View source for Social choice theory

{{Short description|Study of rational collective decision-making}}
{{Electoral systems sidebar|expanded=Social and collective choice}}
{{Economics sidebar}}
'''Social choice theory''' is a branch of [[welfare economics]] that extends the [[Decision theory|theory of rational choice]] to collective decision-making.<ref>{{Cite book |last=Suzumura |first=Kotaro |url=https://www.google.com/books/edition/Rational_Choice_Collective_Decisions_and/GVg5yfYNMD8C |title=Rational Choice, Collective Decisions, and Social Welfare |date=1983 |publisher=Cambridge University Press |isbn=978-0-521-12255-9 |language=en}}</ref> Social choice studies the behavior of different mathematical procedures ([[social welfare function|social welfare functions]]) used to combine individual preferences into a coherent whole.<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><ref>{{cite journal |url=https://epubs.siam.org/doi/abs/10.1137/1016005 |doi=10.1137/1016005 |title=Social Choice Functions |date=1974 |last1=Fishburn |first1=Peter C. |journal=SIAM Review |volume=16 |pages=63–90 }}</ref><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> It contrasts with [[political science]] in that it is a [[Normative economics|normative]] field that studies how a society can make good decisions, whereas political science is a [[Positive economics|descriptive]] field that observes how societies actually do make decisions. While social choice began as a branch of economics and decision theory, it has since received substantial contributions from [[mathematics]], [[philosophy]], [[political science]], and [[game theory]]. 

Real-world examples of social choice rules include [[constitution|constitutions]] and [[Parliamentary procedure|parliamentary procedures]] for voting on laws, as well as [[Electoral system|electoral systems]];<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> as such, the field is occasionally called '''voting theory'''.<ref name=":1" /><ref>{{Citation |last=Nurmi |first=Hannu |title=Voting Theory |date=2010 |work=e-Democracy: A Group Decision and Negotiation Perspective |series=Advances in Group Decision and Negotiation |volume=5 |pages=101–123 |editor-last=Rios Insua |editor-first=David |url=https://doi.org/10.1007/978-90-481-9045-4_7 |access-date=2024-06-20 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-90-481-9045-4_7 |isbn=978-90-481-9045-4 |editor2-last=French |editor2-first=Simon}}</ref><ref>{{Cite book |last=Coughlin |first=Peter J. |url=https://books.google.com/books?id=dWjV-e4xrXYC |title=Probabilistic Voting Theory |date=1992-10-30 |publisher=Cambridge University Press |isbn=978-0-521-36052-4 |language=en}}</ref> It is closely related to [[mechanism design]], which uses [[game theory]] to model social choice with [[imperfect information]] and [[Homo economicus|self-interested]] citizens.

Social choice differs from [[decision theory]] in that the latter is concerned with how ''individuals'', rather than ''societies'', can make rational decisions.

== History ==
The earliest work on social choice theory comes from the writings of the [[Marquis de Condorcet]], who formulated several key results including his [[Condorcet's jury theorem|jury theorem]] and [[Condorcet paradox|his example]] showing the impossibility of [[majority rule]]. His work was prefigured by [[Ramon Llull]]'s 1299 manuscript ''Ars Electionis'' (''The Art of Elections''), which discussed many of the same concepts, but was lost in the [[Late Middle Ages]] and only rediscovered in the early 21st century.<ref>{{Cite journal |last=Colomer |first=Josep M. |date=2013-02-01 |title=Ramon Llull: from 'Ars electionis' to social choice theory |url=https://doi.org/10.1007/s00355-011-0598-2 |journal=Social Choice and Welfare |language=en |volume=40 |issue=2 |pages=317–328 |doi=10.1007/s00355-011-0598-2 |issn=1432-217X}}</ref>

