electowiki

Coombs' method: Difference between revisions

Page Discussion

Coombs' method (view source)

Revision as of 13:03, 28 September 2023

1,452 bytes removed ,  7 months ago
m
no edit summary
VisualWikitext
mNo edit summary
mNo edit summary
 
(7 intermediate revisions by 3 users not shown)
Line 1:
{{Wikipedia}}
'''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.
The '''Coombs' method''', created by Clyde Coombs, is a [[voting system]] used for single-winner elections [[preferential voting|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.
 
==ProceduresProperties==
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.
 
Coombs' method fails the [[Condorcet criterion]], the [[monotonicity criterion]], and the [[participation criterion]].
==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):
<div style="float:right; padding:2px; text-align:center">
[[Image:CondorcetTennesee.png]]</div>
 
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:
* Memphis (Shelby County): 826,330
* Nashville (Davidson County): 510,784
* Chattanooga (Hamilton County): 285,536
* Knoxville (Knox County): 335,749
 
=== Condorcet criterion ===
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:
 
Even though Coombs' frequently selects the [[Condorcet winner criterion|Condorcet winner]], it sometimes fails to do so. For example:
<table class="wikitable" border="1">
{{ballots|
<tr>
7: A>C>D>B
<td>
6: A>D>B>C
'''Group A: 42% of voters <br>(close to Memphis)'''<br>
3: B>A>C>D
1. Memphis<br>
7: B>C>A>D
2. Nashville<br>
9: B>C>D>A
3. Chattanooga<br>
4: C>A>D>B
4. Knoxville
6: D>A>B>C
</td>
3: A>C>B>D
<td valign="top">
}}
'''Group B: 26% of voters <br>(close to Nashville)'''<br>
1. Nashville<br>
2. Chattanooga<br>
3. Knoxville<br>
4. Memphis
</td>
<td>
'''Group C: 15% of voters <br>(close to Chattanooga)'''<br>
1. Chattanooga<br>
2. Knoxville<br>
3. Nashville<br>
4. Memphis
</td>
<td>
'''Group D: 17% of voters <br>(close to Knoxville)'''<br>
1. Knoxville<br>
2. Chattanooga<br>
3. Nashville<br>
4. Memphis
</td>
</tr>
</table>
 
This example, placed in [[Online_poll#Online polling sites|Rob LeGrand's voting calculator]], shows that Coombs arrives at a different result than Condorcet.
Assuming all of the voters vote sincerely (strategic voting is discussed below), the results would be as follows, by percentage:
 
=== Monotonicity criterion ===
<table border="1">
<caption>Coombs' Method Election Results</caption>
<tr>
<th rowspan="2">City</th>
<th colspan="2">Round 1</th>
<th colspan="2">Round 2</th>
</tr>
<tr>
<th>First</th>
<th>Last</th>
<th>First</th>
<th>Last</th>
</tr>
<tr>
<th bgcolor="#ffc0c0">Memphis</th>
<td bgcolor="#ffc0c0">42</td>
<td bgcolor="#ffc0c0">58</td>
<td bgcolor="#e0e0ff"><strike>42</strike> 0 </td>
<td bgcolor="#c0c0c0" rowspan="4"></td>
</tr>
<tr>
<th bgcolor="#ffc0c0">Nashville</th>
<td bgcolor="#ffc0c0">26</td>
<td bgcolor="#ffc0c0">0</td>
<td bgcolor="#ffc0c0"><strike>26</strike> 68</td>
</tr>
<tr>
<th bgcolor="#ffc0c0">Chattanooga</th>
<td bgcolor="#ffc0c0">15</td>
<td bgcolor="#ffc0c0">0</td>
<td bgcolor="#ffc0c0">15</td>
</tr>
<tr>
<th bgcolor="#ffc0c0">Knoxville</th>
<td bgcolor="#ffc0c0">17</td>
<td bgcolor="#ffc0c0">42</td>
<td bgcolor="#ffc0c0">17</td>
</tr>
</table>
 
In the election
* 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.
 
{{ballots|
Note that although Coomb's method chose the [[Condorcet winner]] here, this is not necessarily the case.
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.
==Potential for tactical voting==
The Coombs' method is vulnerable to three strategies: [[tactical voting|compromising, push-over]] and [[strategic nomination|teaming]].
 
=== Participation criterion ===
==External links==
*[http://condorcet.org/emr/methods.shtml#Coombs Condorcet.org EMR: Coombs' method]
 
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 we get the Condorcet failure election where B wins.
 
==Links==
 
*1996
**http://lists.electorama.com/pipermail/election-methods-electorama.com/1996-March/thread.html#65497 - "'Spokane' method"
*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
**https://www.reddit.com/r/EndFPTP/comments/js1qlt/wouldnt_a_rcv_method_where_you_eliminated_the/
 
==Footnotes==
<references />
 
{{fromwikipedia}}
[[Category:PreferentialRanked voting methods]]
[[Category:Sequential loser-elimination methods]]
Kristomun
1,196

edits

Retrieved from "https://electowiki.org/wiki/Special:MobileDiff/8357...17701"