Bottom-Two-Runoff IRV

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Bottom-Two-Runoff IRV (BTR-IRV) is a voting system that selects a single winner using votes that express ranked preferences.

The process: take the two options with the fewest first preference votes. The pairwise loser out of those two options is eliminated, and the next preferences from those ballots are redistributed. This process repeats until there is only one option remaining, and that remaining option is the winner.

BTR-IRV was originally proposed by Rob LeGrand in 2006.[1][2] It was conceived as a modification to standard Instant-runoff voting (IRV) which ensures the runoff doesn't ever eliminate a Condorcet Winner (and in fact, never eliminates all candidates in the Smith set, since a Smith set member can never be eliminated in a runoff against a non-Smith set member). Thus, the method passes the Condorcet Criterion and the Smith criterion, ensuring it functions as a Condorcet method.

This system is a form of single transferable vote (STV), and may be referred to by the more general name BTR-STV, thought the multi-winner variant was not originally recommended by LeGrand.[1]

An example[edit | edit source]

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
City Round 1 Round 2 Round 3
Memphis 42 42 42
Nashville 26 26 26 58
Chattanooga 15 15 32 32 0
Knoxville 17 17 0 0

First elimination round

The two options with the fewest first preferences are Chattanooga (with the fewest - 15%) and Knoxville (with the second fewest - 17%). So Chattanooga and Knoxville are the options which have a possibility of being eliminated in the first round.

Chattanooga is preferred to Knoxville by Memphis voters (42%), Nashville voters (26%), and Chattanooga voters (15%). This means that Chattanooga is preferred to Knoxville by 83% of voters (43% + 26% + 15%). Knoxville is preferred to Chattanooga by Knoxville voters (17%), so 17% of voters prefer Knoxville to Chattanooga.

As there are more voters who prefer Chattanooga to Knoxville (83%) than there are voters who prefer Knoxville to Chattanooga (17%), Knoxville is the pairwise loser. That means that Knoxville is eliminated in the first round. All of the votes for Knoxville have Chattanooga as a second choice, so they are transferred to Chattanooga.

Second elimination round

Nashville now has the fewest first preferences (26%), with Chattanooga having the second fewest first preferences (32%). So Nashville and Chattanooga are the options which have a possibility of being eliminated in the second round.

Nashville is preferred to Chattanooga by Memphis voters (42%), and Nashville voters (26%). This means that Nashville is preferred to Chattanooga by 68% of voters (43% + 26%). Chattanooga is preferred to Nashville by Chattanooga voters (15%), and by Knoxville voters (17%). This means that Chattanooga is preferred to Nashville by 32% of voters (15% + 17%).

As there are more voters who prefer Nashville to Chattanooga (68%) than there are voters who prefer Chattanooga to Nashville (32%), Chattanooga is the pairwise loser. That means that Chattanooga is eliminated in the second round. All of the votes for Chattanooga and Knoxville have Nashville as their third choice, so they are transferred to Nashville.

Nashville now has a majority of the vote (58%: 26% + 32%), and is declared the winner.

In a real election, of course, voters would show greater variation in the rankings they cast, which could influence the result.

Notes[edit | edit source]

BTR-IRV only requires eliminations to be done until one candidate remains who pairwise beats all other uneliminated candidates, at which point that candidate can be declared the winner; this is because that candidate is guaranteed not to be eliminated in any remaining BTR-IRV pairwise matchups. This trick can be used to save time in counting if a pairwise comparison table has already been made, and also means BTR-IRV can be phrased analagously to Benham's method, though in terms of BTR-IRV itself instead of IRV.

BTR-IRV can be thought of as directly related to IRV in the sense that both focus on eliminating one of the two candidates with the fewest 1st choices in each round; the only difference is that BTR-IRV can eliminate the candidate with the 2nd-fewest 1st choices if they lose the pairwise matchup against the candidate with the fewest 1st choices, whereas IRV always eliminates the candidate with the fewest 1st choices.

BTR-IRV is not immune to clones. Example:

Chris Benham's BTR-IRV cloning-failure example (before cloning D). Winner is A after B,C,D eliminated in that order.
#voters their vote
2 B>A>D>C
3 D>C>B>A
4 A>C>B>D
Benham's BTR-IRV cloning-failure example (after cloning D). Winner is B after C,D1,D2,A eliminated in that order.
#voters their vote
2 B>A>D1>D2>C
2 D1>D2>C>B>A
1 D2>D1>C>B>A
4 A>C>B>D2>D1

There are likely to be many candidates tied for having the fewest 1st choices; one possible non-random tiebreaker is to look for those among the tied candidates that have the fewest 2nd choices, then 3rd choices, etc.

Variations of BTR-IRV could be considered to parallel other Condorcet-IRV hybrid methods; one such variation would be "Repeat both steps until only one candidate remains: Eliminate everyone not in the Smith set, then do a pairwise elimination between the two candidates with the fewest 1st choices".

External links[edit | edit source]

References[edit | edit source]