from [[Marquis de Condorcet|Condorcet]]'s formulation of the [[voting paradox]], though it arguably goes back further to [[Ramon Llull]]'s 1299 publication.

[[Kenneth Arrow]]'s ''[[Social Choice and Individual Values]]'' (1951) and [[Arrow's impossibility theorem]] are often acknowledged as the basis of the modern social choice theory and public choice theory <ref name="Sen2008" />. In addition to Arrow's theorem and the voting paradox, the [[Gibbard–Satterthwaite theorem]], the [[Condorcet jury theorem]], the [[median voter theorem]], and [[May's theorem]] are among the more well known results from social choice theory.

[[Amartya Sen]]'s Nobel Prize winning work was also highly influential. See the [[#Interpersonal utility comparison]] section below for more about Sen's work.

Later work also considers approaches to [[compensation principle|compensations]] and [[fair division|fairness]], liberty and rights, axiomatic [[domain of a function|domain]] [[Domain of a function#Formal definition|restrictions]] on [[Preference (economics)|preferences]] of [[agent (economics)|agents]], variable populations, [[strategy-proof]]ing of social-choice mechanisms, [[Natural resource economics|natural resources]],<ref name="Sen2008" /><ref>Walter Bossert and John A. Weymark (2008). "Social Choice (New Developments)," ''The New Palgrave Dictionary of Economics'', 2nd Edition, [http://www.dictionaryofeconomics.com/article?id=pde2008_S000503&q=social%20choice&topicid=&result_number=1 Abstract & TOC.]</ref> [[capabilities approach|capabilities]] and [[capabilities approach#Functionings|functionings]],<ref>{{cite book|last1=Kaushik|first1=Basu|title=Functionings and Capabilities|last2=Lòpez-Calva|first2=Luis F.|year=2011|isbn=9780444508942|series=Handbook of Social Choice and Welfare|volume=2|pages=153–187|doi=10.1016/S0169-7218(10)00016-X}}</ref> and [[Well-being|welfare]],<ref>{{cite book|last1=d'Aspremont|first1=Claude|title=Chapter 10 Social welfare functionals and interpersonal comparability|last2=Gevers|first2=Louis|year=2002|isbn=9780444829146|series=Handbook of Social Choice and Welfare|volume=1|pages=459–541|doi=10.1016/S1574-0110(02)80014-5}}</ref> [[justice (economics)|justice]],<ref>Amartya Sen ([1987] 2008). "Justice," ''The New Palgrave Dictionary of Economics,'' 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_J000033&edition=current&q=justice&topicid=&result_number=1 Abstract & TOC.]   Bertil Tungodden (2008). "Justice (New Perspectives)," ''The New Palgrave Dictionary of Economics,'' 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_J000052&q=justice&topicid=&result_number=2 Abstract.]   Louis Kaplow (2008). "Pareto Principle and Competing Principles," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_P000315&edition=current&q= Abstract.]   Amartya K. Sen (1979 [1984]). ''Collective Choice and Social Welfare'', New York: Elsevier, [http://www.citeulike.org/user/rlai/article/681900 (description)]:     ch. 9, "Equity and Justice," pp. 131-51.     ch. 9*, "Impersonality and Collective Quasi-Orderings," pp. 152-160.   Kenneth J. Arrow (1983). ''Collected Papers'', v. 1, ''Social Choice and Justice'', Cambridge, MA: Belknap Press, [http://www.hup.harvard.edu/catalog/ARRCO1.html Description], [http://www.hup.harvard.edu/catalog/ARRCO1.html?show=contents contents], and chapter-preview [https://books.google.com/books?id=iTgmSNjTdjUC&pg=PP11 links.]   Charles Blackorby, Walter Bossert, and David Donaldson, 2002. "Utilitarianism and the Theory of Justice", in ''Handbook of Social Choice and Welfare'', edited by Kenneth J. Arrow, Amartya K. Sen, and Kotaro Suzumura, v. 1, ch. 11, pp. 543–596. [http://www.sciencedirect.com/science/article/pii/S1574011002800157 Abstract.]</ref> and [[poverty]].<ref>{{cite book|last1=Dutta|first1=Bhaskar|title=Chapter 12 Inequality, poverty and welfare|year=2002|isbn=9780444829146|series=Handbook of Social Choice and Welfare|volume=1|pages=597–633|doi=10.1016/S1574-0110(02)80016-9}}</ref>

==Interpersonal utility comparison==

Social choice theory is the study of theoretical and practical methods to aggregate or combine individual preferences into a collective social welfare function. The field generally assumes that individuals have [[preference]]s, and it follows that they can be modeled using [[utility function]]s. But much of the research in the field assumes that those utility functions are internal to humans, lack a meaningful unit of measure and ''cannot'' be compared across different individuals<ref name="Robbins">Lionel Robbins (1932, 1935, 2nd ed.). ''An Essay on the Nature and Significance of Economic Science'', London: Macmillan. Links for [https://books.google.com/books?id=nySoIkOgWQ4C&pg=PR10 1932 HTML] and [https://www.scribd.com/doc/14242989/An-Essay-on-the-Nature-and-Signicance-of-Economic-Science-Lionel-Robbins- 1935 facsimile].</ref> Whether this type of ''interpersonal utility comparison'' is possible or not significantly alters the available mathematical structures for social welfare functions and social choice theory.

In one perspective, following [[Jeremy Bentham]], [[utilitarianism|utilitarians]] have argued that preferences and utility functions of individuals are interpersonally comparable and may therefore be added together to arrive at a measure of aggregate utility. Utilitarian ethics call for maximizing this aggregate.

In contrast many twentieth century economists, following [[Lionel Robbins]], questioned whether mental states, and the utilities they reflect, can be measured and, ''a fortiori'', interpersonal ''comparisons'' of [[utility]] as well as the social choice theory on which it is based. Consider for instance the [[law of diminishing marginal utility]], according to which utility of an added quantity of a good decreases with the amount of the good that is already in possession of the individual. It has been used to defend transfers of wealth from the "rich" to the "poor" on the premise that the former do not derive as much utility as the latter from an extra unit of income. Robbins ([[An Essay on the Nature and Significance of Economic Science|1935]], pp. 138–40) argues that this notion is beyond [[positivism|positive science]]; that is, one cannot measure changes in the utility of someone else, nor is it required by positive theory.