[[Kenneth Arrow]]'s book ''[[Social Choice and Individual Values]]'' is often recognized as inaugurating the modern era of social choice theory.<ref name="Sen2008" /> Later work has also considered approaches to [[compensation principle|legal compensation]], [[fair division]], [[Population ethics|variable populations]],{{Cn|date=July 2024}} partial [[strategy-proof]]ing of social-choice [[Mechanism design|mechanisms]],<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> [[Natural resource economics|natural resources]],<ref name="Sen2008" /> [[capabilities approach|capabilities and functionings approaches]],<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 measures of [[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><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.]<br />&nbsp; 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.]<br />&nbsp; 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.]<br />&nbsp; Amartya K. Sen (1979 [1984]). ''Collective Choice and Social Welfare'', New York: Elsevier, [http://www.citeulike.org/user/rlai/article/681900 (description)]:<br />&nbsp;&nbsp;&nbsp; ch. 9, "Equity and Justice," pp. 131-51.<br />&nbsp;&nbsp;&nbsp; ch. 9*, "Impersonality and Collective Quasi-Orderings," pp. 152-160.<br />&nbsp; 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.]<br />&nbsp; 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><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>

== Key results ==

=== Arrow's impossibility theorem ===
{{Main|Arrow's impossibility theorem}}
Arrow's impossibility theorem is a key result showing that [[Social choice function|social choice functions]] based only on [[Ordinal utility|ordinal comparisons]], rather than [[cardinal utility]], will behave [[Money pump|incoherently]] (unless they are [[Dictatorship mechanism|dictatorial]]). Such systems violate [[independence of irrelevant alternatives]], meaning they suffer from [[Spoiler effect|spoiler effects]]—the system can behave erratically in response to changes in the quality or popularity of one of the options.

=== Condorcet cycles ===
{{Main|Condorcet cycle}}
Condorcet's example demonstrates that [[democracy]] cannot be thought of as being the same as simple [[majority rule]] or majoritarianism; otherwise, it will be [[paradox|self-contradictory]] when three or more options are available. Majority rule can create cycles that violate the [[Transitive relation|transitive property]]: Attempting to use majority rule as a social choice function creates situations where we have A better than B and B better than C, but C is also better than A.

This contrasts with [[May's theorem]], which shows that simple majority is the optimal voting mechanism when there are only two outcomes, and only ordinal preferences are allowed.

=== Harsanyi's theorem ===
{{Main|Harsanyi's utilitarian theorem}}
[[John Harsanyi|Harsanyi]]'s utilitarian theorem shows that if individuals have preferences that are [[Von Neumann–Morgenstern axioms|well-behaved under uncertainty]] (i.e. [[Coherence (philosophical gambling strategy)|coherent]]), the only [[Coherence (philosophical gambling strategy)|coherent]] and [[Pareto efficiency|Pareto efficient]] social choice function is the [[utilitarian rule]]. This lends some support to the viewpoint expressed of [[John Stuart Mill]], who identified democracy with the ideal of maximizing the [[common good]] (or [[utility]]) of society as a whole, under an [[equal consideration of interests]].

=== Manipulation theorems ===
{{Main|Gibbard's theorem|Gibbard–Satterthwaite theorem}}
Gibbard's theorem provides limitations on the ability of any voting rule to elicit honest preferences from voters, showing that no voting rule is [[Strategyproofness|strategyproof]] (i.e. does not depend on other voters' preferences) for elections with 3 or more outcomes.

The Gibbard–Satterthwaite theorem proves a stronger result for [[Ranked voting|ranked-choice voting systems]], showing that no such voting rule can be sincere (i.e. free of reversed preferences). 

=== Median voter theorem ===
{{Main|Median voter theorem}}

=== Mechanism design ===
{{Main|Mechanism design}}
The field of [[mechanism design]], a subset of social choice theory, deals with the identification of rules that preserve while incentivizing agents to honestly reveal their preferences. One particularly important result is the [[revelation principle]], which is almost a reversal of Gibbard's theorem: for any given social choice function, there exists a mechanism that obtains the same results but incentivizes participants to be completely honest.

Because mechanism design places stronger assumptions on the behavior of participants, it is sometimes possible to design mechanisms for social choice that accomplish apparently-"impossible" tasks. For example, by allowing agents to compensate each other for losses with transfers, the [[Vickrey–Clarke–Groves mechanism|Vickrey–Clarke–Groves (VCG) mechanism]] can achieve the "impossible" according to Gibbard's theorem: the mechanism ensures honest behavior from participants, while still achieving a [[Pareto efficiency|Pareto efficient]] outcome. As a result, the VCG mechanism can be considered a "better" way to make decisions than voting (though only so long as monetary transfers are possible).

