Borda Count — Definition, Formula & Examples

Borda count is a voting method in which each voter ranks all the candidates, each rank position earns a fixed number of points, and the candidate with the most total points wins.

Given $n$ candidates and a profile of complete strict rankings, the Borda count assigns $n-k+1$ points to a candidate for each ballot that places it in position $k$ , where $k = 1, \dots, n$ . The Borda winner is any candidate whose total score is maximal. Many sources use $n-k$ points instead, so last place scores 0. Both conventions produce the same ranking of candidates. Borda count belongs to the family of positional scoring rules in social choice theory.

Key Formula

B(X) = \sum_{k=1}^{n} (n - k + 1)\, v_k(X), \qquad \sum_{\text{all } X} B(X) = V \cdot \frac{n(n+1)}{2}

Where:

$B(X)$ = Borda score of candidate X
$n$ = number of candidates
$k$ = rank position on a ballot (1 = first choice)
$v_k(X)$ = number of voters who rank X in position k
$V$ = total number of voters

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. Use the Borda count with 3 points for 1st, 2 for 2nd, and 1 for 3rd to find the winner.

Step 1: Score A. A is 1st on 40 ballots and 3rd on the other 60.

B(A) = 3(40) + 1(35 + 25) = 120 + 60 = 180

Step 2: Score B and C the same way, using each group's ranking.

B(B) = 2(40) + 3(35) + 2(25) = 235, \qquad B(C) = 1(40) + 2(35) + 3(25) = 185

Step 3: Check the arithmetic. The scores must add up to

V \cdot \frac{n(n+1)}{2}

180 + 235 + 185 = 600 = 100 \cdot \frac{3 \cdot 4}{2}

Answer: B wins with 235 points. C has 185 points and A has 180. A has the most first-place votes (40), but A finishes last under the Borda count.

Why It Matters

Borda count is a standard case study in college social choice and liberal-arts math courses. It is used to show how different voting methods can pick different winners from the same ballots. It can violate the majority criterion and independence of irrelevant alternatives, two properties central to Arrow's impossibility theorem. Variants of positional scoring are used in real awards, such as the Heisman Trophy's 3-2-1 point system.

Common Mistakes

Mistake: Assuming the candidate with the most first-place votes must win the Borda count.

Correction: Borda count rewards broad support across all rank positions. In the example, A has the most first-place votes but finishes last. A candidate can even hold a majority of first-place votes and still lose under Borda.

Related Terms

Game Theory — Studies strategic voting and collective decision rules
Fair Game — Another formal notion of fairness in outcomes
Optimization — Borda winner maximizes a total score function
Graph Theory — Pairwise vote comparisons form tournament graphs
Number Theory — Points per ballot form a triangular number