Apologists of the interpersonal comparison of utility have argued that Robbins claimed too much. [[John Harsanyi]] agrees that full comparability of mental states such as utility is never possible but believes, however, that human beings are able to make some interpersonal comparisons of utility because they share some common backgrounds, cultural experiences, etc. In the example from [[Amartya Sen]] (1970, p. 99), it should be possible to say that [[Emperor Nero]]'s gain from burning Rome was outweighed by the loss incurred by the rest of the Romans. Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption.

Sen proposes, however, that comparability of interpersonal utility need not be partial. Under Sen's theory of informational broadening, even complete interpersonal comparison of utility would lead to socially suboptimal choices because mental states are malleable. A starving peasant may have a particularly sunny disposition and thereby derive high utility from a small income. This fact should not nullify, however, his claim to compensation or equality in the realm of social choice.

Social decisions should accordingly be based on immalleable factors. Sen proposes interpersonal utility comparisons based on a wide range of data. His theory is concerned with access to advantage, viewed as an individual's access to goods that satisfy basic needs (e.g., food), freedoms (in the [[labor market]], for instance), and [[capability approach|capabilities]]. We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. Sen's method of informational broadening allows social choice theory to escape the objections of Robbins, which looked as though they would permanently harm social choice theory.

Additionally, since the seminal results of Arrow's impossibility theorem and the [[Gibbard–Satterthwaite theorem]], many positive results focusing on the restriction of the domain of preferences of individuals have elucidated such topics as optimal voting. The initial results emphasized the impossibility of satisfactorily providing a social choice function free of [[dictatorship]] and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties.{{citation needed|date=August 2015}}

== Empirical studies ==

Since Arrow social choice analysis has primarily been characterized by being extremely theoretical and formal in character. However, since ca. 1960 attention began to be paid to empirical applications of social choice theoretical insights, first and foremost by American political scientist [[William H. Riker]].

The vast majority of such studies have been focused on finding empirical examples of the [[Condorcet paradox]].<ref>{{Cite journal|last=Kurrild-Klitgaard|first=Peter|date=2014|title=Empirical social choice: An introduction|journal=Public Choice|language=en|volume=158|issue=3–4|pages=297–310|doi=10.1007/s11127-014-0164-4|s2cid=148982833|issn=0048-5829}}</ref><ref name=":0">{{Cite journal|last=Van Deemen|first=Adrian|date=2014|title=On the empirical relevance of Condorcet's paradox|journal=Public Choice|language=en|volume=158|issue=3–4|pages=311–330|doi=10.1007/s11127-013-0133-3|s2cid=154862595|issn=0048-5829}}</ref>

A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%<ref name=":0" />{{Rp|325}} (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without). On the other hand, the empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.

While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.<ref>{{Cite journal|last=Kurrild-Klitgaard|first=Peter|date=2014|title=An empirical example of the Condorcet paradox of voting in a large electorate|journal=Public Choice|language=en|volume=107|issue=1/2 |pages=135–145|doi=10.1023/A:1010304729545|s2cid=152300013|issn=0048-5829}}</ref>

== Rules ==
Let <math>X</math> be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from <math>X</math>. For example, in a [[single-winner election]], <math>X</math> may represent the set of candidates; in a [[resource allocation]] setting, <math>X</math> may represent all possible allocations.

Let <math>I</math> be a finite set, representing a collection of individuals. For each <math>i \in I</math>, let <math>u_i:X\longrightarrow\mathbb{R}</math> be a ''[[Utility|utility function]]'', describing the amount of happiness an individual ''i'' derives from each possible state.

A ''social choice rule'' is a mechanism which uses the data <math>(u_i)_{i \in I}</math> to select some element(s) from <math>X</math> which are 'best' for society. The question of what 'best' means is the basic question of social choice theory. The following rules are most common:
*The [[Utilitarian social choice rule|''utilitarian rule'']] - also called the ''max-sum rule'' - aims to maximize the sum of utilities, thus maximizing the efficiency.
*The [[Egalitarian social choice rule|''egalitarian rule'']] - also called the ''max-min rule'' - aims to maximize the smallest utility, thus maximizing the fairness.
*The ''[[proportional-fair rule]]'' - sometimes called the ''max-product rule'' - aims to balance between the previous two rules, attaining a balance between efficiency and fairness.

== Functions ==
A social choice function or a voting rule takes an individual's complete and transitive preferences over a set of candidates (also called alternatives), and returns some subset of (possible singular) the candidates. We can think of this subset as the winners of an election. This is different from social welfare function, which returns a linear order of the set of alternatives as opposed to simply selecting some subset. We can compare different social choice functions based on which axioms or mathematical properties they fulfill.<ref name=":1" /> For example, [[Instant-runoff voting]] satisfies the [[Independence of clones criterion]], whereas the [[Borda count]] does not; conversely, [[Borda Count]] satisfies the [[Monotonicity criterion]] whereas IRV does not.

=== Theorems ===
[[Arrow's impossibility theorem]] is what often comes to mind when one thinks about impossibility theorems in voting. However, Arrow was concerned with social welfare functions, not social choice functions. There are several famous theorems concerning social choice functions. The [[Gibbard–Satterthwaite theorem]] states that all non-dictatorial voting rules that is resolute (it always returns a single winner no matter what the ballots are) and non-imposed (every alternative could be chosen) with more than three alternatives (candidates) is manipulable. That is, a voter can cast a ballot that misrepresents their preferences to obtain a result that is more favorable to them under their sincere preferences. The Campbell-Kelley theorem states that, if there exists a [[Condorcet winner criterion|Condorcet winner]], then selecting that winner is the unique resolute, [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and non-manipulable voting rule.<ref name=":1" /> [[May's theorem]] states that when there are only two candidates, [[Simple majority vote]] is the unique [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and positively responsive voting rule.<ref>{{Cite journal|last=May|first=Kenneth O.|date=October 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}}</ref>

==See also==
{{div col|colwidth=20em}}
* [[Compensation principle]]
* [[Computational social choice]]
* [[Condorcet paradox]]
* [[Emotional choice theory]]
* [[Extended sympathy]]
* [[Game theory]]
* [[Group decision-making]]
* [[Justice (economics)]]
* [[Liberal paradox]]
* [[Mechanism design]]
* [[Nakamura number]]
* [[Rational choice theory]]
* [[Rule according to higher law]]
* [[Voting system]]
{{Div col end}}

