Condorcet Consistent Voting Rule

comsoc

Definition

Condorcet Consistent Voting Rule

In COMSOC, a Condorcet consistent voting rule is a voting rule that always selects the Condorcet winner whenever one exists.

Formally, let

$$\lambda :\mathcal{L}(C{)}^n\to 2^C\setminus \{\varnothing \}$$

be a voting rule on a set of alternatives $C$, where $\mathcal{L}(C)$ is the set of preference rankings over $C$. The rule $\lambda$ is Condorcet consistent if, for every preference profile $R$, whenever an alternative $c\in C$ satisfies

$$\forall y\in C\setminus \{c\}:N_R(c,y)>\frac{n}{2},$$

then

$$\lambda (R)=\{c\}.$$

Here $N_R(c,y)$ is the number of voters who rank $c$ above $y$. Thus the rule must agree with pairwise majority whenever pairwise majority identifies a unique alternative that defeats every other alternative.

Intuition

Majority dominance fixes the answer

Condorcet consistency says that a rule may use its own mechanism only when pairwise majority does not already force an answer. If one alternative beats every other alternative in head-to-head contests, the rule must return that alternative.

Properties

Conditional requirement

Condorcet consistency is only a requirement on profiles that have a Condorcet winner. If the majority relation is cyclic, as in the Condorcet paradox, the condition imposes no winner.

Ranking rules

For voting rules that return a collective ranking rather than only a winning set, Condorcet consistency means that the Condorcet winner, if it exists, must be ranked first. The lecture notes state this form for Kemeny’s rule: the Condorcet winner is first-ranked in every Kemeny consensus.

Examples

Positive examples from the lecture notes

The lecture notes state that Lull’s Rule, Dodgson’s Rule, Young’s Rule, and Kemeny’s Rule are Condorcet consistent.