Condorcet Winner — Definition, Formula & Examples

A Condorcet winner is the candidate who would win a one-on-one majority vote against every other candidate. If you ran every possible two-candidate race, this candidate would win all of them.

Let $C$ be a finite set of candidates, and let each voter hold a strict ranking of $C$ . For $x, y \in C$ , let $N(x,y)$ be the number of voters who rank $x$ above $y$ . A candidate $x$ is the Condorcet winner if $N(x,y) > N(y,x)$ for every $y \in C$ with $y \neq x$ . A profile has at most one Condorcet winner, and it may have none.

Key Formula

x \text{ is the Condorcet winner} \iff N(x,y) > N(y,x) \quad \text{for all } y \neq x

Where:

$x$ = The candidate being tested
$y$ = Any other candidate
$N(x,y)$ = Number of voters who rank x above y

Worked Example

Problem: 100 voters rank three candidates. 40 voters rank A > B > C, 35 voters rank B > C > A, and 25 voters rank C > B > A. Is there a Condorcet winner?

Step 1: Compare B with A. The 35 voters in the second group and the 25 voters in the third group rank B above A. Only the first group of 40 ranks A above B.

N(B,A) = 35 + 25 = 60 > 40 = N(A,B)

Step 2: Compare B with C. The first two groups rank B above C. Only the third group of 25 ranks C above B.

N(B,C) = 40 + 35 = 75 > 25 = N(C,B)

Step 3: B beats both rivals, so B is the Condorcet winner. For contrast, compare A with C. A also loses to C, so the plurality leader A loses every head-to-head race.

N(C,A) = 35 + 25 = 60 > 40 = N(A,C)

Answer: B is the Condorcet winner. B beats A 60 to 40 and beats C 75 to 25. A has the most first-place votes (40), yet A loses to both B and C.

Why It Matters

Social choice courses use the Condorcet criterion to judge voting rules. A rule satisfies the criterion if it always elects the Condorcet winner whenever one exists. Plurality, the Borda count, and instant-runoff voting can all fail this test, while methods such as Copeland and Schulze are built to pass it.

Common Mistakes

Mistake: Assuming every election has a Condorcet winner.

Correction: Majority preferences can form a cycle. Suppose three voters rank A > B > C, B > C > A, and C > A > B. Then A beats B, B beats C, and C beats A, each by 2 to 1. No candidate beats everyone. This is the Condorcet paradox, so check every pairwise race before you name a winner.

Related Terms

Borda Count — Ranked voting rule that can miss the Condorcet winner
Game Theory — Broader field that includes social choice and voting
Graph Theory — Pairwise results form a directed majority graph
Optimization — Choosing a best outcome from competing preferences
Fair Game — Another formal notion of fairness in games