# Quick Runoff

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Quick Runoff or QR or Instant Chain Runoff is a method devised by Kevin Venzke which satisfies Later-no-harm but, unlike IRV, can sometimes elect the candidate with the fewest first preferences.

The method is so named because in the three-candidate case it is not as "instant" as IRV, as instead of a single top-two runoff, there is first a simulated runoff between the top two candidates, and then potentially the bottom two candidates.

The method may be badly named because the winner is actually found without doing any vote transfers. The results of the "runoffs" are inferred from the pairwise contests.

## Definition

1. The voter submits a ranked ballot. Equal-ranking is not allowed; truncation is.
2. Elect the candidate with the most first preferences who does not have a majority-strength pairwise loss (i.e. >50% of all ballots) to the very next candidate in descending order of first-preferences.
3. If every candidate but the last has such a pairwise loss, then the last candidate is elected.

QR satisfies Later-no-harm because if candidate X is elected, the addition of a lower preference can only cause X to lose if it creates a majority defeat of X by the candidate that follows X in descending first preference order. This cannot be done by a voter who prefers X to the new preference.

QR fails monotonicity (Mono-raise and Mono-add-top at least) and is not clone-independent. It is quite possible for a candidate to wish he had a different position in the first-preference ordering, even if it means receiving fewer votes.

QR also doesn't satisfy Later-no-help. In particular, in a three candidate scenario where A has the most first preferences and C the fewest, C may have incentive to insincerely create a majority win for B over A, if the C voters believe C has the second-preferences of the A voters. So, while the A voters technically have both Later-no-harm and Later-no-help guarantees with respect to A, there may be an incentive to withhold their second preference if they feel other factions are relying on those second preferences in order to try to steal the race from A.

Note that the supporters of the top two candidates (by first preferences) have both Later-no-harm and Later-no-help assurances. Labeling the candidates A, B, and C, the A voters' lower preference is only counted once A is known to have lost. The B voters' lower preference is not regarded at all, because candidates are only compared to their "adjacent" candidates in first-preference order, and A vs. C (the race that B voters can influence) are not adjacent.

## 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 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)
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 first preference ordering is Memphis, Nashville, Knoxville, Chattanooga. Since all the voters listed all of their preferences, all of the pairwise losses have majority strength.

Memphis has a majority loss to Nashville, so Memphis will not win. Nashville does not have a majority loss to Knoxville. The method thus ends here, with the election of Nashville.