Borda Winner (COMSOC)

Definition

Borda Winner ( COMSOC)

A Borda winner is an alternative whose Borda score is maximal in a profile $R$ over $C$:

$$x\in {\lambda }_{\mathrm{Borda}}(R)⟺\forall y\in C:{\mathrm{Borda}}_R(x)\ge {\mathrm{Borda}}_R(y).$$

Borda count returns all such alternatives. If several share the maximum, they are co-winners; $x$ is the unique winner precisely when its score is strictly greater than every other alternative’s score.

Winner Determination

Compute all scores and check whether the candidate’s score equals the maximum. For $n$ explicitly listed ballots over $m$ alternatives, this takes $O(nm+m)$ time under unit-cost arithmetic and is therefore polynomial. See the counting procedure.

Example

Winner from the course profile

The 100-voter profile gives

$$\underset{B}{\underbrace{194}}>\underset{C}{\underbrace{180}}>\underset{A}{\underbrace{118}}>\underset{D}{\underbrace{108}}.$$

Thus ${\lambda }_{\mathrm{Borda}}(R)=\{B\}$: $B$ is the unique Borda winner, even though $A$ has more first-place votes.