==Notes==
{{Reflist|30em}}

== References ==
*[[Kenneth Arrow|Arrow, Kenneth J.]] (1951, 2nd ed., 1963). ''[[Social Choice and Individual Values]]'', New York: Wiley. {{ISBN|0-300-01364-7}}
*_____, (1972). "General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice", Nobel Prize Lecture, [http://nobelprize.org/economics/laureates/1972/arrow-lecture.html Link to text], with Section 8 on the theory and background.
*_____, (1983). ''Collected Papers'', v. 1, ''Social Choice and Justice'', Oxford: Blackwell {{ISBN|0-674-13760-4}}
* Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. (1997). ''Social Choice Re-Examined'', 2 vol., London: Palgrave Macmillan {{ISBN|0-312-12739-1}} & {{ISBN|0-312-12741-3}}
*_____, eds. (2002). ''Handbook of Social Choice and Welfare'', v. 1. Chapter-preview [http://www.sciencedirect.com/science/handbooks/15740110/1 links].
*_____, ed. (2011). ''Handbook of Social Choice and Welfare'', v. 2, Amsterdam: Elsevier. Chapter-preview [http://www.sciencedirect.com/science/handbooks/15740110/2 links].
* Bossert, Walter and John A. Weymark (2008). "Social Choice (New Developments)," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition, London: Palgrave Macmillan [http://www.dictionaryofeconomics.com/article?id=pde2008_S000503&q=social%20choice&topicid=&result_number=1 Abstract.]
* [[Dryzek, John S.]] and Christian List (2003). "Social Choice Theory and Deliberative Democracy: A Reconciliation," ''British Journal of Political Science'', 33(1), pp. 1–28, https://www.jstor.org/discover/10.2307/4092266?uid=3739936&uid=2&uid=4&uid=3739256&sid=21102056001967, 2002 PDF [http://personal.lse.ac.uk/LIST/PDF-files/SCTBJPS.pdf link.]
* Feldman, Allan M. and Roberto Serrano (2006). ''Welfare Economics and Social Choice Theory'', 2nd ed., New York: Springer {{ISBN|0-387-29367-1}}, {{ISBN|978-0-387-29367-7}} [https://books.google.com/books?id=7pZV5O5n_WwC&q=%22arrow's+theorem%22 Arrow-searchable chapter previews.]
* [[Marc Fleurbaey|Fleurbaey, Marc]] (1996). Théories économiques de la justice, Paris: Economica.
*{{Cite book | last=Gaertner | first=Wulf | author-link=Gaertner Wulf | title=A primer in social choice theory | date=2006 | publisher=Oxford University Press | location=Oxford | isbn=978-0-19-929751-1 }}
* [[John Harsanyi|Harsanyi, John C.]] (1987). "Interpersonal Utility Comparisons," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 2, London: Palgrave, pp. 955–58.
*{{Cite book | last=Moulin | first=Herve | author-link=Herve Moulin | title=Axioms of cooperative decision making | date=1988 | publisher=Cambridge University Press | location=Cambridge | isbn=978-0-521-42458-5 }}
* {{Cite journal | last = Myerson | first = Roger B. | author-link = Roger Myerson | title = Fundamentals of social choice theory | journal = [[Quarterly Journal of Political Science]] | volume = 8 | issue = 3 | pages = 305–337 | doi = 10.1561/100.00013006 | date = June 2013 | citeseerx = 10.1.1.297.6781 }}
*{{cite book |author=Nitzan, Shmuel | author-link=Shmue Nitzan |title=Collective Preference and Choice |publisher=Cambridge University Press |location=Cambridge, UK |year=2010 |isbn=978-0-521-72213-1 }}
*[[Lionel Robbins|Robbins, Lionel]] (1935). ''[[An Essay on the Nature and Significance of Economic Science]]'', 2nd ed., London: Macmillan, ch. VI
* ____, (1938). "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41.
*[[Amartya Sen|Sen, Amartya K.]] (1970 [1984]). ''Collective Choice and Social Welfare'', New York: Elsevier {{ISBN|0-444-85127-5}} [http://www.citeulike.org/user/rlai/article/681900 Description.]
*_____, (1998). "The Possibility of Social Choice", Nobel Prize Lecture [https://www.nobelprize.org/uploads/2018/06/sen-lecture.pdf].
* _____, (1987). "Social Choice," ''The New Palgrave: A Dictionary of Economics'', v. 4, London: Palgrave, pp. 382–93.
* _____, (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition, London: Palgrave [http://www.dictionaryofeconomics.com/article?id=pde2008_S000164&q=Social%20choice&topicid=&result_number=2 Abstract.]
* {{Cite book | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=[[Cambridge University Press]] | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | location=New York}}. A comprehensive reference from a computational perspective; see Chapter 9. [http://www.masfoundations.org/download.html Downloadable free online].
* [[Kotaro Suzumura|Suzumura, Kotaro]] (1983). ''Rational Choice, Collective Decisions, and Social Welfare'', Cambridge: Cambridge University Press {{ISBN|0-521-23862-5}}
*{{Cite book | last=Taylor | first=Alan D. | author-link=Alan D. Taylor | title=Social choice and the mathematics of manipulation | url=https://archive.org/details/socialchoicemath0000tayl | url-access=registration | date=2005 | publisher=Cambridge University Press | location=New York | isbn=978-0-521-00883-9 }}

==External links==
*{{cite SEP |url-id=social-choice |title=Social Choice Theory |last=List |first=Christian}}
*[http://faculty.maxwell.syr.edu/jskelly/biblioho.htm Social Choice Bibliography by J. S. Kelly] {{Webarchive|url=https://web.archive.org/web/20171223005623/http://faculty.maxwell.syr.edu/jskelly/biblioho.htm |date=2017-12-23 }}
*[https://electowiki.org/wiki/Main_Page Electowiki], a [[wiki]] covering many subjects of social choice and voting theory

{{Industrial and applied mathematics}}
{{Authority control}}

[[Category:Social choice theory| ]]
[[Category:Ethical theories]]
[[Category:Political science]]
[[Category:Public economics]]
[[Category:Applied mathematics]]
[[Category:Collective intelligence]]
[[Category:Social epistemology]]
[[Category:Mathematical economics]]
[[Category:Law and economics]]

Social choice theory

2023-04-01T21:26:05Z

92.237.98.153: /* Overlap with public choice theory */

{{Short description|Academic discipline}}
{{Citation style|date=April 2021}}
{{Economics sidebar}}
{{Politics sidebar}}

