Solution
The Plurality Winner Loses the Count
Show the problem again
Nineteen voters rank four candidates (Ash, Birch, Cedar, Dogwood) from most to least preferred. The ballots fall into four blocs: - 7 voters: Ash > Birch > Cedar > Dogwood - 5 voters: Birch > Cedar > Dogwood > Ash - 4 voters: Cedar > Birch > Dogwood > Ash - 3 voters: Dogwood > Birch > Cedar > Ash Plurality would elect the candidate with the most first-place votes. Instead the election uses the Borda count: each ballot gives 3 points to its top choice, 2 to its second, 1 to its third and 0 to its last. Which candidate wins the Borda count? Answer with the candidate's name.
Worked solution
Compute each candidate's total. Ash: 7×3 + 0 + 0 + 0 = 21. Birch: 7×2 + 5×3 + 4×2 + 3×2 = 14 + 15 + 8 + 6 = 43. Cedar: 7×1 + 5×2 + 4×3 + 3×1 = 7 + 10 + 12 + 3 = 32. Dogwood: 0 + 5×1 + 4×1 + 3×3 = 18. The total is 114 = 19×6, as it must be. Birch wins with 43 points, although Ash wins under plurality (7 first places against Birch's 5). Birch is also the Condorcet winner here: it beats Ash 12 to 7 and beats Cedar and Dogwood by clear majorities. The answer is **Birch**.
Source: Classic illustration of plurality versus Borda count, a theme going back to Borda (1781) and Condorcet; numbers chosen for AxiomIQ. Statement written for AxiomIQ.