Social Choice Theory

Social choice theory is the study of collective decision procedures and mechanisms.

Central Questions

How can a group of individuals choose a winning outcome (e.g., policy, electoral candidate) from a given set of options?

What are the properties of different voting systems?

When is a voting system democratic?

How can a collective (e.g., electorate, legislature, collegial court, expert panel, or committee) arrive at coherent collective preferences or judgments on some issues, on the basis of its members’ individual preferences or judgments?

How can we rank different social alternatives in an order of social welfare?

Theory

Voting Systems

Rational Voting System

Every rational voting system should satisfy the following 5 conditions:

1. Unanimity
2. No Dictators
3. Unrestricted Domain
4. Transitive
5. Independence of Irrelevant Alternatives

Kenneth Arrow - Social Choice and Individual Values (1951)

Types of Voting Systems

• Ranked Voting
• Rated Voting
• Approval Voting

Ranked Voting

Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner or multiple winners.

Example of a ranked ballot:

No ranked voting system can be rational.

This theorem was proved by Kenneth Archer in 1951. Satisfying all conditions in a ranked voting system with 3 or more candidates is impossible.

Condorcet Cycles

The Condorcet winner is the candidate preferred over all others by a majority of voters. However, it is possible for an election to have no Condorcet winner, a situation called a Condorcet Cycle.

Example: 3 Candidates, 3 Voters. The Voters votes are:

1. A-C-B
2. B-A-C
3. C-B-A

A cannot be the Condorcet winner, because 2/3 of voters prefer B over A. Nor can B be the Condorcet winner, because 2/3 of voters prefer C over B. Similar logic applies for C.

Rated Voting Systems

Approval Voting Systems

Sources

• Fundamentals of Social Choice Theory - https://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Myerson96.pdf

• Arrow, K. J. (2012). Social choice and individual values (Vol. 12). Yale university press. - https://ve42.co/Arrow2012
• Arrow, K. J. (1950). A Difficulty in the Concept of Social Welfare. Journal of Political Economy - https://ve42.co/Arrow1950
• Black, D. (1948). On the rationale of group decision-making. Journal of political economy, 56(1), 23-34. - https://ve42.co/Black1948
• Black, D. (1969). On Arrow's impossibility theorem. The Journal of Law and Economics, 12(2), 227-248. - https://ve42.co/Arrow1969
• Maskin, E., & Sen, A. (2014). The Arrow impossibility theorem. Columbia University Press. - https://ve42.co/Maskin2014
• Gehrlein, W. V., & Valognes, F. (2001). Condorcet efficiency: A preference for indifference. Social Choice and Welfare - https://ve42.co/Gehrlein2001
• Dardanoni, V. (2001). A pedagogical proof of Arrow's Impossibility Theorem. Social Choice and Welfare, 18(1), 107-112. - https://ve42.co/Dardanoni2001
• McCune, D., & Wilson, J. (2023). Ranked-choice voting and the spoiler effect. Public Choice, 196(1), 19-50. - https://ve42.co/McCune2023
• Santucci, J. (2021). Variants of ranked-choice voting from a strategic perspective. Politics and Governance, 9(2), 344-353. - https://ve42.co/Santucci2021
• Kaminski, M. M. (2018). Spoiler effects in proportional representation systems: evidence from eight Polish parliamentary elections, 1991–2015. Public Choice, 176(3), 441-460. - https://ve42.co/Kaminski2018
• Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review, 72(3), 831-847. - https://ve42.co/Brams1978