'''Social choice theory''' or '''social choice''' is a [[Theory|theoretical]] framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.<ref name="Sen2008">[[Amartya Sen]] (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition, [http://www.dictionaryofeconomics.com/article?id=pde2008_S000164&q=Social%20choice&topicid=&result_number=2 Abstract & TOC.]</ref> Whereas choice theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with how to translate the preferences of individuals into the preferences of a group. A non-theoretical example of a collective decision is enacting a law or set of laws under a [[constitution]]. Another example is voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences.<ref name=":1">{{Citation|last1=Zwicker|first1=William S.|title=Introduction to the Theory of Voting|date=2016|url=https://www.cambridge.org/core/product/identifier/CBO9781107446984A019/type/book_part|work=Handbook of Computational Social Choice|pages=23–56|editor-last=Brandt|editor-first=Felix|place=Cambridge|publisher=Cambridge University Press|doi=10.1017/cbo9781107446984.003|isbn=978-1-107-44698-4|access-date=2021-12-24|last2=Moulin|first2=Herve|editor2-last=Conitzer|editor2-first=Vincent|editor3-last=Endriss|editor3-first=Ulle|editor4-last=Lang|editor4-first=Jerome}}</ref>

Social choice blends elements of [[welfare economics]] and [[public choice]] theory. It is [[Methodological individualism|methodologically individualistic]], in that it aggregates preferences and behaviors of individual members of society. Using elements of [[formal logic]] for generality, analysis proceeds from a set of seemingly reasonable [[axiom]]s of social choice to form a ''[[social welfare function]]'' (or [[Social welfare function#Arrow social welfare function (constitution)|''constitution'']]).<ref>For example, in Kenneth J. Arrow (1951). ''Social Choice and Individual Values'', New York: Wiley, ch. [http://cowles.econ.yale.edu/P/cm/m12-2/m12-2-02.pdf II], section 2, A Notation for Preferences and Choice, and ch. [http://cowles.econ.yale.edu/P/cm/m12-2/m12-2-03.pdf III], "The Social Welfare Function".</ref> Results uncovered the logical incompatibility of various axioms, as in [[Arrow's theorem]], revealing an [[aggregation problem]] and suggesting reformulation or theoretical [[triage]] in dropping some axiom(s).<ref name="Sen2008"/>

== Overlap with public choice theory ==
{{See also|Public choice}}
"Public choice" and "social choice" are heavily overlapping fields of endeavor.

Social choice and [[public choice]] theory may overlap but are disjoint if narrowly construed. The [[JEL classification codes|Journal of Economic Literature classification codes]] place Social Choice under [[Microeconomics]] at [[JEL classification codes#Microeconomics JEL: D Subcategories|JEL D71]] (with Clubs, Committees, and Associations) whereas most Public Choice subcategories are in JEL D72 (Economic Models of Political Processes: [[rent-seeking|Rent-Seeking]], Elections, Legislatures, and Voting Behavior).

Social choice theory (and public choice theory) dates from [[Marquis de Condorcet|Condorcet]]'s formulation of the [[voting paradox]], though it arguably goes back further to [[Ramon Llull]]'s 1299 publication.

[[Kenneth Arrow]]'s ''[[Social Choice and Individual Values]]'' (1951) and [[Arrow's impossibility theorem]] are often acknowledged as the basis of the modern social choice theory and public choice theory{{Citation needed}}.<ref name="Sen2008" /> In addition to Arrow's theorem and the voting paradox, the [[Gibbard–Satterthwaite theorem]], the [[Condorcet jury theorem]], the [[median voter theorem]], and [[May's theorem]] are among the more well known results from social choice theory.

[[Amartya Sen]]'s Nobel Prize winning work was also highly influential. See the [[#Interpersonal utility comparison]] section below for more about Sen's work.

Later work also considers approaches to [[compensation principle|compensations]] and [[fair division|fairness]], liberty and rights, axiomatic [[domain of a function|domain]] [[Domain of a function#Formal definition|restrictions]] on [[Preference (economics)|preferences]] of [[agent (economics)|agents]], variable populations, [[strategy-proof]]ing of social-choice mechanisms, [[Natural resource economics|natural resources]],<ref name="Sen2008" /><ref>Walter Bossert and John A. Weymark (2008). "Social Choice (New Developments)," ''The New Palgrave Dictionary of Economics'', 2nd Edition, [http://www.dictionaryofeconomics.com/article?id=pde2008_S000503&q=social%20choice&topicid=&result_number=1 Abstract & TOC.]</ref> [[capabilities approach|capabilities]] and [[capabilities approach#Functionings|functionings]],<ref>{{cite book|last1=Kaushik|first1=Basu|title=Functionings and Capabilities|last2=Lòpez-Calva|first2=Luis F.|year=2011|isbn=9780444508942|series=Handbook of Social Choice and Welfare|volume=2|pages=153–187|doi=10.1016/S0169-7218(10)00016-X}}</ref> and [[Well-being|welfare]],<ref>{{cite book|last1=d'Aspremont|first1=Claude|title=Chapter 10 Social welfare functionals and interpersonal comparability|last2=Gevers|first2=Louis|year=2002|isbn=9780444829146|series=Handbook of Social Choice and Welfare|volume=1|pages=459–541|doi=10.1016/S1574-0110(02)80014-5}}</ref> [[justice (economics)|justice]],<ref>Amartya Sen ([1987] 2008). "Justice," ''The New Palgrave Dictionary of Economics,'' 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_J000033&edition=current&q=justice&topicid=&result_number=1 Abstract & TOC.]   Bertil Tungodden (2008). "Justice (New Perspectives)," ''The New Palgrave Dictionary of Economics,'' 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_J000052&q=justice&topicid=&result_number=2 Abstract.]   Louis Kaplow (2008). "Pareto Principle and Competing Principles," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_P000315&edition=current&q= Abstract.]   Amartya K. Sen (1979 [1984]). ''Collective Choice and Social Welfare'', New York: Elsevier, [http://www.citeulike.org/user/rlai/article/681900 (description)]:     ch. 9, "Equity and Justice," pp. 131-51.     ch. 9*, "Impersonality and Collective Quasi-Orderings," pp. 152-160.   Kenneth J. Arrow (1983). ''Collected Papers'', v. 1, ''Social Choice and Justice'', Cambridge, MA: Belknap Press, [http://www.hup.harvard.edu/catalog/ARRCO1.html Description], [http://www.hup.harvard.edu/catalog/ARRCO1.html?show=contents contents], and chapter-preview [https://books.google.com/books?id=iTgmSNjTdjUC&pg=PP11 links.]   Charles Blackorby, Walter Bossert, and David Donaldson, 2002. "Utilitarianism and the Theory of Justice", in ''Handbook of Social Choice and Welfare'', edited by Kenneth J. Arrow, Amartya K. Sen, and Kotaro Suzumura, v. 1, ch. 11, pp. 543–596. [http://www.sciencedirect.com/science/article/pii/S1574011002800157 Abstract.]</ref> and [[poverty]].<ref>{{cite book|last1=Dutta|first1=Bhaskar|title=Chapter 12 Inequality, poverty and welfare|year=2002|isbn=9780444829146|series=Handbook of Social Choice and Welfare|volume=1|pages=597–633|doi=10.1016/S1574-0110(02)80016-9}}</ref>

