Borda Score (COMSOC)

Definition

Borda Score ( COMSOC)

The Borda score of an alternative is its total points under Borda count. For a profile of complete, strict rankings over alternatives,

where is ‘s position on ballot , with position best. Each ballot gives points to its first choice and to its last. Alternatives with the largest total are Borda winners.

Intuition

Start with one ballot. There are four alternatives. For , the alternatives below it are and : two alternatives, hence two points. In general, position leaves alternatives below it.

The points belong to the positions: first gets , second , third , last . Now suppose 30 voters submit this same ranking. Each gives two points, so their contribution to is .

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

Read the contributions beneath the boxed squares and add them:

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 and consider the profile from the lecture notes:

voters	ranking	points to	points to	points to	points to
total

Hence

is the largest score, so is the Borda winner.