Coombs' method

Coombs' method or the Coombs rule is a ranked voting system which uses a ballot counting method for ranked voting created by Clyde Coombs. Coombs' method can be thought of as a cross between instant-runoff voting and anti-plurality voting (and as such, inherits many of the pathologies of both).

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).

Procedures

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 resemble 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. 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).

An example

Tennessee and its four major cities: Memphis in the south-west; Nashville in the centre, Chattanooga in the south, and Knoxville in the east

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 to 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 the state
  • 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

Assuming all of the voters vote sincerely (strategic voting is discussed below), the results would be as follows, by percentage:

Coombs' method election results
City Round 1 Round 2
First Last First Last
Memphis 42 58 42 0
Nashville 26 0 26 68
Chattanooga 15 0 15
Knoxville 17 42 17
  • In the first round, no candidate has an absolute majority of first-place votes (51).
  • Memphis, having the most last-place votes (26+15+17=58), is therefore eliminated.
  • In the second round, Memphis is out of the running, and so must be factored out. Memphis was ranked first on Group A's ballots, so the second choice of Group A, Nashville, gets an additional 42 first-place votes, giving it an absolute majority of first-place votes (68 versus 15+17=32), and making it the winner.
  • Note that the last-place votes are only used to eliminate a candidate in a voting round where no candidate achieves an absolute majority; they are disregarded in a round where any candidate has 51% or more. Thus last-place votes play no role in the final round.

Use

The voting rounds used in the reality television program Survivor could be considered a variation of Coombs' method, 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.

Potential for strategic voting

Like anti-plurality voting and the Borda count, Coombs' rule is extremely vulnerable to strategic voting, and as a result is known more as an example of a pathological social choice function than as a serious voting rule.

Coombs' method is extremely sensitive to incomplete ballots, compromising, push-over, and teaming; the vast majority of voters' effects on the election comes from how they fill out the bottom of their ballots. As a result, voters have a strong incentive to rate the strongest candidates last to defeat them early on. This results in a Keynesian beauty pageant, where the Nash equilibrium results in candidates being chosen completely at random. However, naively strategic voters (i.e. boundedly rational voters) will often elect the worst candidate, analogously to the behavior in the "2/3 of the average" game.

See also


This page was last updated at 2024-02-05 02:11 UTC. Update now. View original page.

All our content comes from Wikipedia and under the Creative Commons Attribution-ShareAlike License.


Top

If mathematical, chemical, physical and other formulas are not displayed correctly on this page, please useFirefox or Safari