==Interpersonal utility comparison==

Social choice theory is the study of theoretical and practical methods to aggregate or combine individual preferences into a collective social welfare function. The field generally assumes that individuals have [[preference]]s, and it follows that they can be modeled using [[utility function]]s. But much of the research in the field assumes that those utility functions are internal to humans, lack a meaningful unit of measure and ''cannot'' be compared across different individuals<ref name="Robbins">Lionel Robbins (1932, 1935, 2nd ed.). ''An Essay on the Nature and Significance of Economic Science'', London: Macmillan. Links for [https://books.google.com/books?id=nySoIkOgWQ4C&pg=PR10 1932 HTML] and [https://www.scribd.com/doc/14242989/An-Essay-on-the-Nature-and-Signicance-of-Economic-Science-Lionel-Robbins- 1935 facsimile].</ref> Whether this type of ''interpersonal utility comparison'' is possible or not significantly alters the available mathematical structures for social welfare functions and social choice theory.

In one perspective, following [[Jeremy Bentham]], [[utilitarianism|utilitarians]] have argued that preferences and utility functions of individuals are interpersonally comparable and may therefore be added together to arrive at a measure of aggregate utility. Utilitarian ethics call for maximizing this aggregate.

In contrast many twentieth century economists, following [[Lionel Robbins]], questioned whether mental states, and the utilities they reflect, can be measured and, ''a fortiori'', interpersonal ''comparisons'' of [[utility]] as well as the social choice theory on which it is based. Consider for instance the [[law of diminishing marginal utility]], according to which utility of an added quantity of a good decreases with the amount of the good that is already in possession of the individual. It has been used to defend transfers of wealth from the "rich" to the "poor" on the premise that the former do not derive as much utility as the latter from an extra unit of income. Robbins ([[An Essay on the Nature and Significance of Economic Science|1935]], pp. 138–40) argues that this notion is beyond [[positivism|positive science]]; that is, one cannot measure changes in the utility of someone else, nor is it required by positive theory.

Apologists of the interpersonal comparison of utility have argued that Robbins claimed too much. [[John Harsanyi]] agrees that full comparability of mental states such as utility is never possible but believes, however, that human beings are able to make some interpersonal comparisons of utility because they share some common backgrounds, cultural experiences, etc. In the example from [[Amartya Sen]] (1970, p. 99), it should be possible to say that [[Emperor Nero]]'s gain from burning Rome was outweighed by the loss incurred by the rest of the Romans. Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption.

Sen proposes, however, that comparability of interpersonal utility need not be partial. Under Sen's theory of informational broadening, even complete interpersonal comparison of utility would lead to socially suboptimal choices because mental states are malleable. A starving peasant may have a particularly sunny disposition and thereby derive high utility from a small income. This fact should not nullify, however, his claim to compensation or equality in the realm of social choice.

Social decisions should accordingly be based on immalleable factors. Sen proposes interpersonal utility comparisons based on a wide range of data. His theory is concerned with access to advantage, viewed as an individual's access to goods that satisfy basic needs (e.g., food), freedoms (in the [[labor market]], for instance), and [[capability approach|capabilities]]. We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. Sen's method of informational broadening allows social choice theory to escape the objections of Robbins, which looked as though they would permanently harm social choice theory.

Additionally, since the seminal results of Arrow's impossibility theorem and the [[Gibbard–Satterthwaite theorem]], many positive results focusing on the restriction of the domain of preferences of individuals have elucidated such topics as optimal voting. The initial results emphasized the impossibility of satisfactorily providing a social choice function free of [[dictatorship]] and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties.{{citation needed|date=August 2015}}

== Empirical studies ==

Since Arrow social choice analysis has primarily been characterized by being extremely theoretical and formal in character. However, since ca. 1960 attention began to be paid to empirical applications of social choice theoretical insights, first and foremost by American political scientist [[William H. Riker]].

The vast majority of such studies have been focused on finding empirical examples of the [[Condorcet paradox]].<ref>{{Cite journal|last=Kurrild-Klitgaard|first=Peter|date=2014|title=Empirical social choice: An introduction|journal=Public Choice|language=en|volume=158|issue=3–4|pages=297–310|doi=10.1007/s11127-014-0164-4|s2cid=148982833|issn=0048-5829}}</ref><ref name=":0">{{Cite journal|last=Van Deemen|first=Adrian|date=2014|title=On the empirical relevance of Condorcet's paradox|journal=Public Choice|language=en|volume=158|issue=3–4|pages=311–330|doi=10.1007/s11127-013-0133-3|s2cid=154862595|issn=0048-5829}}</ref>

A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%<ref name=":0" />{{Rp|325}} (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without). On the other hand, the empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.

While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.<ref>{{Cite journal|last=Kurrild-Klitgaard|first=Peter|date=2014|title=An empirical example of the Condorcet paradox of voting in a large electorate|journal=Public Choice|language=en|volume=107|issue=1/2 |pages=135–145|doi=10.1023/A:1010304729545|s2cid=152300013|issn=0048-5829}}</ref>

== Rules ==
Let <math>X</math> be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from <math>X</math>. For example, in a [[single-winner election]], <math>X</math> may represent the set of candidates; in a [[resource allocation]] setting, <math>X</math> may represent all possible allocations.

Let <math>I</math> be a finite set, representing a collection of individuals. For each <math>i \in I</math>, let <math>u_i:X\longrightarrow\mathbb{R}</math> be a ''[[Utility|utility function]]'', describing the amount of happiness an individual ''i'' derives from each possible state.

A ''social choice rule'' is a mechanism which uses the data <math>(u_i)_{i \in I}</math> to select some element(s) from <math>X</math> which are 'best' for society. The question of what 'best' means is the basic question of social choice theory. The following rules are most common:
*The [[Utilitarian social choice rule|''utilitarian rule'']] - also called the ''max-sum rule'' - aims to maximize the sum of utilities, thus maximizing the efficiency.
*The [[Egalitarian social choice rule|''egalitarian rule'']] - also called the ''max-min rule'' - aims to maximize the smallest utility, thus maximizing the fairness.
*The ''[[proportional-fair rule]]'' - sometimes called the ''max-product rule'' - aims to balance between the previous two rules, attaining a balance between efficiency and fairness.