=== Others ===
If the domain of preferences is restricted to those that include a [[Condorcet winner criterion|majority-strength Condorcet winner]], then selecting that winner is the unique [[Resolvability criterion|resolvable]], [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and non-manipulable voting rule.<ref name=":1" />{{Explain|reason=More details--what does non-manipulable mean?|date=March 2024}}

== 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, by the [[Von Neumann–Morgenstern utility theorem|VNM theorem]]. 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.<ref name="Robbins" />

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 such measures of utility could be measured, or even considered meaningful. Following arguments similar to those espoused by [[Behaviorism|behaviorists in psychology]], Robbins argued concepts of [[utility]] were unscientific and [[unfalsifiable]]. 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 argued 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.<ref name="Robbins2">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>

Apologists for the interpersonal comparison of utility have argued that Robbins claimed too much. [[John Harsanyi]] agreed that perfect comparisons of mental states are not practically possible, but people can still make ''some'' comparisons thanks to their similar backgrounds, cultural experiences, and psychologies. [[Amartya Sen]] argues that even if interpersonal comparisons of utility are imperfect, we can still say that (despite being positive for [[Nero]]) the [[Great Fire of Rome]] had a negative overall value. Harsanyi and Sen thus argue that at least ''partial'' comparability of utility is possible, and social choice theory should proceed under that assumption.

== Relationship to public choice theory ==
{{See also|Public choice}}

Despite the similar names, "public choice" and "social choice" are two distinct fields that are only weakly related. Public choice deals with the modeling of [[Political system|political systems]] as they actually exist in the real world, and is primarily limited to [[positive economics]] (predicting how politicians and other stakeholders will act). It is therefore often thought of as the application of [[Microeconomics|microeconomic]] models to [[political science]], in order to predict the behavior of [[Politician|political actors]]. By contrast, social choice has a much more [[Normative economics|normative]] bent, and deals with the abstract study of decision procedures and their properties. 

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 Public Choice falls under JEL D72 (Economic Models of Political Processes: [[rent-seeking|Rent-Seeking]], Elections, Legislatures, and Voting Behavior).{{Cn|date=June 2024}}

== Empirical research ==
Since Arrow, social choice theory has been characterized by being predominantly [[Mathematics|mathematical]] and [[theoretical]], but some research has aimed at estimating the frequency of various voting paradoxes, such as 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}} While examples of the paradox seem to occur often in small settings like parliaments, very few examples have been found in larger groups (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> However, the frequency of such paradoxes depends heavily on the number of options and other factors.

== Rules ==
{{Main|Social welfare function}}
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 a common question in social choice theory. The following rules are most common:
*[[Utilitarian social choice rule|''Utilitarian rule'']] – sometimes called the ''max-sum rule'' or [[Benthamite|''Benthamite welfare'']] – aims to maximize the sum of utilities. 
*[[Egalitarian social choice rule|''Egalitarian rule'']] – sometimes called the ''max-min rule'' or [[Rawlsian|''Rawlsian welfare'']] – aims to maximize the smallest utility.

== Social choice functions ==
{{Main|Social choice function}}
A social choice function, sometimes called a [[voting system]] in the context of politics, is a rule that takes an individual's complete and transitive preferences over a set of outcomes and returns a single chosen outcome (or a set of tied outcomes). We can think of this subset as the winners of an election, and compare different social choice functions based on which axioms or mathematical properties they fulfill.<ref name=":1" />

[[Arrow's impossibility theorem]] is what often comes to mind when one thinks about impossibility theorems in voting. There are several famous theorems concerning social choice functions. The [[Gibbard–Satterthwaite theorem]] implies that the only rule satisfying non-imposition (every alternative can be chosen) and [[strategyproofness]] when there are more than two candidates is the [[dictatorship mechanism]]. That is, a voter may be able to cast a ballot that misrepresents their preferences to obtain a result that is more favorable to them under their sincere preferences. [[May's theorem]] shows that when there are only two candidates and [[ordinal utility|only rankings of options are available]], the [[simple majority vote]] is the unique [[Neutrality (social choice)|neutral]], [[Anonymity (social choice)|anonymous]], and [[Positive response|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}}
* [[Arrow's impossibility theorem]]
* [[Compensation principle]]
* [[Computational social choice]]
* [[Condorcet paradox]]
* [[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.&nbsp;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.&nbsp;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.&nbsp;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

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