Social Choice Theory

Overview

Social choice theory is a subfield of political theory concerned with the logic of collective decision-making, whether voting or multicriterion decision-making. As such it is a subset of rational choice theory, which includes the logic of individual as well as collective choice. Much of the writing in the field is abstract, using formal mathematical logic. The origins of social choice theory lie in voting under a single preference criterion, but the social choice model has been extended to multidimensional decision-making involving multiple decision criteria (see Arrow and Reynaud, 1986).

Key Concepts and Terms

The general impossibility theorem, also known as Arrow's possibility theorem, demonstrates that it is impossible to construct a perfect system of voting when there are more than two alternatives. See Arrow (1951). Subsequent research by Riker (1982) used social choice analysis pioneered by Arrow to explore the pros and cons of alternative voting methods. More specifically, Arrow's theorem holds that if there are more than two alternatives, there is no social preference function which can satisfy four properties (see Johnson, 1998: 16-20 for proof of Arrow's theorem):
1. Universal domain: All preference orderings are an acceptable basis for voting, and the outcome of a vote is rational only if it is one of the preference orderings of the voters.
2. Pareto efficiency: A voting system is irrational if voters prefer one policy, but the voting system makes another policy the winning choice.
3. Nondictatorship: No one person can dictate the outcome.
4. Independence from irrelevant alternatives: The order of preferences among the given alternatives in the vote is not affected by alternatives not being voted on. That is, the preference order of A and B will be the same whether or not C is also being considered.
The voters' paradox shows that when there are three or more alternatives, voters' preference orders may be intransitive, leading to a cyclical majority. Consider three alternatives, A, B, and C. Let Voter 1 prefer A to B to C. Let voter 2 prefer C to A to B. Let Voter 3 prefer B to C to A. If there is a vote on A and B, A wins. If there is a vote on B and C, B wins. If there is a vote on A and C, C wins. Under these conditions, the winner of the runoff election will depend on the sequence of votes. If there is a vote on A and C first, the runoff will be C and B, and B will win. If there is a vote on A and B first, the runoff will be A and C, and C will win. If there is a vote on B and C first, the runoff will be A and B, and A will win. The reality, of course, is that all candidates are equal, but the majority voting system will result in a purely arbitrary winner anyway.
A Condorcet winner is an alternative which will win in a pairwise majority vote, for all other alternatives. That is, there may be a cyclical majority for the remaining choices, but the Condorcet winner is the clear first choice in all votes.
The median voter theorem holds that the median of the most-preferred points of the voters is an equilibrium point. That is, majority voting leads to a topical (transitive) preference ordering among alternatives if the number of choices is odd and preferences are single-peaked (there is only one criterion or dimension for ranking) and in one direction (all voters rank the same from low to high, agreeing on what is "low"). The median of voters' choices is a Condorcet winner.
- Take it or leave it votes and agenda setters. Voters can be forced to accept an alternative other than the median one under certain conditions. If an "agenda setter" is in a position to demand policy y (which is other than the median choice) or else the outcome goes to an undesirable "reversion point," r, then the worse r is, the more deviant y may be from the median choice and still have voters agreeing to y.

Assumptions

Universal domain, Pareto efficiency, nondictatorship, and independence from irrelevant alternatives are assumed, as discussed above in the section on Arrow's theorem.

Illustrative Hypotheses

Hypotheses below are illustrative and not all authors associated with this theory would subscribe to all hypotheses listed.

Voting systems with three or more alternatives may generate arbitrary or suboptimal outcomes, and therefore lower voter satisfaction.
Borda count voting will be associated with higher voter satisfaction than simple majority voting on multiple alternatives.

Frequently Asked Questions

Aren't there other voting systems besides simple majority vote? What is Borda count voting?

Bibliography

Arrow, Kenneth (1951). Social choice and individual values. NY: John Wiley. Second edition, 1963. This is the seminal work in the field.
Arrow, Kenneth J. and H. Reynaud (1986). Social choice and multicriterion decision-making. Cambridge, MA: MIT Press.
Johnson, Paul E. (1998). Social choice theory and research. Quantitative Applications in the Social Sciences Series, No. 123. Thousand Oaks, CA: Sage Publications.
Kelly, J. S. (1978). Arrow impossibility theorems. NY: Academic Press.
Riker, William H. (1982). Liberalism against populism: A confrontation between the theory of democracy and the theory of social choice. San Francisco, CA: W. H. Freeman.