# Single Contest

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Single Contest is a method devised by Kevin Venzke which, according to his simulations, has very little incentive for insincere ranking and very good sincere Condorcet efficiency. It uses a rank ballot with explicitly-placed approval cutoff, expecting voters to employ approval strategy.

## Definition

1. Voters fill out rank ballots, with an explicitly-placed approval cutoff. Equal ranking and truncation are allowed.
2. If there is a strict majority favorite (not counting equal-top rankings to the majority) then this candidate wins.
3. Otherwise, elect the winner of the pairwise contest between the pair of candidates who minimize the number of voters that approved neither one. For example, if no one approved A or B, there are no votes for the AB pair. If everyone voted for at least one of A and B, all votes are for the AB pair.
4. If pairs are tied on this measure, break the tie for the pair whose winner has the most voters on his side in that contest (i.e. winning votes). Secondarily, break ties in favor of the pair whose winner has the most approval. (This order might get reversed in the future, since WV is based on rankings and the idea is to minimize their importance.)

The method essentially uses the approval scores to guess at the most important two-way contest. It is an approval election for a pair, where you vote for a pair by approving either candidate in that pair. Then the rankings are used to resolve this contest. Two-candidate races don't have strategy incentives.

There is potentially strategy with the majority favorite rule, as creating a majority favorite will end the method prematurely and possibly in a way that a voter prefers.

Manipulating which contest will be selected as the "single contest" is not straightforward. The only way to vote against a contest (because one foresees that it won't resolve favorably) is to approve neither candidate involved. Similarly, there is no way to prop up a contest (foreseeing that it will be easily won by one's preferred candidate) when one was already going to approve one of the candidates.

Single Contest satisfies Condorcet Loser, but not Condorcet:

```40 A>B | C  (i.e. only C is disapproved)
25 B | A>C
35 C | A>B
100
```

The Condorcet winner is A. The BC pair is selected as the most important contest, because all voters distinguish between B and C using approval. Then B is elected.

The method is not clone-independent due to the (admittedly inelegant) majority favorite rule. But the approval and final contest components don't introduce additional clone issues. It's not monotone, because raising the winner from disapproved to approved on some ballots which already counted to the selected pair (that the winner was part of) could add votes for a different pair where the original winner is pairwise beaten.

## Example

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

The voters need to decide where to place their approval cutoffs. They might look at the polling information and decide that the winner will probably be either Memphis or Nashville if the votes are strategically reasonable. So, suppose that the Memphis voters approve only Memphis, and the other voters approve everyone as good as or better than Nashville.

No candidate has a majority of first preferences, so the "votes" for each pair are counted:

```100 Memphis & Nashville
74  Memphis & Chattanooga
74  Memphis & Knoxville
58  Nashville & Chattanooga
58  Nashville & Knoxville
32  Chattanooga & Knoxville
```

Memphis vs. Nashville is selected as the single contest. According to the rankings, Nashville beats Memphis 58-42, and so Nashville wins.