Social choice theory
Social choice theory or social choice is a branch of welfare economics that studies the processes of collective decision-making.[1] 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.[2][3] Social choice theory includes the closely-related field of voting theory,[4][5][6] and is strongly tied to the field of mechanism design, which can be thought of as the combination of social choice with game theory.
Whereas decision theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with groups making decisions, 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.[4]
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 in the Late Middle Ages and only rediscovered in the early 21st century.[7]
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 compensation, fair division, variable populations, strategy-proofing of social-choice mechanisms,[8] natural resources,[1] capabilities and functionings approaches,[9] and measures of welfare.[10][11][12]
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.[15][16] 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%.[16]: 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.[17] 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:
