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. BTR-IRV passes the Condorcet Criterion, and is by definition a Condorcet method.

BTR-IRV was conceived as a sort of Instant-runoff voting-Condorcet hybrid, in that the runoff it uses is very similar to IRV, but the runoff doesn't ever eliminate a Condorcet Winner.

The process is as follows: 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.

It was originally proposed by Rob LeGrand.[1][2]

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.

External links[edit | edit source]

References[edit | edit source]