Coombs' method: Difference between revisions

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'''Coombs' method''' (or the '''Coombs rule''')<ref>Grofman, Bernard, and Scott L. Feld (2004) [https://dx.doi.org/10.1016/j.electstud.2003.08.001 "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.</ref> is a [[ranked voting systems|ranked voting system]] created by [[wikipedia:Clyde Coombs|Clyde Coombs]] used for single-winner [[election]]s. 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.

==~~Example~~ ==

==Properties==

Coombs' method ~~frequently selects~~ the [[Condorcet ~~winner~~ criterion~~|Condorcet winner~~]]~~. However~~, ~~this~~ ~~does~~ ~~not~~ ~~always~~ ~~happen.~~ ~~For~~ ~~example:~~

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

⚫

The following examples are due to Felsenthal and Tideman<ref name="FT">{{Cite journal |last=Felsenthal |first=Dan |last2=Tideman |first2=Nicolaus |date=2013 |title=Varieties of failure of monotonicity and participation under five voting methods |url=https://www.researchgate.net/profile/Dan-Felsenthal/publication/257667897_Varieties_of_failure_of_monotonicity_and_participation_under_five_voting_methods/links/54aec0fb0cf21670b35870a6/Varieties-of-failure-of-monotonicity-and-participation-under-five-voting-methods.pdf?origin=publication_detail|journal=Theory and Decision |language=en |volume=75 |issue=1 |pages=59–77}}</ref> unless otherwise noted:

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

=== Condorcet criterion ===

⚫

~~This example, placed in [https://cs.angelo.edu/~rlegrand/rbvote/calc.html Rob LeGrand's voting calculator], shows that Coombs arrives at a different result than Condorcet.~~ The ~~example~~ is ~~pulled~~ ~~from~~ a Felsenthal and Tideman ~~paper.~~<ref>{{Cite journal |last=Felsenthal |first=Dan |last2=Tideman |first2=Nicolaus |date=2013 |title=Varieties of failure of monotonicity and participation under five voting methods |url=https://www.researchgate.net/profile/Dan-Felsenthal/publication/257667897_Varieties_of_failure_of_monotonicity_and_participation_under_five_voting_methods/links/54aec0fb0cf21670b35870a6/Varieties-of-failure-of-monotonicity-and-participation-under-five-voting-methods.pdf?origin=publication_detail|journal=Theory and Decision |language=en |volume=75 |issue=1 |pages=59–77}}</ref>

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

{{ballots|

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 [[Online_poll#Online polling sites|Rob LeGrand's voting calculator]], shows that Coombs arrives at a different result than Condorcet.

=== Monotonicity criterion ===

In the election

{{ballots|

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

{{ballots|

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 B wins.

==Links==