Economics & Market Microstructure
Borda Count vs. Condorcet Winner Formula: Ranked Ballot Points & Pairwise Majority
Evaluate ranked-choice voter profiles by computing Jean-Charles de Borda’s positional score alongside Marquis de Condorcet’s pairwise head-to-head majority test—exposing when plurality, Borda, and Condorcet winners diverge!
Why a candidate who beats EVERY rival head-to-head (Condorcet Winner) can lose a Borda or plurality vote!
In 18th-century French social-choice theory, Jean-Charles de Borda (1770) proposed awarding m - 1, m - 2, ..., 0 positional points across m ranked options (2, 1, 0 points for 1st, 2nd, 3rd among 3 candidates), while the Marquis de Condorcet (1785) argued that an option defeating EVERY rival by > 50% in one-on-one pairwise comparisons is the true majority Condorcet Winner!
Remarkably, on a 100-voter ranked profile where 51 voters rank A > B > C, 2 voters rank B > A > C, and 47 voters rank B > C > A: Candidate A wins 51% outright (beating B 51-to-49 and C 53-to-47, making A the undisputed Condorcet Winner!), YET Candidate B scores 2*(2+47) + 1*(51) = 149 Borda points versus Candidate A’s 2*(51) + 1*(2) = 104 Borda points! Computing both side by side in your governance or product-prioritization sheet reveals consensus compromises versus strict majority winners!
| Candidate Across 100 Ranked Ballots (51: A>B>C, 2: B>A>C, 47: B>C>A) | Borda Score (1st=2, 2nd=1, 3rd=0) | Pairwise Head-to-Head Record |
|---|---|---|
| Candidate A | 2*51 + 1*2 + 0*47 = 104 pts | 2-0 (Beats B 51–49 & C 53–47 -> Condorcet Winner!) |
| Candidate B | 2*(2+47) + 1*51 = 149 pts (Borda Winner!) | 1-1 (Loses to A 49–51, Beats C 100–0) |
| Candidate C | 1*47 = 47 pts | 0-2 (Condorcet Loser) |
Candidate B Borda Count Score (2 pts per 1st-Place Rank + 1 pt per 2nd-Place Rank)
Weights voter bloc counts B2:B7 across the 6 strict rankings of {A, B, C} by Candidate B’s positional points (1, 2, 2, 1, 0, 0).
=IF(SUM(B2:B7) <= 0, "No Ballots Cast", B2 * 1 + B3 * 0 + B4 * 2 + B5 * 2 + B6 * 0 + B7 * 1)Condorcet Winner Identifier Across All 6 Voter-Bloc Counts (A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, C>B>A)
Tests pairwise head-to-head majorities (A vs B, A vs C, B vs C) to identify the unique Condorcet Winner—or detect a Condorcet Paradox cycle!
=IF(SUM(B2:B7) <= 0, "No Ballots Cast", IF(AND(B2 + B3 + B6 > (SUM(B2:B7) / 2), B2 + B3 + B4 > (SUM(B2:B7) / 2)), "Candidate A (Condorcet Winner)", IF(AND(B4 + B5 + B7 > (SUM(B2:B7) / 2), B2 + B4 + B5 > (SUM(B2:B7) / 2)), "Candidate B (Condorcet Winner)", IF(AND(B5 + B6 + B7 > (SUM(B2:B7) / 2), B3 + B6 + B7 > (SUM(B2:B7) / 2)), "Candidate C (Condorcet Winner)", "No Condorcet Winner (Tie or Majority Cycle!)"))))Try the example
Use the original values or change them to test your understanding.