== Functions ==
A social choice function or a voting rule takes an individual's complete and transitive preferences over a set of candidates (also called alternatives), and returns some subset of (possible singular) the candidates. We can think of this subset as the winners of an election. This is different from social welfare function, which returns a linear order of the set of alternatives as opposed to simply selecting some subset. We can compare different social choice functions based on which axioms or mathematical properties they fulfill.<ref name=":1" /> For example, [[Instant-runoff voting]] satisfies the [[Independence of clones criterion]], whereas the [[Borda count]] does not; conversely, [[Borda Count]] satisfies the [[Monotonicity criterion]] whereas IRV does not.

=== Theorems ===
[[Arrow's impossibility theorem]] is what often comes to mind when one thinks about impossibility theorems in voting. However, Arrow was concerned with social welfare functions, not social choice functions. There are several famous theorems concerning social choice functions. The [[Gibbard–Satterthwaite theorem]] states that all non-dictatorial voting rules that is resolute (it always returns a single winner no matter what the ballots are) and non-imposed (every alternative could be chosen) with more than three alternatives (candidates) is manipulable. That is, a voter can cast a ballot that misrepresents their preferences to obtain a result that is more favorable to them under their sincere preferences. The Campbell-Kelley theorem states that, if there exists a [[Condorcet winner criterion|Condorcet winner]], then selecting that winner is the unique resolute, [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and non-manipulable voting rule.<ref name=":1" /> [[May's theorem]] states that when there are only two candidates, [[Simple majority vote]] is the unique [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and positively responsive voting rule.<ref>{{Cite journal|last=May|first=Kenneth O.|date=October 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}}</ref>

==See also==
{{div col|colwidth=20em}}
* [[Compensation principle]]
* [[Computational social choice]]
* [[Condorcet paradox]]
* [[Emotional choice theory]]
* [[Extended sympathy]]
* [[Game theory]]
* [[Group decision-making]]
* [[Justice (economics)]]
* [[Liberal paradox]]
* [[Mechanism design]]
* [[Nakamura number]]
* [[Rational choice theory]]
* [[Rule according to higher law]]
* [[Voting system]]
{{Div col end}}

==Notes==
{{Reflist|30em}}

== References ==
*[[Kenneth Arrow|Arrow, Kenneth J.]] (1951, 2nd ed., 1963). ''[[Social Choice and Individual Values]]'', New York: Wiley. {{ISBN|0-300-01364-7}}
*_____, (1972). "General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice", Nobel Prize Lecture, [http://nobelprize.org/economics/laureates/1972/arrow-lecture.html Link to text], with Section 8 on the theory and background.
*_____, (1983). ''Collected Papers'', v. 1, ''Social Choice and Justice'', Oxford: Blackwell {{ISBN|0-674-13760-4}}
* Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. (1997). ''Social Choice Re-Examined'', 2 vol., London: Palgrave Macmillan {{ISBN|0-312-12739-1}} & {{ISBN|0-312-12741-3}}
*_____, eds. (2002). ''Handbook of Social Choice and Welfare'', v. 1. Chapter-preview [http://www.sciencedirect.com/science/handbooks/15740110/1 links].
*_____, ed. (2011). ''Handbook of Social Choice and Welfare'', v. 2, Amsterdam: Elsevier. Chapter-preview [http://www.sciencedirect.com/science/handbooks/15740110/2 links].
* Bossert, Walter and John A. Weymark (2008). "Social Choice (New Developments)," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition, London: Palgrave Macmillan [http://www.dictionaryofeconomics.com/article?id=pde2008_S000503&q=social%20choice&topicid=&result_number=1 Abstract.]
* [[Dryzek, John S.]] and Christian List (2003). "Social Choice Theory and Deliberative Democracy: A Reconciliation," ''British Journal of Political Science'', 33(1), pp. 1–28, https://www.jstor.org/discover/10.2307/4092266?uid=3739936&uid=2&uid=4&uid=3739256&sid=21102056001967, 2002 PDF [http://personal.lse.ac.uk/LIST/PDF-files/SCTBJPS.pdf link.]
* Feldman, Allan M. and Roberto Serrano (2006). ''Welfare Economics and Social Choice Theory'', 2nd ed., New York: Springer {{ISBN|0-387-29367-1}}, {{ISBN|978-0-387-29367-7}} [https://books.google.com/books?id=7pZV5O5n_WwC&q=%22arrow's+theorem%22 Arrow-searchable chapter previews.]
* [[Marc Fleurbaey|Fleurbaey, Marc]] (1996). Théories économiques de la justice, Paris: Economica.
*{{Cite book | last=Gaertner | first=Wulf | author-link=Gaertner Wulf | title=A primer in social choice theory | date=2006 | publisher=Oxford University Press | location=Oxford | isbn=978-0-19-929751-1 }}
* [[John Harsanyi|Harsanyi, John C.]] (1987). "Interpersonal Utility Comparisons," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 2, London: Palgrave, pp. 955–58.
*{{Cite book | last=Moulin | first=Herve | author-link=Herve Moulin | title=Axioms of cooperative decision making | date=1988 | publisher=Cambridge University Press | location=Cambridge | isbn=978-0-521-42458-5 }}
* {{Cite journal | last = Myerson | first = Roger B. | author-link = Roger Myerson | title = Fundamentals of social choice theory | journal = [[Quarterly Journal of Political Science]] | volume = 8 | issue = 3 | pages = 305–337 | doi = 10.1561/100.00013006 | date = June 2013 | citeseerx = 10.1.1.297.6781 }}
*{{cite book |author=Nitzan, Shmuel | author-link=Shmue Nitzan |title=Collective Preference and Choice |publisher=Cambridge University Press |location=Cambridge, UK |year=2010 |isbn=978-0-521-72213-1 }}
*[[Lionel Robbins|Robbins, Lionel]] (1935). ''[[An Essay on the Nature and Significance of Economic Science]]'', 2nd ed., London: Macmillan, ch. VI
* ____, (1938). "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41.
*[[Amartya Sen|Sen, Amartya K.]] (1970 [1984]). ''Collective Choice and Social Welfare'', New York: Elsevier {{ISBN|0-444-85127-5}} [http://www.citeulike.org/user/rlai/article/681900 Description.]
*_____, (1998). "The Possibility of Social Choice", Nobel Prize Lecture [https://www.nobelprize.org/uploads/2018/06/sen-lecture.pdf].
* _____, (1987). "Social Choice," ''The New Palgrave: A Dictionary of Economics'', v. 4, London: Palgrave, pp. 382–93.
* _____, (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition, London: Palgrave [http://www.dictionaryofeconomics.com/article?id=pde2008_S000164&q=Social%20choice&topicid=&result_number=2 Abstract.]
* {{Cite book | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=[[Cambridge University Press]] | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | location=New York}}. A comprehensive reference from a computational perspective; see Chapter 9. [http://www.masfoundations.org/download.html Downloadable free online].
* [[Kotaro Suzumura|Suzumura, Kotaro]] (1983). ''Rational Choice, Collective Decisions, and Social Welfare'', Cambridge: Cambridge University Press {{ISBN|0-521-23862-5}}
*{{Cite book | last=Taylor | first=Alan D. | author-link=Alan D. Taylor | title=Social choice and the mathematics of manipulation | url=https://archive.org/details/socialchoicemath0000tayl | url-access=registration | date=2005 | publisher=Cambridge University Press | location=New York | isbn=978-0-521-00883-9 }}

