Borda Score (COMSOC)

comsoc

Definition

Borda Score

In COMSOC, the Borda score of an alternative is the total number of Borda points it receives under Borda Count in a preference profile.

Let $C$ be a set of $m$ alternatives and let

$$R:=({\succ }_1,\dots ,{\succ }_n)$$

be a preference profile. If ${\mathrm{pos}}_i(x)$ is the position of alternative $x$ in voter $v_i$‘s ranking, then

$${\mathrm{Borda}}_R(x):=\sum _{i=1}^n(m-{\mathrm{pos}}_i(x)).$$

The summand $m-{\mathrm{pos}}_i(x)$ is the number of points that voter $v_i$ gives to $x$. Higher rankings contribute more points.

Derivation

Borda Count uses the positional score vector

$$(m-1,m-2,\dots ,1,0).$$

If $x$ is ranked in position $j$, then there are $m-j$ alternatives below it. Borda Count gives one point for each alternative ranked below $x$, so the contribution from that voter is $m-j$.

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.