Condorcet Winner (COMSOC)

Definition

Condorcet Winner ( COMSOC)

A Condorcet winner is an alternative that defeats every other alternative by strict majority under the Condorcet method. For a profile of complete, strict rankings over , let be its pairwise support function. Alternative is a Condorcet winner exactly when

A tie against even one opponent is insufficient. A Condorcet winner need not exist, but is unique whenever it exists: two alternatives cannot each be preferred to the other by more than half the voters.

Majority Graph

In the majority graph , let and let be the number of outgoing arcs from . Then

There must be an outgoing arc to every other alternative, because each arc records a strict head-to-head victory. Having no incoming arcs alone is insufficient: a tied contest produces no arc, but prevents from defeating every opponent.

Dodgson Score

An alternative is already a Condorcet winner exactly when no adjacent swaps are needed to make it one. Thus its Dodgson score satisfies

Young Score

Under the minimum-deletions convention, an alternative is already a Condorcet winner exactly when no voters need to be removed. Thus its Young score satisfies

Under the alternative maximum-retained convention, the corresponding score is , not zero: all voters can be retained.