# Social choice theory

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Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a collective decision or social welfare in some sense. [1] 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. [2]

## Contents

Social choice blends elements of welfare economics 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 for generality, analysis proceeds from a set of seemingly reasonable axioms of social choice to form a social welfare function (or constitution). [3] 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). [1]

## 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, 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's Social Choice and Individual Values (1951), Arrow's impossibility theorem and often acknowledged as the basis of the modern social choice theory and public choice theory. [1] 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 compensations and fairness, liberty and rights, axiomatic domain restrictions on preferences of agents, variable populations, strategy-proofing of social-choice mechanisms, natural resources, [1] [4] capabilities and functionings, [5] and welfare, [6] justice, [7] and poverty. [8]

## 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 functions. 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 [9] 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, 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 (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 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 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 ]

## 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. [10] [11]

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% [11] :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. [12]

## Rules

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 a single-winner election, ${\displaystyle X}$ may represent the set of candidates; in a resource 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 ${\displaystyle u_{i}:X\longrightarrow \mathbb {R} }$ be a 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 ${\displaystyle (u_{i})_{i\in I}}$ to select some element(s) from ${\displaystyle X}$ which are 'best' for society. The question of what 'best' means is 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 - 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. [2] 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, then selecting that winner is the unique resolute, neutral, anonymous, and non-manipulable voting rule. [2] May's theorem states that when there are only two candidates, Simple majority vote is the unique neutral, anonymous, and positively responsive voting rule. [13]

## Notes

1. Amartya Sen (2008). "Social Choice,". The New Palgrave Dictionary of Economics, 2nd Edition, Abstract & TOC.
2. Zwicker, William S.; Moulin, Herve (2016), Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jerome (eds.), "Introduction to the Theory of Voting", Handbook of Computational Social Choice, Cambridge: Cambridge University Press, pp. 23–56, doi:10.1017/cbo9781107446984.003, ISBN   978-1-107-44698-4 , retrieved 2021-12-24
3. For example, in Kenneth J. Arrow (1951). Social Choice and Individual Values, New York: Wiley, ch. II, section 2, A Notation for Preferences and Choice, and ch. III, "The Social Welfare Function".
4. Walter Bossert and John A. Weymark (2008). "Social Choice (New Developments)," The New Palgrave Dictionary of Economics, 2nd Edition, Abstract & TOC.
5. Kaushik, Basu; Lòpez-Calva, Luis F. (2011). Functionings and Capabilities. Handbook of Social Choice and Welfare. Vol. 2. pp. 153–187. doi:10.1016/S0169-7218(10)00016-X. ISBN   9780444508942.
6. d'Aspremont, Claude; Gevers, Louis (2002). Chapter 10 Social welfare functionals and interpersonal comparability. Handbook of Social Choice and Welfare. Vol. 1. pp. 459–541. doi:10.1016/S1574-0110(02)80014-5. ISBN   9780444829146.
7. Amartya Sen ([1987] 2008). "Justice," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract & TOC.
Bertil Tungodden (2008). "Justice (New Perspectives)," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
Louis Kaplow (2008). "Pareto Principle and Competing Principles," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
Amartya K. Sen (1979 [1984]). Collective Choice and Social Welfare, New York: Elsevier, (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, Description, contents, and chapter-preview 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. Abstract.
8. Dutta, Bhaskar (2002). Chapter 12 Inequality, poverty and welfare. Handbook of Social Choice and Welfare. Vol. 1. pp. 597–633. doi:10.1016/S1574-0110(02)80016-9. ISBN   9780444829146.
9. Lionel Robbins (1932, 1935, 2nd ed.). An Essay on the Nature and Significance of Economic Science, London: Macmillan. Links for 1932 HTML and 1935 facsimile.
10. Kurrild-Klitgaard, Peter (2014). "Empirical social choice: An introduction". Public Choice. 158 (3–4): 297–310. doi:10.1007/s11127-014-0164-4. ISSN   0048-5829. S2CID   148982833.
11. Van Deemen, Adrian (2014). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3–4): 311–330. doi:10.1007/s11127-013-0133-3. ISSN   0048-5829. S2CID   154862595.
12. Kurrild-Klitgaard, Peter (2014). "An empirical example of the Condorcet paradox of voting in a large electorate". Public Choice. 107: 135–145. doi:10.1023/A:1010304729545. ISSN   0048-5829. S2CID   152300013.
13. May, Kenneth O. (October 1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. JSTOR   1907651.

## Related Research Articles

The Condorcet paradox in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other: Suppose majorities prefer, for example, candidate A over B, B over C, and yet C over A. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

Kenneth Joseph Arrow 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 social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In welfare economics, a social welfare function is a function that ranks social states as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income.

The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it always attempts to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulation differs widely in both language and exact content.

In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner. It states that for every voting rule, one of the following three things must hold:

1. The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
2. The rule limits the possible outcomes to two alternatives only; or
3. The rule is susceptible to tactical voting: in certain conditions, a voter's sincere ballot may not best defend their opinion.

Welfare economics is a branch of economics that uses microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level.

The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences.

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 mild conditions:

1. The Unrestrictedness Condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,
2. The Pareto Condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and
3. Liberalism, or L : all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, you can exert your freedom of choice at least in some decision with tangible results.

Lexicographic preferences or lexicographic orderings describe comparative preferences where an economic agent prefers any amount of one good (X) to any amount of another (Y). Specifically, if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie between bundles with regard to the number of units of X will the agent start comparing the number of units of Y across bundles. Lexicographic preferences extend utility theory analogously to the way that nonstandard infinitesimals extend the real numbers. With lexicographic preferences, the utility of certain goods is infinitesimal in comparison to others.

Kenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somewhat formally, the "social choice" in the title refers to Arrow's representation of how social values from the set of individual orderings would be implemented under the constitution. Less formally, each social choice corresponds to the feasible set of laws passed by a "vote" under the constitution even if not every individual voted in favor of all the laws.

The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social choice theory and microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept was introduced by Sen (1969) to study the consequences of Arrow's theorem.

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. Here any characteristics that define each person are distinguished from the rest of the social state and put on a par with conventional measures of wealth insofar as they affect an extended sympathy judgment.

Justice in economics is a subcategory of welfare economics. It is a "set of moral and ethical principles for building economic institutions." Economic justice aims to create opportunities for every person to have a dignified, productive and creative life that extends beyond simple economics.

In social choice and operations research, the utilitarian rule is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the sum of the utilities of all individuals in society. It is a formal mathematical representation of the utilitarian philosophy.

Kevin W. S. Roberts was the Sir John Hicks Professor of Economics at the University of Oxford until his retirement in 2020.

Prasanta Kumar Pattanaik, is emeritus professor at the Department of Economics at the University of California. He is a Fellow of the Econometric Society.

A jury theorem is a mathematical theorem proving that, under certain assumptions, a decision attained using majority voting in a large group is more likely to be correct than a decision attained by a single expert. It serves as a formal argument for the idea of wisdom of the crowd, for decision of questions of fact by jury trial, and for democracy in general.

Fractional social choice is a branch of social choice theory in which the collective decision is not a single alternative, but rather a weighted sum of two or more alternatives. For example, if society has to choose between three candidates: A B or C, then in standard social choice, exactly one of these candidates is chosen, while in fractional social choice, it is possible to choose "2/3 of A and 1/3 of B". A common interpretation of the weighted sum is as a lottery, in which candidate A is chosen with probability 2/3 and candidate B is chosen with probability 1/3. Due to this interpretation, fractional social choice is also called random social choice, probabilistic social choice, or stochastic social choice. But it can also be interpreted as a recipe for sharing, for example: