Bucklin voting

Bucklin is a voting system that can be used for single-member districts and also multi-member districts. It is also known as the Grand Junction system after Grand Junction, Colorado, where it was first proposed. It is closely related to the class of graded Bucklin systems, in which equal and/or skipped rankings are allowed, which includes such systems as Majority Choice Approval (MCA) or Majority Judgment. Modern theorists tend to prefer graded Bucklin systems over ungraded ones, as they usually comply better with criteria such as FBC.

How did it work?
Voters were allowed rank preference ballots - first, second, third. In some cases, voters were allowed multiple rankings at the third rank, although there is no record of the use of MCA, which allows equal ranking at all levels.

First choice votes were first counted. If one candidate had a majority, that candidate won. Otherwise the second choices were added to the first choices. Again, if a candidate with a majority vote was found, the winner was the candidate with the most votes in that round. Lower rankings were added as needed.

A majority was defined as half the number of voters, similar to absolute majority. Since after the first round there were more votes cast than voters, it was possible more than one candidate to have majority support.

For multi-member districts, voters marked as many first choices as there are seats to be filled. Voters marked the same number of second and further choices. In some localities, the voter was required to mark a full set of first choices for his or her ballot to be valid.

Where was it used?
This method was apparently first used in Geneva during the French Revolution, in the period from 1792 to 1798, at the suggestion of the Marquis de Condorcet. This was a time of upheaval and experiment, and this usage has only recently come to light again.

It was later reinvented and used in many political elections in the United States in the early 20th Century. In most states it was repealed and in a few states it was found to violate the state constitution.

Satisfied criteria
Bucklin satisfies the Plurality criterion, the Majority criterion for solid coalitions, Monotonicity, Later-no-help, and Minimal Defense (which implies satisfaction of the Strong Defensive Strategy criterion). It fails the Condorcet criterion, Clone Independence, Participation, and Later-no-harm.

An example
The first round has no majority winner. Therefore the second rank votes are added. This moves Nashville and Chattanooga above 50%, so a winner can be determined. Since Nashville is supported by a higher majority (68% versus 58%), Nashville is the winner.

An alternative way to demonstrate it:

Voter Strategy
Voters supporting a strong candidate have an advantage to "Bullet Vote" (Only offer one ranking), in hopes that other voters will add enough votes to help their candidate win. This strategy is most secure if the supported candidate appears likely to gain many second rank votes.

In the above example, Memphis voters have the most first place votes and might not offer a second preference in hopes of winning, but this attempted strategy fails because they are not a second favorite from competitors.

Fallback voting
Fallback voting (FV) is a voting method strongly related to Bucklin voting, invented by Brams and Sanver. It is a ranked method which (with a slight modification) works by examining the 1st rank, and electing the candidate with the largest majority of voters ranking them 1st, if such a candidate exists. If no such candidate exists, then it examines both the 1st and 2nd ranks, and elects the candidate ranked either 1st or 2nd by the largest majority of voters. If none exists, the procedure continues to sequentially examine an additional rank at a time until either some candidate has the largest majority of ballots ranking them within the examined ranks, in which case they win, or until all ranks have been added in, at which point the candidate ranked on the most ballots wins.

Fallback voting is equivalent to the Expanding Approvals Rule in the single-winner case under certain conditions."Remark 3 [...] For k = 1 and under linear orders for all but a subset of equally least preferred candidates applying the tweak in Remark 2 leads to the EAR [Expanding Approvals Rule] being equivalent to the Fallback voting rule (Brams and Sanver, 2009)."

Related systems
There are a number of variations of Bucklin voting such as Iterated Bucklin, Expanding Approvals Rule, Maximum Constrained Approval Bucklin, Venzke Bucklin Variant and Majority Choice Approval/ER-Bucklin. Majority Judgment is a cardinal voting system which is similar.