Condorcet-consistent Voting Rule

Definition

Condorcet-consistent Voting Rule ( COMSOC)

A voting rule is Condorcet consistent if it always selects the Condorcet winner whenever one exists. For a rule $\lambda$ returning a non-empty winner set and any profile $R$,

$$c\text{ is the Condorcet winner of }R⟹\lambda (R)=\{c\}.$$

The rule must select $c$ alone, not merely include it among tied winners. For a single-winner rule $f$, the requirement is $f(R)=c$. When no Condorcet winner exists, this condition imposes no restriction on the outcome.

Intuition

The rule need not compute pairwise contests internally. Its output must nevertheless agree with them whenever one alternative defeats every opponent. Agreement on one profile is not enough: consistency requires agreement on every profile with a Condorcet winner. One counterexample disproves it.

Ranking Rules

For a rule returning a collective ranking, the corresponding requirement places the Condorcet winner first in every returned ranking. In particular, it is first in every Kemeny consensus when it exists.