Borda Score (COMSOC)

Definition

Borda Score ( COMSOC)

The Borda score of an alternative $x$ is its total points under Borda count. For a profile $R=({\succ }_1,\dots ,{\succ }_n)$ of $n$ complete, strict rankings over $m$ alternatives,

$${\mathrm{Borda}}_R(x):=\sum _{i=1}^n\underset{\text{points from ballot }i}{\underbrace{(m-j_i(x))}},$$

where $j_i(x)\in \{1,\dots ,m\}$ is $x$‘s position on ballot $i$, with position $1$ best. Each ballot gives $m-1$ points to its first choice and $0$ to its last. Alternatives with the largest total are Borda winners.

Intuition

Start with one ballot. There are four alternatives. For $B$, the alternatives below it are $C$ and $D$: two alternatives, hence two points. In general, position $j$ leaves $m-j$ alternatives below it.

The points belong to the positions: first gets $3$, second $2$, third $1$, last $0$. Now suppose 30 voters submit this same ranking. Each gives $B$ two points, so their contribution to $B$ is $30\cdot 2=60$.

Repeat for the other voter groups. Their rankings differ, so $B$ occupies different positions. The boxes below follow the same candidate $B$, not the same column.

Read the contributions beneath the boxed $B$ squares and add them:

$${\mathrm{Borda}}_R(B)=\underset{60}{\underbrace{30\cdot 2}}+\underset{54}{\underbrace{18\cdot 3}}+\underset{56}{\underbrace{28\cdot 2}}+\underset{24}{\underbrace{24\cdot 1}}=194.$$

Each row multiplies points per ballot by the number of voters; the final sum collects all points awarded to the same alternative.

Example

Scores for the course profile

Let $C:=\{A,B,C,D\}$ and consider the profile from the lecture notes:

voters	ranking	points to $A$	points to $B$	points to $C$	points to $D$
$30$	$A\succ B\succ C\succ D$	$90$	$60$	$30$	$0$
$18$	$B\succ D\succ C\succ A$	$0$	$54$	$18$	$36$
$28$	$C\succ B\succ A\succ D$	$28$	$56$	$84$	$0$
$24$	$D\succ C\succ B\succ A$	$0$	$24$	$48$	$72$
total		$118$	$194$	$180$	$108$

Hence

$${\mathrm{Borda}}_R(B)=194$$

is the largest score, so $B$ is the Borda winner.