Social choice theory
Social choice theory or social choice is a branch of economics that analyzes mechanisms and procedures for collective decisions.[1] Social choice incorporates insights from welfare economics, mathematics, and political science to find the best ways to combine individual opinions, preferences, or beliefs into a single coherent measure of well-being.
Whereas decision 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]
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]
History[edit]
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 until being rediscovered in the early 21st century.[4]
Kenneth Arrow's book Social Choice and Individual Values is often recognized as inaugurating modern social choice theory.[1] Later work also considers approaches to compensations, fair division, variable populations, strategy-proofing of social-choice mechanisms,[5] natural resources,[1] capabilities and functionings approaches,[6] and measures of welfare.[7][8][9]
Interpersonal utility comparison[edit]
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, by the 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.[10] 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.
Empirical research[edit]
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.[11][12] 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%.[12]: 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.[13] 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:
Functions[edit]
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.[2]
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 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. May's theorem states that when there are only two candidates, simple majority vote is the unique neutral, anonymous, and positively responsive voting rule.[14]