# Coombs' method

**Coombs' method** (or the **Coombs rule**)^{[1]} is a ranked voting system created by Clyde Coombs used for single-winner elections. Similarly to instant-runoff voting, it uses candidate elimination and redistribution of votes cast for that candidate until one candidate has a majority of votes. Its difference from IRV lies in its elimination criterion: instead of eliminating the candidate ranked first by the fewest voters, it eliminates the candidate ranked last by the most.

## Properties

Coombs' method fails the Condorcet criterion, the monotonicity criterion, and the participation criterion.

The following examples are due to Felsenthal and Tideman^{[2]} unless otherwise noted:

### Condorcet criterion

Even though Coombs' frequently selects the Condorcet winner, it sometimes fails to do so. For example:

7: A>C>D>B 6: A>D>B>C 3: B>A>C>D 7: B>C>A>D 9: B>C>D>A 4: C>A>D>B 6: D>A>B>C 3: A>C>B>D

This example, placed in Rob LeGrand's voting calculator, shows that Coombs arrives at a different result than Condorcet.

### Monotonicity criterion

In the election

1: A>B>C 10: A>C>B 11: B>A>C 11: B>C>A 10: C>A>B 2: C>B>A

C wins, but if the 11 B>A>C voters raise C and vote B>C>A, then B wins.

### Participation criterion

In the election

7: A>C>D>B 6: A>D>B>C 3: B>A>C>D 7: B>C>A>D 9: B>C>D>A 4: C>A>D>B 6: D>A>B>C

A is the CW and wins. But if three additional voters vote A>C>B>D then we get the Condorcet failure election where B wins.

## Links

- 1996
- 2005
- https://web.archive.org/web/20050909092356/http://condorcet.org/emr/methods.shtml#Coombs - 2005 archive of Condorcet.org glossary of terminology

- 2019
- https://imgur.com/gallery/SLTHgCO - Diagram of Coombs' and center squeeze

- 2020

## Footnotes

- ↑ Grofman, Bernard, and Scott L. Feld (2004) "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule,"
*Electoral Studies***23**:641-59. - ↑ Felsenthal, Dan; Tideman, Nicolaus (2013). "Varieties of failure of monotonicity and participation under five voting methods" (PDF).
*Theory and Decision*.**75**(1): 59–77.