Social choice theory is a branch ofwelfare economics that extends thetheory of rational choice to collective decision-making.^[1] Social choice studies the behavior of different mathematical procedures (social welfare functions) used to combine individual preferences into a coherent whole.^[2][3][4] It contrasts withpolitical science in that it is anormative field that studies how a society can make good decisions, whereas political science is adescriptive 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 frommathematics,philosophy,political science, andgame theory.

Real-world examples of social choice rules includeconstitutions andparliamentary procedures for voting on laws, as well aselectoral systems;^[5] as such, the field is occasionally calledvoting theory.^[5][6][7] It is closely related tomechanism design, which usesgame theory to model social choice withimperfect information andself-interested citizens.

Social choice differs fromdecision theory in that the latter is concerned with howindividuals, rather thansocieties, can make rational decisions.

The earliest work on social choice theory comes from the writings of theMarquis de Condorcet, who formulated several key results including hisjury theorem andhis example showing the impossibility ofmajority rule. His work was prefigured byRamon Llull's 1299 manuscriptArs Electionis (The Art of Elections), which discussed many of the same concepts, but was lost in theLate Middle Ages and only rediscovered in the early 21st century.^[8]

Kenneth Arrow's bookSocial Choice and Individual Values is often recognized as inaugurating the modern era of social choice theory.^[4] Later work has also considered approaches tolegal compensation,fair division,variable populations,^{[citation needed]} partialstrategy-proofing of social-choicemechanisms,^[9]natural resources,^[4]capabilities and functionings approaches,^[10] and measures ofwelfare.^[11][12][13]

Arrow's impossibility theorem is a key result showing thatsocial choice functions based only onordinal comparisons, rather thancardinal utility, will behaveincoherently (unless they aredictatorial). Such systems violateindependence of irrelevant alternatives, meaning they suffer fromspoiler effects—the system can behave erratically in response to changes in the quality or popularity of one of the options.

Condorcet's example demonstrates thatdemocracy cannot be thought of as being the same as simplemajority rule or majoritarianism; otherwise, it will beself-contradictory when three or more options are available. Majority rule can create cycles that violate thetransitive 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 withMay'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 utilitarian theorem shows that if individuals have preferences that arewell-behaved under uncertainty (i.e.coherent), the onlycoherent andPareto efficient social choice function is theutilitarian rule. This lends some support to the viewpoint expressed ofJohn Stuart Mill, who identified democracy with the ideal of maximizing thecommon good (orutility) of society as a whole, under anequal consideration of interests.

Gibbard's theorem provides limitations on the ability of any voting rule to elicit honest preferences from voters, showing that no voting rule isstrategyproof (i.e. does not depend on other voters' preferences) for elections with 3 or more outcomes.

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

The field ofmechanism 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 therevelation 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, theVickrey–Clarke–Groves (VCG) mechanism can achieve the "impossible" according to Gibbard's theorem: the mechanism ensures honest behavior from participants, while still achieving aPareto 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).

If the domain of preferences is restricted to those that include amajority-strength Condorcet winner, then selecting that winner is the uniqueresolvable,neutral,anonymous, and non-manipulable voting rule.^{[5][further explanation needed]}

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 havepreferences, and it follows that they can be modeled usingutility functions, by theVNM theorem. But much of the research in the field assumes that those utility functions are internal to humans, lack a meaningful unit of measure andcannot be compared across different individuals.^[14] Whether this type ofinterpersonal utility comparison is possible or not significantly alters the available mathematical structures for social welfare functions and social choice theory.^[14]

In one perspective, followingJeremy Bentham,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, followingLionel Robbins, questioned whether such measures of utility could be measured, or even considered meaningful. Following arguments similar to those espoused bybehaviorists in psychology, Robbins argued concepts ofutility were unscientific andunfalsifiable. Consider for instance thelaw 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 beyondpositive science; that is, one cannot measure changes in the utility of someone else, nor is it required by positive theory.^[15]

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 makesome 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 forNero) theGreat Fire of Rome had a negative overall value. Harsanyi and Sen thus argue that at leastpartial comparability of utility is possible, and social choice theory should proceed under that assumption.

Despite the similar names, "public choice" and "social choice" are two distinct fields that are only weakly related. Public choice deals with the modeling ofpolitical systems as they actually exist in the real world, and is primarily limited topositive economics (predicting how politicians and other stakeholders will act). It is therefore often thought of as the application ofmicroeconomic models topolitical science, in order to predict the behavior ofpolitical actors. By contrast, social choice has a much morenormative bent, and deals with the abstract study of decision procedures and their properties.

