# Plurality criterion

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

#### Statement of Criterion

If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference, then B must not be elected.

#### Complying Methods

First-Preference Plurality, Approval voting, IRV, and many Condorcet methods (using winning votes as defeat strength) satisfy the Plurality criterion. Condorcet methods using margins as the measure of defeat strength fail it, as does Raynaud (using either winning votes or margins as the measure of defeat strength), and also Minmax(pairwise opposition).

#### Commentary

When the Plurality criterion requires that B not be elected, it means that even if all the voters who gave B some ranking were to elevate him to the top position, he would still not be the First Preference Plurality winner.

It also means that A has a stronger pairwise victory over B than B has even a path of victories to any other candidate.

It is conceivable that if B were elected, voters might not consider this a legitimate result.

One connection the Plurality criterion has to most voting methods is that it implies that when all voters bullet vote (in Score voting, also max-scoring the bullet-voted candidate), the candidate bullet voted by the most voters (i.e. the FPTP winner) will win. Most voting methods do this. An extension of this is to check whether, for a given voting method, when all voters vote some candidates 1st and all other candidates last (and Min-max vote in Score), then the candidate marked 1st on the most ballots (the Approval voting winner) wins. This isn't passed by all voting methods; for example, IRV with its most common equal-ranking implementation, fractional equal-ranking, doesn't necessarily elect such a candidate.

## Plurality criterion for rated ballots

Example where Score voting fails if the definition of the criterion is extended to scored ballots:

3 A:1

1 C:5 B:4

1 D:5 B:4

Scores are A 3, C 5, B 8, D 5, making B the winner. Yet when looking at the rankings:

3 A

1 C>B

1 D>B

B is preferred on 2 ballots, while A is preferred 1st on 3 ballots. However, Score voting passes a related criterion: "If the number of ballots giving A maximal support is greater than the number of ballots on which another candidate B is given any support, then B must not be elected." This is because candidate B can't get more points than A, since even if all voters who score B give B the maximum score, candidate A will have more ballots giving them the maximum score than B, and thus more points.