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Social choice theory or social choice is a
theoretical A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom and the United States. Sen has made contributions to welfare economics, social choice theory, econom ...
(2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition
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 A constitution is the aggregate of fundamental principles or established precedents that constitute the legal basis of a polity, organisation or other type of entity and commonly determine how that entity is to be governed. When these princ ...
. Another example is voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences. Social choice blends elements of
welfare economics Welfare economics is a branch of economics that uses microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of welfare economics gives rise to the field of public ec ...
and public choice theory. It is methodologically individualistic, in that it aggregates preferences and behaviors of individual members of society. Using elements of
formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
for generality, analysis proceeds from a set of seemingly reasonable axioms of social choice to form a ''
social welfare function In welfare economics, a social welfare function is a function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the f ...
'' (or ''constitution''). Results uncovered the logical incompatibility of various axioms, as in Arrow's theorem, revealing an
aggregation problem An ''aggregate'' in economics is a summary measure. It replaces a vector that is composed of many real numbers by a single real number, or a scalar. Consequently there occur various problems that are inherent in the formulations that use aggregate ...
and suggesting reformulation or theoretical
triage In medicine, triage () is a practice invoked when acute care cannot be provided for lack of resources. The process rations care towards those who are most in need of immediate care, and who benefit most from it. More generally it refers to prio ...
in dropping some axiom(s).


Overlap with public choice theory

"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 Journal of Economic Literature classification codes place Social Choice under Microeconomics at 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 is the act of growing one's existing wealth without creating new wealth by manipulating the social or political environment. Rent-seeking activities have negative effects on the rest of society. They result in reduced economic effic ...
, Elections, Legislatures, and Voting Behavior). Social choice theory (and public choice theory) dates from Condorcet's formulation of the voting paradox, though it arguably goes back further to Ramon Llull's 1299 publication.
Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist. He was the joint winner of the Nobel Memorial Prize in Economic Sciences with John Hicks in 1972. In economics ...
's ''
Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963, 3rd ed., 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. ...
'' (1951),
Arrow's impossibility theorem Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syst ...
and often acknowledged as the basis of the modern social choice theory and public choice theory. 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 Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom and the United States. Sen has made contributions to welfare economics, social choice theory, econom ...
'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 compensations and
fairness Fairness or being fair can refer to: * Justice * The character in the award-nominated musical comedy '' A Theory of Justice: The Musical.'' * Equity (law), a legal principle allowing for the use of discretion and fairness when applying justice ...
, liberty and rights, axiomatic domain restrictions on preferences of agents, variable populations,
strategy-proof In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information abou ...
ing of social-choice mechanisms,
natural resources Natural resources are resources that are drawn from nature and used with few modifications. This includes the sources of valued characteristics such as commercial and industrial use, aesthetic value, scientific interest and cultural value. ...
, capabilities and functionings, and
welfare Welfare, or commonly social welfare, is a type of government support intended to ensure that members of a society can meet basic human needs such as food and shelter. Social security may either be synonymous with welfare, or refer specifical ...
,
justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and perspective ...
, and poverty.


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 preferences, and it follows that they can be modeled using
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
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 individualsLionel Robbins (1932, 1935, 2nd ed.). ''An Essay on the Nature and Significance of Economic Science'', London: Macmillan. Links fo
1932 HTML
an
1935 facsimile
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 Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747">Old_Style_and_New_Style_dates.htm ...
,
utilitarians In ethical philosophy, utilitarianism is a family of normative ethical theories that prescribe actions that maximize happiness and well-being for all affected individuals. Although different varieties of utilitarianism admit different charact ...
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 Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist, and prominent member of the economics department at the London School of Economics (LSE). He is known for his leadership at LSE, his proposed def ...
, questioned whether mental states, and the utilities they reflect, can be measured and, ''a fortiori'', interpersonal ''comparisons'' of
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
as well as the social choice theory on which it is based. Consider for instance the
law of diminishing marginal utility In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a good or service describes how much pleasure or satisfaction is gained by consumers as a result of the increase or decrease in consumpt ...
, 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 ( 1935, pp. 138–40) argues that this notion is beyond 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 John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994. He is best known for his contributions to the ...
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 Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom and the United States. Sen has made contributions to welfare economics, social choice theory, econom ...
(1970, p. 99), it should be possible to say that
Emperor Nero Nero Claudius Caesar Augustus Germanicus ( ; born Lucius Domitius Ahenobarbus; 15 December AD 37 – 9 June AD 68), was the fifth Roman emperor and final emperor of the Julio-Claudian dynasty, reigning from AD 54 unt ...
'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 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 A dictatorship is a form of government which is characterized by a leader, or a group of leaders, which holds governmental powers with few to no limitations on them. The leader of a dictatorship is called a dictator. Politics in a dictatorship a ...
and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties.


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. 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% (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.


Rules

Let X be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from X. For example, in a single-winner election, X may represent the set of candidates; in a
resource allocation In economics, resource allocation is the assignment of available resources to various uses. In the context of an entire economy, resources can be allocated by various means, such as markets, or planning. In project management, resource allocati ...
setting, X may represent all possible allocations. Let I be a finite set, representing a collection of individuals. For each i \in I, let u_i:X\longrightarrow\mathbb be a ''
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
'', describing the amount of happiness an individual ''i'' derives from each possible state. A ''social choice rule'' is a mechanism which uses the data (u_i)_ to select some element(s) from X 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 rule'' - also called the ''max-sum rule'' - aims to maximize the sum of utilities, thus maximizing the efficiency. *The ''egalitarian rule'' - also called the ''max-min rule'' - aims to maximize the smallest utility, thus maximizing the fairness. *The ''
proportional-fair rule In operations research and social choice, the proportional-fair (PF) rule is a rule saying that, among all possible alternatives, one should pick an alternative that cannot be improved, where "improvement" is measured by the sum of relative improv ...
'' - 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 individuals' 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. For example, Instant-runoff voting satisfies the
Independence of clones criterion In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the ...
, whereas the Borda count does not; conversely, Borda Count satisfies the
Monotonicity criterion The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them h ...
whereas IRV does not.


