View source for Instant-Runoff Voting with Bottom Two Runoff

'''Instant-Runoff Voting with Bottom Two Runoff''' (abbreviated as '''IRV-BTR''') is an election vote-counting method that uses ranked ballots. In each round the two candidates with the fewest first place votes are put in a simulated runoff where it is determined which of the two would lose in a head-to-head election using the rankings from all ballots.

This method modifies [[Instant-Runoff Voting|instant runoff voting]] (IRV) by adding a runoff between the bottom two candidates and can be viewed as an expansion of [[Ranked Choice Including Pairwise Elimination|RCIPE]] by providing a broader criteria for when the voter with the fewest first rank votes is not eliminated. This addition makes it satisfy the [[Condorcet winner criterion|condorcet criterion]] and would have prevented the failure of instant-runoff voting to elect the most popular candidate in the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election 2009 mayoral election in Burlington, Vermont].

== Description ==

Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.

This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.

In each round the two candidates are put into a simulated runoff. Each of the candidates receives a simulated vote for every ballot on which they are ranked higher than their opponent. The candidate with the fewer simulated votes is eliminated.

The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Small political parties would not qualify to move their runner-up candidate to the runoff or general election.

Importantly, the runner-up candidate does not deserve to win any kind of elected seat. This means this method is not suitable for filling multiple seats, such as on a city council or in a multi-member district.

To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. Such ballots would count for neither candidate in the case of a simulated runoff between the two.

Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.

If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.

The ranking level below the lowest ranking level is reserved for write-in candidates whose names do not appear on the ballot being counted.


== Example ==
{{Tenn_voting_example}}
These ballot preferences are converted into pairwise counts and displayed in the following tally table.

{| class="wikitable"
|-
! rowspan=2" | All possible pairs<br/>of choice names
! colspan=3 | Number of votes with indicated preference
|-
! style="text-align:left" | Prefer X over Y
! style="text-align:left" | Equal preference
! style="text-align:left" | Prefer Y over X
|-
| align="left" | X = Memphis<br/>Y = Nashville
| align="left" | 42%
| align="left" | 0
| align="left" | 58%
|-
| align="left" | X = Memphis<br/>Y = Chattanooga
| align="left" | 42%
| align="left" | 0
| align="left" | 58%
|-
| align="left" | X = Memphis<br/>Y = Knoxville
| align="left" | 42%
| align="left" | 0
| align="left" | 58%
|-
| align="left" | X = Nashville<br/>Y = Chattanooga
| align="left" | 68%
| align="left" | 0
| align="left" | 32%
|-
| align="left" | X = Nashville<br/>Y = Knoxville
| align="left" | 68%
| align="left" | 0
| align="left" | 32%
|-
| align="left" | X = Chattanooga<br/>Y = Knoxville
| align="left" | 83%
| align="left" | 0
| align="left" | 17%
|}

The initial rankings are 42% Memphis, 26% Nashville, 17% Knoxville, and 15% Chattanooga.

In the first elimination round Knoxville and Chattanooga are compared to each other. Knoxville is eliminated because it loses the pairwise comparison to Chattanooga (83%-17%).

If there had not been a runoff, Chattanooga would have been eliminated because it has the smallest number of ballots that rank Chattanooga as the first choice (15% - 17%).

The rankings would now be 42% Memphis, 32% Chattanooga, and 26% Nashville.

In the second elimination round, Chattanooga and Nashville are compared. In this case Chattanooga is eliminated because it loses the pairwise comparison (32%-68%). Nashville would have been eliminated in a normal IRV election due to it having fewer first rank votes. 

In the third and final elimination round, Memphis is eliminated by Nashville (42% - 58%). This final round is indistinguishable from a standard IRV final round due to the existence of only two candidates.


== Mathematical criteria ==
The frequency with which this method passes or fails each of the following criteria have not been estimated.

This method always passes the following criteria.

* [[Condorcet criterion|Condorcet]]: pass
* [[Condorcet loser criterion|Condorcet loser]]: pass
* [[Smith criterion|Smith]]/[[ISDA]]: pass


The below methods have not been evaluated at this point.

* Resolvable: pass
* Polytime: pass
* [[Majority criterion|Majority]]: fail
* [[Majority loser criterion|Majority loser]]: fail
* Mutual majority: fail

* LIIA: fail
* IIA: fail
* Cloneproof: fail
* Monotone: fail
* Consistency: fail
* Reversal symmetry: fail
* Later no harm: fail
* Later no help: fail
* Burying: fail
* Participation: fail
* No favorite betrayal: fail

It is [[Summability criterion|summable]] with O(N<sup>2</sup>).

[[Category:Sequential loser-elimination methods]]
[[Category:Ranked voting methods]]