Social choice theory

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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

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]

Arrow's impossibility theorem is a key result showing that

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 theorem

Harsanyi's utilitarian theorem shows that if individuals have preferences that are

Gibbard's theorem provides limitations on the ability of any voting rule to elicit honest , showing that no voting rule is strategyproof (i.e. does not depend on other voters' choices) 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).

Median voter theorem

Mechanism design

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 democratic voting when monetary transfers are available.

Others

The Campbell-Kelley theorem states that, if there exists a Condorcet winner, then selecting that winner is the unique resolvable, neutral, anonymous, and non-manipulable voting rule.^{[2][further explanation needed]}

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

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

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.^{[citation needed]} Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption.

Relationship to public choice theory

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).

Empirical research

Since Arrow, social choice theory has been characterized by being predominantly

However, the frequency of such paradoxes depends heavily on the number of options and other factors.

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

${\displaystyle X}$

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 a common question in social choice theory. The following rules are most common:

• Benthamite welfare
  – aims to maximize the sum of utilities.
• Rawlsian welfare
  – aims to maximize the smallest utility.

Functions

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]

Notes