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Academic discipline

Social choice theory or social choice is a branch of welfare economics that analyzes mechanisms and procedures for collective decision making. Social choice incorporates insights from economics, mathematics, and game theory to find the best ways to combine individual opinions, preferences, or beliefs into a single coherent measure of the quality of different outcomes, called a social welfare function. Whereas decision theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with groups making choices based on the preferences of individuals. Real world examples include enacting laws under a constitution or voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences.

The earliest work on social choice theory comes from the writings of the Marquis de Condorcet, who formulated several key results including his jury theorem and 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. Kenneth Arrow's book Social Choice and Individual Values is often recognized as inaugurating modern social choice theory. natural resources, and measures of welfare.

Arrow's impossibility theorem is a key result showing that social choice functions based only on ordinal comparisons, rather than cardinal utility, will behave incoherently (unless they are dictatorial or violate Pareto efficiency). Such systems violate independence of irrelevant alternatives, i. e. the system will behave erratically when outcomes are added or removed.

Condorcet's example demonstrates that democracy cannot be thought of as being the same as simple majority rule or majoritarianism; otherwise, it will be self contradictory when 3 or more options are available. Majority rule can create cycles that violate the 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 still better than A. This contrasts with May's theorem, which shows that simple majority is the optimal voting mechanism when there are only 2 outcomes, and only ordinal preferences are allowed.

Harsanyi's utilitarian theorem shows that if individuals have preferences that are well behaved under uncertainty (i. e. coherent), the only coherent and 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.

Gibbard's theorem provides limitations on the ability of any voting rule to elicit honest preferences from voters, showing that no voting rule is 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 choice voting systems, showing that no such voting rule can be sincere (i. e. free of reversed preferences).

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 drops some of the assumptions made by voting theory, it is possible to design mechanisms for social choice to accomplish "impossible" tasks. For example, by allowing agents to compensate each other for losses with transfers, the 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 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).

The Campbell Kelley theorem states that, if there exists a majority strength Condorcet winner, then selecting that winner is the unique resolvable, neutral, anonymous, and non manipulable voting rule. 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 such measures of utility could be measured, or even considered meaningful. Following arguments similar to those espoused by 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 (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 not practically possible, but believes human beings can make some interpersonal comparisons of utility because they have similar backgrounds, cultural experiences, and psychology. Amartya Sen (1970, p. 99) 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 proceeds under that assumption.

Despite the similar names, "public choice" and "social choice" are two distinct fields, though the two are closely related. 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).

Since Arrow, social choice theory has been characterized by being predominantly mathematical and theoretical, but some research has aimed at estimating the frequency of various voting paradoxes, such as 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%. However, the frequency of such paradoxes depends heavily on the number of options and other factors.

Let be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from For example, in a single winner election, may represent the set of candidates; in a resource allocation setting, may represent all possible allocations. Let be a finite set, representing a collection of individuals. For each , let 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 to select some element(s) from 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 the max sum rule or Benthamite welfare – aims to maximize the sum of utilities. Egalitarian rule – sometimes called the max min rule or Rawlsian welfare – aims to maximize the smallest utility.

A social choice function or a voting rule 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.