Coombs' method is a ranked voting system popularized by Clyde Coombs.[1] It was described by Edward J. Nanson as the "Venetian method",[2] but should not be confused with the Republic of Venice's use of score voting in elections for Doge. Coombs' method can be thought of as a cross between instant-runoff voting and anti-plurality voting.
Like instant runoff, Coombs' method candidate elimination and redistribution of votes cast for that candidate until one candidate has a majority of votes. However, unlike instant-runoff, each round eliminates the candidate rated last by the most voters (instead of first by the fewest voters).[1]
The method fails most voting system criteria, including Condorcet's majority criterion, monotonicity, participation, and clone-independence.[3][4] However, it does satisfy the median voter property.
Each voter rank-orders all of the candidates on their ballot. Otherwise, the candidate ranked last by the largest number (plurality) of voters is eliminated, making each individual round equivalent to anti-plurality voting. Conversely, under instant-runoff voting, the candidate ranked first (among non-eliminated candidates) by the fewest voters is eliminated.
In some sources, the elimination proceeds regardless of whether any candidate is ranked first by a majority of voters, and the last candidate to be eliminated is the winner.[5] This variant of the method can result in a different winner than the former one (unlike in instant-runoff voting, where checking to see if any candidate is ranked first by a majority of voters is only a shortcut that does not affect the outcome).
Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:
The preferences of each region's voters are:
Assuming all of the voters vote sincerely (strategic voting is discussed below), the results would be as follows, by percentage:
The voting rounds used in the reality television program Survivor could be considered a variation of Coombs' method but with sequential voting rounds. Everyone votes for one candidate they support for elimination each round, and the candidate with a plurality of that vote is eliminated. A strategy difference is that sequential rounds of voting means the elimination choice is fixed in a ranked ballot Coombs' method until that candidate is eliminated.
Like anti-plurality voting, Coombs' rule is extremely vulnerable to strategic voting. As a result, it is more often considered as an example of a pathological voting rule than it is a serious rule.[6] Coombs' method is extremely sensitive to incomplete ballots, compromising, push-over, and teaming, and the vast majority of voters' effects on the election come from how they fill out the bottom of their ballots.[6] As a result, voters have a strong incentive to rate the strongest candidates last to defeat them in earlier rounds.[7]
This results in a Keynesian beauty pageant that is extremely sensitive to minor variations in the perceived strengths of candidates.