==External links==
*{{cite SEP |url-id=social-choice |title=Social Choice Theory |last=List |first=Christian}}
*[http://faculty.maxwell.syr.edu/jskelly/biblioho.htm Social Choice Bibliography by J. S. Kelly] {{Webarchive|url=https://web.archive.org/web/20171223005623/http://faculty.maxwell.syr.edu/jskelly/biblioho.htm |date=2017-12-23 }}
*[https://electowiki.org/wiki/Main_Page Electowiki], a [[wiki]] covering many subjects of social choice and voting theory

{{Industrial and applied mathematics}}
{{Authority control}}

[[Category:Social choice theory| ]]
[[Category:Ethical theories]]
[[Category:Political science]]
[[Category:Public economics]]
[[Category:Applied mathematics]]
[[Category:Collective intelligence]]
[[Category:Social epistemology]]
[[Category:Mathematical economics]]
[[Category:Law and economics]]

Five-a-side football

2023-02-08T23:01:07Z

92.237.98.153: Added a citation request for a (probable) troll comment

{{short description|Variant of association football}}
{{Multiple issues|
{{More citations needed|date=December 2009}}
{{Lead too short|date=July 2021}}
}}
[[Image:Indoor soccer singapore z.JPG|thumb|240px|alt=Men playing football on artificial grass pitch.|Five-a-side game on artificial turf pitch, [[Singapore]]]]

'''Five-a-side football''' is a version of [[minifootball]], in which each team fields five players (four [[outfield#In football (soccer)|outfield]] players and a [[goalkeeper (association football)|goalkeeper]]). Other differences from [[association football|football]] include a smaller pitch, smaller goals, and a reduced game duration. Matches are played indoors, or outdoors on [[artificial grass]] pitches that may be enclosed within a barrier or "cage" to prevent the ball from leaving the playing area and keep the game constantly flowing.

5-a-side 2022 world cup winners are Rather be at the Pub.{{Citation needed|reason=What world cup? Who are this team? Was this a troll edit?|date=January 2023}}

==Rules==
The penalty area is significantly different from football: it is semi-circular in shape, only the goalkeeper is allowed to touch the ball within it, and they may or may not be allowed out.
Goalkeepers are only allowed to give the ball out to another player through hands. The goalkeeper may only kick the ball to effect a save.
There are no [[Offside (association football)|offside]] rules.
Headers are allowed. There is no protocol of deliberate handball versus accidental handball – the referee needs to make a decision based on the distance from where the ball was hit.
Yellow cards may result in the offending player being sent to the "[[sin bin]]" for a predetermined length of time. Red cards work in the same way as the 11-a-side game, the offending player being dismissed from the match.
Charging/sliding tackles are awarded a yellow card.

Additionally, metal studded boots cannot be worn, as this would damage the playing surface. Players are also required to wear shin guards, but enforcement of this is usually at the discretion of the referee.

Five-a-side is commonly played informally, and the rules are therefore flexible and are sometimes decided immediately before play begins; this is in contrast to [[futsal]], for which official laws are published by FIFA.

The [[The Football Association|English FA]] have drawn up a full list of laws for the small-sided game which expands upon the rules outlined above and includes minimum/maximum pitch dimensions as well as technicalities on free-kicks and other parts of the game.<ref>{{cite web|url=http://www.thefa.com/~/media/files/pdf/the-fa-2013-14/fa-rules-2013-14/fa-ssf-laws-of-the-game_2012.ashx|title=TheFA.com - Small Sided Football - Laws of the Game |access-date=2016-05-20|publisher=The FA}}</ref><ref>{{cite news|author=David Conn |url=https://www.theguardian.com/football/2012/may/28/fa |title=FA votes for smaller-sided matches for young footballers | Football |work=The Guardian |date= 2012-05-28|access-date=2012-07-16 |location=London}}</ref><ref>{{cite news|author=Football |url=https://www.telegraph.co.uk/sport/football/9296434/Football-Association-make-historic-decision-on-future-of-youth-football-for-the-future-good-of-England.html |title=Football Association make historic decision on future of youth football for the future good of England |publisher=Telegraph |date= 2012-05-28|access-date=2012-07-16 |location=London}}</ref><ref>{{cite web|last=Roan |first=Dan |url=https://www.bbc.co.uk/sport/0/football/18236981 |title=BBC Sport - Football Association vote in favour of youth football changes |publisher=Bbc.co.uk |date=2012-05-28 |access-date=2012-07-16}}</ref>

==Variations==

===Futsal===
{{Main|Futsal}}

[[Futsal]] is a version of indoor five-a-side football developed by the [[Asociación Mundial de Fútbol de Salón]] (AMF / English: World Futsal Association). It currently has two governing bodies: the AMF and the association football international body; the [[Fédération Internationale de Football Association]] (FIFA).

===Indoor soccer===
{{Main|Indoor soccer}}

[[Indoor soccer]] is an indoor variant played primarily in North America, typically with six-a-side teams on an [[ice hockey]]-sized pitch.

===Jorkyball===
[[Jorkyball]] is a 2vs2 format of football played in a plexiglass cage of 10m x 5m. The players can use the walls to pass and to score. The governing body is the Jorkyball International Federation.

===Beach soccer===
{{Main|Beach soccer}}
[[Beach soccer]] is a variation on five-a-side football in that it is played on a sandy surface. Rules do not greatly differ from those found in regular five-a-side football.
[[Image:7asidesoccerpitch.jpg|thumb|250px|alt=Diagram of seven-a-side football pitch showing pitch markings and dimensions.|Seven-a-side pitch markings. Dimensions and shape of penalty area may differ for other variants.]]

