Borda Rule (COMSOC)

Definition

Borda Rule ( COMSOC)

Borda rule is a voting rule that awards $m-j$ points to an alternative ranked in position $j$, where $m$ is the number of alternatives alternatives and position $1$ is best. For a profile $R=({\succ }_1,\dots ,{\succ }_n)$ of complete, strict rankings over $C$, let $j_i(x)$ be the position of $x$ on voter $i$‘s ballot. Then

$$\begin{array}{rl}{\mathrm{Borda}}_R(x)& :=\sum _{i=1}^n\underset{\text{points from voter }i}{\underbrace{(m-j_i(x))}},\\ {\lambda }_{\mathrm{Borda}}(R)& :=\arg \underset{x\in C}{\max }{\mathrm{Borda}}_R(x).\end{array}$$

The alternatives with the highest total Borda score are the Borda winners. Tied leaders are all co-winners unless a tie-breaking rule selects one. The rule counts points, not just first-place votes.

Intuition

An alternative in position $j$ has exactly $m-j$ alternatives below it. Borda gives it one point for each of them. Thus a ballot $A\succ B\succ C\succ D$ gives $A$ three points, $B$ two, $C$ one and $D$ none.

Unlike plurality, lower positions contribute too. Every ballot distributes the same total:

$$\sum _{j=1}^m(m-j)=0+1+\cdots +(m-1)=\frac{m(m-1)}{2}.$$

Moving an alternative up one position adds one point to its score and takes one point from the alternative it passes. The rankings specify order, not how strongly a voter prefers one alternative to another; the equal spacing of points is the rule’s choice.

Winner Determination

Read each of the $n$ ballots once. At position $j$, add $m-j$ to the listed alternative’s score. After $nm$ score updates, scan the $m$ totals for the maximum: $O(nm+m)$ time and $O(m)$ score storage under unit-cost arithmetic. There are no elimination rounds or transfers.

Issues

Condorcet-inconsistency

In the 100-voter example below, $C$ is the Condorcet winner:

$$\begin{array}{rl}{N}_R(C,A)& =18+28+24=70,\\ N_R(C,B)& =28+24=52,\\ N_R(C,D)& =30+28=58.\end{array}$$

All three counts exceed $n/2=100/2=50$.

Nevertheless, Borda selects $B$ with 194 points rather than $C$ with 180. A total score can favour an alternative that loses a head-to-head contest: 52 voters prefer $C$ to $B$.

Indeed, Borda adds support across opponents:

$$\begin{array}{rl}{\mathrm{Borda}}_R(B)& =N_R(B,A)+N_R(B,C)+N_R(B,D)=70+48+76=194,\\ {\mathrm{Borda}}_R(C)& =N_R(C,A)+N_R(C,B)+N_R(C,D)=70+52+58=180.\end{array}$$

$B$‘s larger majority over $D$ outweighs its loss against $C$ in the total. Winning every pairwise contest is therefore not the same as maximising the sum, and Borda is not Condorcet consistent.

Example

The same 100-voter profile

Each row groups voters with the same complete ranking. The copied diagram shows the plurality scores for comparison; below it, we count Borda points instead.

With $m=4$, each ballot awards $(3,2,1,0)$ points from first to last place. Multiply each position’s points by its group’s size:

$$\begin{array}{rl}{\mathrm{Borda}}_R(A)& =30\cdot 3+18\cdot 0+28\cdot 1+24\cdot 0=118,\\ {\mathrm{Borda}}_R(B)& =30\cdot 2+18\cdot 3+28\cdot 2+24\cdot 1=194,\\ {\mathrm{Borda}}_R(C)& =30\cdot 1+18\cdot 1+28\cdot 3+24\cdot 2=180,\\ {\mathrm{Borda}}_R(D)& =30\cdot 0+18\cdot 2+28\cdot 0+24\cdot 3=108.\end{array}$$

Hence ${\lambda }_{\mathrm{Borda}}(R)=\{B\}$. Although only 18 voters rank $B$ first, another $30+28=58$ rank it second. These lower positions let it beat plurality winner $A$.

Check: $118+194+180+108=600=100\cdot (3+2+1+0)$ points.