Condorcet's Theory of Voting

TLDR

Condorcet's criterion posits that an alternative defeating all others by majority is socially optimal, and he argued that the majority winner is statistically most likely the best choice when voters may err. This procedure, known as Kemeny's rule, is the unique social welfare function satisfying a variant of independence of irrelevant alternatives and other standard properties. Although Condorcet's rule is statistically correct for ranking alternatives, it is not always the sharpest estimate, with Bordas rule sometimes providing a sharper estimate.

Abstract

Condcrcet's criterion states that an alternative that defeats every other by a simple majority is the socially optimal choice. Condorcet argued that if the object of voting is to determine the “best” decision for society but voters sometimes make mistakes in their judgments, then the majority alternative (if it exists) is statistically most likely to be the best choice. Strictly speaking, this claim is not true; in some situations Bordas rule gives a sharper estimate of the best alternative. Nevertheless, Condorcet did propose a novel and statistically correct rule for finding the most likely ranking of the alternatives. This procedure, which is sometimes known as “Kemeny's rule,” is the unique social welfare function that satisfies a variant of independence of irrelevant alternatives together with several other standard properties.