===Six-a-side football===
{{Main|International Socca Federation}}
A variation with increased pitch size and number of players on a team. In this variation there are five outfield players and one goalkeeper on the pitch for each team at any time. Other rules do not differ from those found in five-a-side football.

===Seven-a-side football===
{{Main|Seven-a-side football}}
This is another variation with increased pitch and team size; in this case with six outfield players and a goalkeeper on each side. The rules differ from those from five-a-side.<ref>{{cite web|url=http://www.clubwebsite.co.uk/news/2012/05/31/fa-changes-to-youth-football-whats-in-store/ |title=FA changes to youth football – what's in store? « Club Website – News and Updates |publisher=Clubwebsite.co.uk |date=2012-05-31 |access-date=2012-08-03}}</ref>

SUB football is a variation of seven-a-side football primarily played in Australia and New Zealand. The rules have been modified slightly to encourage new players to the game, with strict enforcement of non-contact and two ways to score points: by scoring a goal in the same manner as the other formats, or by scoring a board that is on either side of the goal. The boards are usually 2.5m long and one third of the height of the goal. A goal is 3 points and a board is 1 point. When the ball goes out of play, it may be kicked or thrown in. This applies to the sideline and corners.<ref>{{Cite web|url=https://www.stuff.co.nz/auckland/local-news/manukau-courier/sport/2760323/SUB-football-is-on-the-way|title=SUB football is on the way|website=Stuff|date=31 August 2009|access-date=2017-08-09}}</ref><ref>{{Cite web|url=http://www.scoop.co.nz/stories/CU1106/S00425/sub-football-call-for-teams-for-14th-season.htm|title=Sub Football Call For Teams For 14th Season {{!}} Scoop News|website=www.scoop.co.nz|access-date=2017-08-09}}</ref>

=== Blind football ===
{{Main|Paralympic association football}}

==Different organisations==
There are many operators of five-a-side football in Europe ([[Powerleague]], [[Goals Soccer Centres|Goals Soccer Center]], [[:fr:UrbanSoccer|UrbanSoccer]]), and most of all in the UK.<ref>{{cite news|author=Terry Macalister |url=https://www.theguardian.com/business/2007/sep/04/3 |title=Popularity of five-a-side kicks off profits | Business |work=The Guardian |date= 2007-09-03|access-date=2012-08-03 |location=London}}</ref>

[[World Minifootball Federation]] (WMF) unites 71 national associations, grouped into federations by continent. [[European Minifootball Federation]] consists of 32 member associations. EMF organizes [[EMF miniEURO]] and EMF Champions League competitions.

[[International Socca Federation]] (ISF) is a 6 a side football organization, running yearly Socca World Cup events since 2018. 40 national teams participated in the 2019 Socca World Cup.

The [[F5WC]] is the world's largest amateur five-a-side football tournament in the world with over 48 participating nations.{{citation needed|date=June 2016}}

IFA7 is the international association that promotes seven-a-side football. IFA7 held the first known Seven-a-side Football World Cup in 2017 in [[Guatemala]], with [[Russia]] winning the title.<ref>[[:es:IFA7]]</ref>{{Circular reference|date=October 2019}}<ref>{{cite web |url=https://ifa7.com/index.php/ifa7-tournaments/world-cup |title=World Cup |website=ifa7.com |url-status=dead |archive-url=https://web.archive.org/web/20160525214646/http://www.ifa7.com/index.php/ifa7-tournaments/world-cup |archive-date=2016-05-25}} </ref><ref>{{Cite web|url=https://www.publinews.gt/gt/deportes/2017/10/29/resultado-del-partido-guatemala-vs-rusia-final-la-ifa7-2017.html|title = Rusia es el campeón de la Copa del Mundo IFA 7|date = 30 October 2017}}</ref>

==Youth organisations==
The popularity of five-a-side youth football has grown tremendously within the U.S.<ref>ESPN FC, Relegation Zone, Sep. 20, 2012, [http://espnfc.com/blog/_/name/relegationzone/id/262?cc=5901 Soccer's big takeover]</ref><ref>New York Times, July 23, 2010, [https://www.nytimes.com/2010/07/24/sports/soccer/24soccer.html?_r=0 Soccer's Growth in the U.S. Seems Steady]</ref> Many organisations have chosen this format and modified it slightly to promote an environment where children can excel early in youth sports.<ref>Livestrong.com, Jan. 6, 2011, [http://www.livestrong.com/article/351270-the-history-of-womens-high-school-soccer/ The History of Women's High School Soccer]</ref> The [[American Youth Soccer Organization]] and the [[United States Youth Soccer Association]] are among the largest organisations bringing this format to the regional US-based youth soccer arena.<ref name=autogenerated1>[http://www.ayso.org/Libraries/Resources/national_rules__regs_as_of_8282013cleaned.pdf AYSO National Rules & Regulations] [[American Youth Soccer Organization|AYSO]] 2011–2012 Version</ref><ref>[http://www.usyouthsoccer.org/media_kit/ataglance/ At a Glance | US Youth Soccer]</ref><ref>{{Cite web |url=http://www.ayso.org/AboutAYSO.aspx |title=About AYSO |access-date=2013-09-12 |archive-url=https://web.archive.org/web/20130917040041/http://www.ayso.org/AboutAYSO.aspx |archive-date=2013-09-17 |url-status=dead }}</ref>

==Sponsorships==
In recent years a few five-a-side teams have found themselves with sponsorship deals amounting up to thousand of pounds contracts. Sponsors feel with the vast numbers of participation in five-a-side football rising in the UK that it is a good place to advertise and tend to sponsor competition winners or league winners at local facilities so that they know that their deals are with the best five-a-side teams around the area.<ref>{{cite news|author=Hans Kundnani |url=https://www.theguardian.com/business/2006/oct/03/football.money |title=Five-a-side fever nets increased turnover for Powerleague | Business |work=The Guardian |date= 2006-10-03|access-date=2012-08-03 |location=London}}</ref>

==See also==
*[[Masters football]]
*[[Street football]]
*[[Rugby sevens]]
*[[3v3 Soccer]]
*[[Tennents' Sixes]]
*[[Paralympic association football]]
*[[Jorkyball]]

==References==
{{Reflist|30em}}

{{commons cat}}
{{Association football}}

[[Category:Association football variants]]
[[Category:Paralympic association football|*]]