Bottom-Two-Runoff IRV

From electowiki

The "bottom-two-runoff-instant-runoff-voting method" (or "BTR-IRV method") is an election method that selects a single winner using votes that express ranked preferences. It is a Condorcet-IRV hybrid distinct from other hybrids like Smith//IRV.

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.

The BTR-IRV method was originally proposed by Rob LeGrand in 2002, and first referred to by that name by Jan Kok in 2005.[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.

A benefit of BTR-IRV is that first choices are honored in the elimination process, so that a polarizing candidate can survive to later rounds until they have a single opponent who they can be individually compared to. This attribute and ease of explaining the system makes it less prone to claims of fraud than other systems for resolving the Condorcet paradox.

This system is a form of single transferable vote (STV), and may be referred to by the more general name BTR-STV, though the multi-winner variant was not originally recommended by LeGrand.[3]

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.

Passed and failed criteria[edit | edit source]

Like IRV, BTR-IRV fails monotonicity and summability. Unlike IRV, BTR-IRV passes the Smith criterion.

If the voters don't produce any Condorcet cycles, then like every other Condorcet method, BTR-IRV is monotone and summable. However, this is not necessarily known in advance.

Clone independence[edit | edit source]

BTR-IRV is not immune to clones. A crowding 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

Note that the example requires two cases of the Condorcet paradox in the base case: b>a, a>c, c>b and also c>b, b>d, d>c, so it is unlikely to occur in practice.

Dominant mutual third candidate burial resistance[edit | edit source]

Unlike many other Condorcet-IRV hybrid methods, BTR-IRV fails dominant mutual third candidate burial resistance.

4: A>B>C
2: B>A>C
3: B>C>A
2: C>A>B

B has five first preferences, A has four, and C has two. A is the Condorcet winner with 36% of the first preferences, and thus the DMT candidate.

Let one B>A>C voter bury A under C:

4: A>B>C
1: B>A>C
4: B>C>A
2: C>A>B

This creates an ABCA cycle. BTR-IRV starts by determining which of the two Plurality losers (A and C) should be eliminated. Since C beats A pairwise, A is eliminated. In the second round, B beats C pairwise and wins.

Thus the burial benefited the B>A>C voter as the winner changed from A to B.

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.

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".

BTR-IRV is not the same as Smith-IRV or Benham's method, as they don't pass the same criteria.

Simplified Variant[edit | edit source]

If you remove the redistribution step, leaving the candidates in the initial 1st choice sort order for the entire process, BTR-IRV becomes precinct summable. Vote counting only requires the 1st choice vote counts and the pairwise preference matrix from each precinct, not the complete ranking counts.

External links[edit | edit source]

References[edit | edit source]

  1. LeGrand, Rob (2002-12-20). "Re: IRV-Condorcet Compromise?". Instantrunoff-freewheeling Yahoo Group archives.
  2. Kok, Jan (2005-07-26). ""Better" IRV?". ApprovalVoting Yahoo Group archives.
  3. LeGrand, Rob (2006-06-08). "An example of BTR-STV". Election-methods mailing list archives.