Vote For and Against
The Vote For and Against or VFA is a simple election method in which the voter votes for one candidate and also against one candidate.
If a ranked ballot were being used, this would be equivalent to the voter having to submit a complete strict ranking of the candidates. The voter would be counted as voting for the first preference and against the last preference.
Just as in First-Preference Plurality, the candidate with the most for votes is elected. The only exception is that if one candidate receives more than half of the against votes, this candidate is not eligible to be elected.
If we suppose ranked ballots are being used, then in the general case VFA satisfies the Monotonicity criterion, and what could be called the Majority Last Preference criterion (i.e., the last preference of a majority cannot win). It fails the Condorcet criterion, the Majority criterion for solid coalitions, Clone Independence, and the Participation criterion.
In the three-candidate case, VFA does satisfy the Majority criterion for solid coalitions, and in one respect also Clone Independence: When the winner in a two-candidate race is cloned, this cannot cause the loser of the two-candidate race to be elected.
Although VFA performs rather poorly with respect to criteria, it is as easy to count as First-Preference Plurality or Approval voting, neither of which satisfies the Majority Last Preference criterion or (even in the three-candidate case) the Majority criterion for solid coalitions.
Variant satisfying Participation
VFA can be made into a version of Descending Solid Coalitions, and so can be made to satisfy the Participation criterion, by making a small rule change: Instead of only disqualifying a candidate receiving more than half of the against votes, disqualify every candidate who receives more against votes than for votes.
When this rule is used, the addition of a ballot can only disqualify the candidate against whom that ballot voted; it cannot cause any candidate to no longer be disqualified except for the candidate voted for; and similarly that's the only candidate whose for vote tally can be increased.
In the original version, adding a ballot could cause a candidate to no longer be disqualified, by altering how many votes constitutes "more than half." This rule has the advantage that if a candidate is disqualified, there is no way to alter the votes that didn't vote against this candidate in such a way that this candidate wouldn't have been disqualified.
The variant rule doesn't share this property, and so voters would have incentive to vote for the strongest candidate they prefer, in attempt to keep him from being disqualified.
Also, under the variant rule it is possible to be elected with extremely few for votes. It's possible that the two strongest candidates could both be disqualified, giving the election to the candidate with the third-most for votes, no matter how few votes that is.
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
- Memphis on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
- Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
- Knoxville on Wikipedia, with 17% of the voters
- Chattanooga on Wikipedia, with 15% of the voters
The preferences of the voters would be divided like this:
| 42% of voters
(close to Memphis)
| 26% of voters
(close to Nashville)
| 15% of voters
(close to Chattanooga)
| 17% of voters|
(close to Knoxville)
In this scenario, Memphis receives the most for votes. However, Memphis also receives more than half of the against votes, so that Memphis can't win. The winner is instead the option with the next-greatest for votes, Nashville.
Nashville also happens to be the Condorcet winner, but VFA doesn't reliably elect these candidates. Nashville only wins over Knoxville and Chattanooga due to possessing more for votes than either of the others.