Borda Count - Positional scoring rule for rank aggregation

Overview

B_i ∈ [0, m−1]. Higher B_i means the alternative is ranked near the top by more criteria/voters. In a symmetric example all alternatives tie; real problems break ties. Weighted Borda allows differential criterion importance.

Output

rank position, higher is better

Data

Crisp, complete rank

Size

2+ alternatives, 3-12 criteria works best

Used for

Rank aggregation, social choice, preference fusion

How it works

1. 1

   Collect K expert rankings R_k over m alternatives.

   $$\mathcal{R}=\{R_1,R_2,\dots ,R_K\},R_k:\{A_1,\dots ,A_m\}\to \{1,\dots ,m\}$$

   de Borda 1781, Sec.1

2. 2

   Borda count B_i = Σ (m − R_k(A_i)).

   $$B_i=\sum _{k=1}^K(m-R_k(A_i))$$

   de Borda 1781, Sec.2

3. 3

   Descending ranking by B_i.

   $$\text{rank}(A_i)\propto -B_i$$

   de Borda 1781

Fits when / Look elsewhere when

Assumptions to verify

• Input is a rank matrix (1=best, m=worst per voter)
• Each voter ranks all alternatives

Edge cases and pitfalls

• Borda is susceptible to the independence-of-irrelevant-alternatives (IIA) violation - adding a new alternative can change the ranking of existing ones.

• Input must be ordinal ranks (integers 1..m), NOT raw criterion values - apply a separate ranking step first.

How to cite