TheJournal of Economic Literature classification codes place Social Choice underMicroeconomics atJEL D71 (with Clubs, Committees, and Associations) whereas Public Choice falls under JEL D72 (Economic Models of Political Processes:Rent-Seeking, Elections, Legislatures, and Voting Behavior).^{[citation needed]}

Since Arrow, social choice theory has been characterized by being predominantlymathematical andtheoretical, but some research has aimed at estimating the frequency of various voting paradoxes, such as theCondorcet paradox.^[16][17] 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%.^[17]: 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.^[18] However, the frequency of such paradoxes depends heavily on the number of options and other factors.

Let${\displaystyle X}$ be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from${\displaystyle X}$. For example, in asingle-winner election,${\displaystyle X}$ may represent the set of candidates; in aresource allocation setting,${\displaystyle X}$ may represent all possible allocations.

Let${\displaystyle I}$ be a finite set, representing a collection of individuals. For each${\displaystyle i\in I}$, let${\displaystyle u_i:X⟶\mathbb{R}}$ be autility function, describing the amount of happiness an individuali derives from each possible state.

Asocial choice rule is a mechanism which uses the data${\displaystyle (u_i{)}_{i\in I}}$ to select some element(s) from${\displaystyle X}$ 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 rule – sometimes called themax-sum rule orBenthamite welfare – aims to maximize the sum of utilities.
• Egalitarian rule – sometimes called themax-min rule orRawlsian welfare – aims to maximize the smallest utility.

A social choice function, sometimes called avoting 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.^[5]

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. TheGibbard–Satterthwaite theorem implies that the only rule satisfying non-imposition (every alternative can be chosen) andstrategyproofness when there are more than two candidates is thedictatorship 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 andonly rankings of options are available, thesimple majority vote is the uniqueneutral,anonymous, andpositively-responsive voting rule.^[19]

• 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,Link to text, with Section 8 on the theory and background.
• _____, (1983).Collected Papers, v. 1,Social Choice and Justice, Oxford: BlackwellISBN 0-674-13760-4
• Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. (1997).Social Choice Re-Examined, 2 vol., London: Palgrave MacmillanISBN 0-312-12739-1 &ISBN 0-312-12741-3
• _____, eds. (2002).Handbook of Social Choice and Welfare, v. 1. Chapter-previewlinks.
• _____, ed. (2011).Handbook of Social Choice and Welfare, v. 2, Amsterdam: Elsevier. Chapter-previewlinks.
• Bossert, Walter and John A. Weymark (2008). "Social Choice (New Developments),"The New Palgrave Dictionary of Economics, 2nd Edition, London: Palgrave MacmillanAbstract.
• 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 PDFlink.
• Feldman, Allan M. and Roberto Serrano (2006).Welfare Economics and Social Choice Theory, 2nd ed., New York: SpringerISBN 0-387-29367-1,ISBN 978-0-387-29367-7Arrow-searchable chapter previews.
• Fleurbaey, Marc (1996). Théories économiques de la justice, Paris: Economica.
• Gaertner, Wulf (2006).A primer in social choice theory. Oxford: Oxford University Press.ISBN 978-0-19-929751-1.
• Harsanyi, John C. (1987). "Interpersonal Utility Comparisons,"TheNew Palgrave: A Dictionary of Economics, v. 2, London: Palgrave, pp. 955–58.
• Moulin, Herve (1988).Axioms of cooperative decision making. Cambridge: Cambridge University Press.ISBN 978-0-521-42458-5.
• Myerson, Roger B. (June 2013). "Fundamentals of social choice theory".Quarterly Journal of Political Science.8 (3):305–337.CiteSeerX 10.1.1.297.6781.doi:10.1561/100.00013006.
• Nitzan, Shmuel (2010).Collective Preference and Choice. Cambridge, UK: Cambridge University Press.ISBN 978-0-521-72213-1.
• 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.
• Sen, Amartya K. (1970 [1984]).Collective Choice and Social Welfare, New York: ElsevierISBN 0-444-85127-5Description.
• _____, (1998). "The Possibility of Social Choice", Nobel Prize Lecture[1].
• _____, (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: PalgraveAbstract.
• Shoham, Yoav; Leyton-Brown, Kevin (2009).Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York:Cambridge University Press.ISBN 978-0-521-89943-7.. A comprehensive reference from a computational perspective; see Chapter 9.Downloadable free online.
• Suzumura, Kotaro (1983).Rational Choice, Collective Decisions, and Social Welfare, Cambridge: Cambridge University PressISBN 0-521-23862-5
• Taylor, Alan D. (2005).Social choice and the mathematics of manipulation. New York: Cambridge University Press.ISBN 978-0-521-00883-9.

• List, Christian."Social Choice Theory". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.ISSN 1095-5054.OCLC 429049174.
• Social Choice Bibliography by J. S. KellyArchived 2017-12-23 at theWayback Machine
• Electowiki, awiki covering many subjects of social choice and voting theory

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