Coombs' method: Difference between revisions

Revision as of 07:44, 8 September 2003

Latest revision as of 15:37, 28 May 2025

(113 intermediate revisions by 66 users not shown)

Line 1:

{{Short description|Single-winner ranked voting rule}}{{Confused|Combs method|text=the [[Combs method]]}}{{Electoral systems sidebar|expanded=Single-winner}}

The '''Coombs' method''' is a [[voting system]] used for single-winner [[election]]s, in which each voter rank-orders the candidates. If there is a simple majority in people's first choice, the candidate who holds that majority wins. But if there is no simple majority, a series of eliminations occurs. The Coombs' method was invented by [[Clyde Coombs]], hence the name.

'''Coombs' method''' is a [[ranked voting systems|ranked voting system]]. Like [[Instant-runoff voting|instant-runoff (IRV-RCV)]], Coombs' method is a [[Sequential elimination method|sequential-loser method]], where the last-place finisher according to one method is eliminated in each round. However, unlike in instant-runoff, each round has electors voting against their least-favorite candidate; the candidate ranked last by the most voters is eliminated.<ref name="Grofman">{{Cite journal |last=Grofman |first=Bernard |last2=Feld |first2=Scott L. |date=2004-12-01 |title=If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule |url=https://www.sciencedirect.com/science/article/abs/pii/S026137940300060X |journal=Electoral Studies |volume=23 |issue=4 |pages=641–659 |doi=10.1016/j.electstud.2003.08.001 |issn=0261-3794|url-access=subscription }}</ref>

The method fails several [[voting system criteria]], including [[Condorcet winner criterion|Condorcet's majority criterion]], [[Monotonicity criterion|monotonicity]], [[Participation criterion|participation]], and [[Independence of clones criterion|clone-independence]].<ref>{{cite journal |last=Nurmi |first=Hannu |title=Voting Procedures: A Summary Analysis |journal=British Journal of Political Science |volume=13 |issue=2 |pages=181-208 |publisher=Cambridge University Press |date=1983-04-01 |language=English |url=https://www.cambridge.org/core/journals/british-journal-of-political-science/article/abs/voting-procedures-a-summary-analysis/67C52E7250EB4B88018B22D59CAA6908 |doi=10.1017/S0007123400003215 |access-date=2024-05-19 |url-access=subscription }}</ref><ref>{{cite book |last=Nurmi |first=Hannu |title=Comparing Voting systems |publisher=Springer Dordrecht |series= Theory and Decision Library A |volume=3 |edition=Illustrated |date=2012-12-06 |pages=209 |language=English |url= https://link.springer.com/book/10.1007/978-94-009-3985-1?utm_medium=referral&utm_source=google_books&utm_campaign=3_pier05_buy_print&utm_content=en_08082017 |doi= 10.1007/978-94-009-3985-1 |isbn= 9789400939851}}</ref> However, it does satisfy Black's single-peaked [[Median voter property|median voter criterion]].<ref name="Grofman" />{{rp|at=prop. 2}}

== History ==

The method was popularized by [[Clyde Coombs]].<ref name="Grofman" /> It was described by [[Edward J. Nanson]] as the "Venetian method"<ref>{{Cite book |last=Royal Society of Victoria (Melbourne |first=Vic ) |url=http://archive.org/details/transactionsproc1719roya |title=Transactions and proceedings of the Royal Society of Victoria .. |date=1864 |publisher=Melbourne : The Society |others=American Museum of Natural History Library}}</ref> (which should not be confused with the [[Republic of Venice]]'s use of [[score voting]] in elections for [[Doge of Venice|Doge]]).

==Procedures==

Each voter rank-orders all of the candidates on their ballot. Otherwise, the candidate ranked last by the largest number ([[plurality (voting)|plurality]]) of voters is eliminated, making each individual round equivalent to [[anti-plurality voting]]. Conversely, under [[instant-runoff voting]], the candidate ranked first (among non-eliminated candidates) by the fewest voters is eliminated.

