# Coombs' method

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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.