Vote For and Against

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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.


Tennessee's four cities are spread throughout the state

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, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville, with 26% of the voters, near the center of Tennessee
  • Knoxville, with 17% of the voters
  • Chattanooga, 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)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

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.