Each voter rank-orders all of the candidates on their ballot.

In some sources, the elimination proceeds regardless of whether any candidate is ranked first by a majority of voters, and the last candidate to be eliminated is the winner.<ref>Pacuit, Eric, [https://plato.stanford.edu/archives/fall2017/entries/voting-methods/ "Voting Methods"], ''The Stanford Encyclopedia of Philosophy'' (Fall 2017 Edition), Edward N. Zalta (ed.)</ref> This variant of the method can result in a different winner than the former one (unlike in instant-runoff voting, where checking to see if any candidate is ranked first by a majority of voters is only a shortcut that does not affect the outcome).

The method works as follows. First, if there is a simple majority in people's first choice, that candidate wins. If there isn't a simple majority, the candidate with a majority of last place votes is eliminated. If no such candidate exists, the candidate with the most first place votes is temporarily eliminated from consideration for perminent elimination. Once a perminently eliminated candidate is found, that candidate is factored out of the ballot, and the process is repeated. This is done until a single candidate remains.

==An example==

{{Tenn_voting_example}}

Imagine an election for 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">

Assuming all of the voters vote sincerely (strategic voting is discussed below), the results would be as follows, by percentage:

[[Image:CondorcetTennesee.png]]</div>

{| border="1"

* Memphis (Shelby County): 826,330

|+'''Coombs' method election results'''

* Nashville (Davidson County): 510,784

|-

* Chattanooga (Hamilton County): 285,536

! rowspan="2" | City

* Knoxville (Knox County): 335,749

! colspan="2" | Round 1

! colspan="2" | Round 2

|-

! First

! Last

! First

! Last

|-

! bgcolor="#ffc0c0" | Memphis

| bgcolor="#ffc0c0" | 42

| bgcolor="#ffc0c0" | 58

| bgcolor="#e0e0ff" | <s>42</s> 0

| bgcolor="#c0c0c0" rowspan="4" |

|-

! bgcolor="#ffc0c0" | Nashville

| bgcolor="#ffc0c0" | 26

| bgcolor="#ffc0c0" | 0

| bgcolor="#ffc0c0" | <s>26</s> 68

|-

! bgcolor="#ffc0c0" | Chattanooga

| bgcolor="#ffc0c0" | 15

| bgcolor="#ffc0c0" | 0

| bgcolor="#ffc0c0" | 15

|-

! bgcolor="#ffc0c0" | Knoxville

| bgcolor="#ffc0c0" | 17

| bgcolor="#ffc0c0" | 42

| bgcolor="#ffc0c0" | 17

|}

* In the first round, no candidate has an absolute majority of first-place votes (51).

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:

* 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 the winner.

* Note that the last-place votes are only used to eliminate a candidate in a voting round where no candidate achieves an absolute majority; they are disregarded in a round where any candidate has more than 50%. Thus last-place votes play no role in the final round.

== In practice ==

<table border=1>

The voting rounds used in the [[reality television]] program ''[[Survivor (TV series)|Survivor]]'' could be considered a variation of Coombs' method but with sequential voting rounds. Everyone votes for one candidate they support for elimination each round, and the candidate with a plurality of that vote is eliminated. A strategy difference is that sequential rounds of voting means the elimination choice is fixed in a ranked ballot Coombs' method until that candidate is eliminated.