Theorems

Arrow's impossibility theorem Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives (options), no ranked voting electoral syst ...
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, then selecting that winner is the unique resolute,
neutral Neutral or neutrality may refer to: Mathematics and natural science Biology * Neutral organisms, in ecology, those that obey the unified neutral theory of biodiversity Chemistry and physics * Neutralization (chemistry), a chemical reaction in ...
, anonymous, and non-manipulable voting rule. May's theorem states that when there are only two candidates,
Simple majority vote Plurality voting refers to electoral systems in which a candidate, or candidates, who poll more than any other counterpart (that is, receive a plurality), are elected. In systems based on single-member districts, it elects just one member per ...
is the unique
neutral Neutral or neutrality may refer to: Mathematics and natural science Biology * Neutral organisms, in ecology, those that obey the unified neutral theory of biodiversity Chemistry and physics * Neutralization (chemistry), a chemical reaction in ...
, anonymous, and positively responsive voting rule.


See also

* Compensation principle *
Computational social choice Computational social choice is a field at the intersection of social choice theory, theoretical computer science, and the analysis of multi-agent systems. It consists of the analysis of problems arising from the aggregation of preferences of a gr ...
* Condorcet paradox *
Emotional choice theory Emotional choice theory (also referred to as the "logic of affect") is a social scientific action model to explain human decision-making. Its foundation was laid in Robin Markwica’s monograph ''Emotional Choices'' published by Oxford University ...
*
Extended sympathy Extended sympathy in welfare economics refers to interpersonal value judgments of the form that social state ''x'' for person ''A'' is ranked better than, worse than, or as good as social state ''y'' for person ''B'' (Arrow, 1963, pp. 114–15) ...
* Game theory * Group decision-making * Justice (economics) *
Liberal paradox The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly ...
*
Mechanism design Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts a ...
* Nakamura number * Rational choice theory * Rule according to higher law *
Voting system An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...


Notes


References

* Arrow, Kenneth J. (1951, 2nd ed., 1963). ''
Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963, 3rd ed., 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. ...
'', New York: Wiley. *_____, (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: Blackwell * Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. (1997). ''Social Choice Re-Examined'', 2 vol., London: Palgrave Macmillan & *_____, eds. (2002). ''Handbook of Social Choice and Welfare'', v. 1. Chapter-previe
links
*_____, ed. (2011). ''Handbook of Social Choice and Welfare'', v. 2, Amsterdam: Elsevier. Chapter-previe
links
* Bossert, Walter and John A. Weymark (2008). "Social Choice (New Developments)," ''
The New Palgrave Dictionary of Economics ''The New Palgrave Dictionary of Economics'' (2018), 3rd ed., is a twenty-volume reference work on economics published by Palgrave Macmillan. It contains around 3,000 entries, including many classic essays from the original Inglis Palgrave Diction ...
'', 2nd Edition, London: Palgrave Macmilla
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 PD
link.
* Feldman, Allan M. and Roberto Serrano (2006). ''Welfare Economics and Social Choice Theory'', 2nd ed., New York: Springer ,
Arrow-searchable chapter previews.
* Fleurbaey, Marc (1996). Théories économiques de la justice, Paris: Economica. * * Harsanyi, John C. (1987). "Interpersonal Utility Comparisons," ''The New Palgrave: A Dictionary of Economics'', v. 2, London: Palgrave, pp. 955–58. * * * * Robbins, Lionel (1935). ''
An Essay on the Nature and Significance of Economic Science Lionel Robbins' ''Essay'' (1932, 1935, 2nd ed., 158 pp.) sought to define more precisely economics as a science and to derive substantive implications. Analysis is relative to "accepted solutions of particular problems" based on best modern practi ...
'', 2nd ed., London: Macmillan, ch. VI * ____, (1938). "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41. * Sen, Amartya K. (1970
984 Year 984 ( CMLXXXIV) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. Events By place Europe * Spring – German boy-king Otto III (4-years old) is seized by the deposed Henry II ...
. ''Collective Choice and Social Welfare'', New York: Elsevier
Description.
*_____, (1998). "The Possibility of Social Choice", Nobel Prize Lectur

* _____, (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: Palgrav
Abstract.
* . A comprehensive reference from a computational perspective; see Chapter 9

* Kotaro Suzumura, Suzumura, Kotaro (1983). ''Rational Choice, Collective Decisions, and Social Welfare'', Cambridge: Cambridge University Press *


External links

*
Social Choice Bibliography by J. S. Kelly

Electowiki
a
wiki A wiki ( ) is an online hypertext publication collaboratively edited and managed by its own audience, using a web browser. A typical wiki contains multiple pages for the subjects or scope of the project, and could be either open to the pub ...
covering many subjects of social choice and voting theory {{Authority control Ethics Political science Public economics Applied mathematics Collective intelligence Social epistemology Mathematical economics Law and economics