Coombs' method: Difference between revisions

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

The '''Coombs' method''', created by [[Clyde Coombs]], is a [[voting system]] used for single-winner [[election]]s [[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.

'''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. IfOtherwise, atthe anycandidate timeranked onelast candidateby isthe rankedlargest firstnumber (among[[plurality non-eliminated candidates(voting)|plurality]]) byof anvoters absoluteis majorityeliminated, ofmaking theeach voters,individual thenround thisequivalent isto the[[anti-plurality winnervoting]]. AsConversely, longunder as[[instant-runoff this is not the casevoting]], the candidate which is ranked lastfirst (again among non-eliminated candidates) by the most (or a [[plurality]] of)fewest voters is eliminated.

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).

==An example==

{{Tenn_voting_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>

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

* Memphis (Shelby County): 826,330

* Nashville (Davidson County): 510,784

* Chattanooga (Hamilton County): 285,536

* Knoxville (Knox County): 335,749

{| border="1"

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:

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

|-

! rowspan="2" | City

! 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).

<table border=1>

* Memphis, having the most last-place votes (26+15+17=58), is therefore eliminated.

<tr>

* 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.

<td>

* 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.

'''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>

== In practice ==

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

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.

== Potential for strategic voting ==

<table border="1">

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>

<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>

==See also==

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

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

* 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.

==Notes==

Note that although Coomb's method chose the [[Condorcet winner]] here, this is not necessarily the case.

<references />

{{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]