<tr>

<td>

'''Group A: 42% of voters (close to Memphis)'''<br>

1. Memphis<br>

2. Nashville<br>

3. Chattanooga<br>

4. Knoxville

</td>

<td valign="top">

'''Group B: 26% of voters (close to Nashville)'''<br>

1. Nashville<br>

2. Chattanooga<br>

3. Knoxville<br>

4. Memphis

</td>

<td>

'''Group C: 15% of voters (close to Chattanooga)'''<br>

1. Chattanooga<br>

2. Knoxville<br>

3. Nashville<br>

4. Memphis

</td>

<td>

'''Group D: 17% of voters (close to Knoxville)'''<br>

1. Knoxville<br>

2. Chattanooga<br>

3. Nashville<br>

4. Memphis

</td>

</tr>

</table>

== Potential for strategic voting ==

Assuming all of the voters vote sincerely (strategic voting is discussed below), the results would be as follows, by percentage. Note, these are the numbers of '''last place''' votes:

Like [[anti-plurality voting]], Coombs' rule is extremely vulnerable to strategic voting. As a result, it is more often used as an example of a [[Pathological (mathematics)|pathological]] voting rule than a serious proposal.<ref name=":0" /> The equilibrium position for Coombs' method is extremely sensitive to [[Exhausted ballot|incomplete ballots]] and [[strategic nomination]] because the vast majority of voters' effects on the election come from how they fill out the bottom of their ballots.<ref name=":0">[http://www.accuratedemocracy.com/l_data.htm "Data on Manipulability"]</ref> As a result, voters have a strong incentive to rate the strongest candidates last to defeat them in earlier rounds.<ref>{{Cite journal |last=Smith |first=Warren D. |date=12 July 2006 |title=Descriptions of single-winner voting systems |url=http://www.9mail.de/m-schulze/votedesc.pdf |journal=Voting Systems}}</ref>

==See also==

<table border="1">

* [[List of democracy and elections-related topics]]

<caption>Coombs Method Election Results (last place votes)</caption>

<tr>

<th>City</th>

<th>Round 1</th>

<th>Round 2</th>

<th>Round 3<br>

</th>

</tr>

<tr>

<th bgcolor="#ffc0c0">Memphis</th>

<td bgcolor="#ffc0c0">58</td>

<td bgcolor="#e0e0ff"><strike>58</strike> 0 </td>

<td bgcolor="#e0e0ff">n/a</td>

</td>

</tr>

<tr>

<th bgcolor="#ffc0c0">Nashville</th>

<td bgcolor="#ffc0c0">0</td>

<td bgcolor="#ffc0c0"><strike>0</strike> 32</td>

<td bgcolor="#ffc0c0">32 &lt;-winner with least last place votes</td>

</tr>

<tr>

<th bgcolor="#ffc0c0">Chattanooga</th>

<td bgcolor="#ffc0c0">0</td>

<td bgcolor="#ffc0c0">0</td>

<td bgcolor="#ffc0c0"><strike>0</strike> 68 </td>

</td>

</tr>

<tr>

<th bgcolor="#ffc0c0">Knoxville</th>

<td bgcolor="#ffc0c0">42</td>

<td bgcolor="#ffc0c0"><strike>42</strike> 68 </td>

<td bgcolor="#e0e0ff"><strike>68</strike> 0 </td>

</tr>

</table>

==Notes==

Starting off, there is no candidate with a simple majority of first place votes. A series of eliminations now occurs:

<references />

* In the first round, Memphis, having a majority of last place votes (26+15+17=58), is eliminated.

* In the second round, Memphis is out of the running, and so must be factored out. The lowest ranked candidate on Group B's ballots is Knoxville, and the lowest ranked candidate on Group C and D's ballots is Nashville, so the votes are transferred accordingly. Knoxville now has a majority of last place votes, and is eliminated.

* In the third round, Knoxville is out of the running, and is factored out of the ballots. Now, the lowest ranked candidate on both Group A and Group B's ballots is Chatanooga, so these ballots now count against Chatanooga. Since Chatanooga has a majority against it, it is eliminated, leaving Nashville the winner.

{{voting systems}}

==Potential for Tactical voting==

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

[[Category:Single-winner electoral systems]]

==External Link==

[[Category:Preferential electoral systems]]

*[http://condorcet.org/emr/methods.shtml#Coombs Condorcet.org EMR: Coombs' method]