# Coombs' method

The Coombs' method, created by Clyde Coombs, is a voting system used for single-winner elections in which each voter rank-orders the candidates. It sort of works like Instant Runoff Voting (a US term; it is known as Preferential Voting in some countries) in reverse.

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## Procedures

Each voter rank-orders all of the candidates on their ballot. If at any time one candidate is ranked first (among non-eliminated candidates) by an absolute majority of the voters, then this is the winner. As long as this is not the case, the candidate which is ranked last (again among non-eliminated candidates) by the most (or a plurality of) voters is eliminated.

## An example

Imagine an election to choose the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 miles northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county):

• Memphis (Shelby County): 826,330
• Nashville (Davidson County): 510,784
• Chattanooga (Hamilton County): 285,536
• Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennesee follows from those population centers, one could easily envision an election where the percentages of sincere preferences would be as follows:

 Group A: 42% of voters (close to Memphis) 1. Memphis 2. Nashville 3. Chattanooga 4. Knoxville Group B: 26% of voters (close to Nashville) 1. Nashville 2. Chattanooga 3. Knoxville 4. Memphis Group C: 15% of voters (close to Chattanooga) 1. Chattanooga 2. Knoxville 3. Nashville 4. Memphis Group D: 17% of voters (close to Knoxville) 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 thus the winner. Note that the last place votes are disregarded in the final round.

Note that although Coomb's method chose the Condorcet winner here, this is not necessarily the case.

## Potential for tactical voting

The Coombs' method is vulnerable to three strategies: compromising, push-over and teaming.

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