Definition
Condorcet Rule ( COMSOC)
The Condorcet rule compares every pair of alternatives by majority vote and selects an alternative that defeats every opponent. For a profile
of
complete, strict rankings over , let be its pairwise support function. Then In each contest, every voter supports whichever of the two alternatives they rank higher. A strict majority wins the contest; a tie is not a victory. The selected alternative is the Condorcet winner.
Unlike plurality or Borda count, this procedure can return no alternative:
is either a singleton or empty. A voting rule that must always return a winner needs an additional convention for the empty case.
Intuition
For one contest, keep only the two alternatives being compared on each ballot. Their relative order stays unchanged. Unlike plurality, a voter need not rank an alternative first overall to support it in this contest.
The course profile has
Thus
Condorcet Consistency
A Condorcet consistent voting rule selects the Condorcet winner whenever one exists. This requirement alone does not specify what to return when none exists; the Condorcet principle is therefore not a complete voting rule on all profiles.
Issues
Non-existence
Pairwise majorities can cycle:
Definition
Link to originalCondorcet Paradox
In COMSOC, the Condorcet paradox is the fact that individually consistent preference rankings can induce a cyclic majority relation over alternatives.
Let
be a preference profile, and define the majority relation by where
is the pairwise support function and is the number of voters. A Condorcet paradox occurs when contains a cycle, for example No alternative on the cycle is a Condorcet winner, since each loses to another member. If the cycle includes every alternative, no Condorcet winner exists; an alternative outside the cycle may still defeat all others.
Majority Graph
Each vertex represents an alternative. An arrow points from the head-to-head winner to the loser:
The label gives winner’s votes : loser’s votes. Here 52 voters prefer
Now focus on
A tied pair has no arrow in either direction. Thus having no incoming arrows is not enough to be a Condorcet winner: an alternative must actually defeat every opponent.
In the course profile,
Do not infer another victory by following a path. In this separate profile, each ranking occurs once, and the three majorities form a cycle. Every alternative has an incoming arrow, so none is a Condorcet winner.
Example
The same 100-voter profile
Each row groups voters with the same complete ranking. The copied diagram shows plurality scores for comparison; Condorcet contests use the relative order of each pair instead.
Count the voters preferring each alternative to its opponent:
contest majority winner votes for winner–loser vs. — vs. — vs. — vs. — vs. — vs. — Since
defeats , , and , the alternative is the Condorcet winner. In particular,
This differs from the same profile’s plurality winner
and Borda winner .