Condorcet winner criterion: Difference between revisions

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Mainly because of [[Condorcet paradox|Condorcet's voting paradox]], a beats-all winner will not always exist in a given set of votes. However, there will always be a smallest group of candidates such that more voters prefer anyone in the group over anyone outside of the group. If the beats-all winner exists, they will be the only candidate in this group, which is called the [[Smith set]]. Voting methods that always elect from the Smith set are known as "[[Smith-efficient]]".  

 

As an example, if there are 3 candidates, with head-to-head matchups indicating a 51% majority prefers the second candidate over the first, and a 43% plurality prefer the second over 37% preferring the third (with 20% of voters having no preference), then the second candidate gets more votes than their competitors in all matchups and so they are the Condorcet winner.

 

Suppose the voters had been polled on their preferences among the candidates, and the following preferences in head-to-head matchups are produced between French Fries (FF), Hamburger (H), and Cookies (C) (FF>C shows the number of voters who prefer FF over C, for example):

 

FF (French Fries) is the CW here.

 

 

 

 

'''Traditional definition of "beat":'''

}}

 

 

The majority Condorcet criterion is the same as the above, but with "beat" replaced by "majority-beat", defined to be "X majority-beats Y iff over 50% voters vote X over Y." Thus, a '''majority Condorcet winner''' is a Condorcet winner who majority-beats all other candidates.

==Commentary==

 

 

On a one-dimensional [[political spectrum]], the beats-all winner will be at the position of the median voter. Example (also see https://i.stack.imgur.com/z4bjy.png):<graph>{

}</graph>Supposing this is the voter distribution, with 3 candidates Leftist, Centrist, Rightist who are each points on the left, center, and right of the distribution respectively, with all voters distributed from left to right, and voters on, for example, the left preferring Leftist>Centrist>Rightist, and the same holding vice versa. The Condorcet winner is the Centrist, because a majority prefer them over the "other side", for whichever side you look at.

 

The Condorcet winner/[[Smith set]] is a common [[equilibrium]] point in many voting methods. This is because a majority/plurality of voters have no incentive to deviate towards another candidate. Example for [[Approval voting]]:

 

B is the CW. If voters approve everyone they ranked before the "|", then B is approved by all voters, and wins. If any of the three groups of voters here raises their approval threshold (only approves their 1st choice), then another group has an incentive to maintain their approval threshold where it is i.e. if C-top voters stop approving B, then the 69 voters who prefer B>C have an incentive to move their approval thresholds between B and C to ensure B is approved by a majority and C is not. Note that this requires both accurate polling and coordinated [[Strategic voting|strategic voting]].

 

Non-ranking methods such as [[plurality voting|plurality]] and [[approval voting|approval]] cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.

 

 

<blockquote><table>

</table></blockquote>

 

See [[Score voting#Majority-related criteria]] to see how Score can fail the Condorcet criterion. In general however, it is expected that the Condorcet winner (and Smith Set candidates in general) will almost always be very high-utility when compared to the utilitarian winner.

 

Note that the Condorcet criterion also implies the following criterion which is somewhat related to Independence of Irrelevant Alternatives: removing losing candidates can't change the result of an election if there is a Condorcet winner. <ref>https://arxiv.org/abs/1804.02973 The Schulze Method of Voting p.351 "The Condorcet criterion for single-winner elections (section 4.7) is important because, when there is a Condorcet winner b ∈ A, then it is still a Condorcet winner when alternatives a1,...,an ∈ A \ {b} are removed. So an alternative b ∈ A doesn’t owe his property of being a Condorcet winner to the presence of some other alternatives. Therefore, when we declare a Condorcet winner b ∈ A elected whenever a Condorcet winner exists, we know that no other alternatives a1,...,an ∈ A \ {b} have changed the result of the election without being elected."</ref> In addition, adding candidates who are pairwise beaten by the Condorcet winner (when one exists) can't change the result of the election.

 

 

Sometimes there is no Condorcet winner, but there may be candidate(s) who are preferred by at least as many voters as all other candidates (i.e. they beat '''or''' tie all other candidates; as many voters rank or score them higher or equally as each of the other candidates as the other way around), who are known as weak Condorcet winners. While it may thus seem reasonable that a Condorcet method should pass a condition of always electing solely from the set of weak Condorcet winners when no regular Condorcet winner exists and at least one weak Condorcet winner exists, this guaranteeably leads to failures of reversal symmetry and clone immunity, and so it may be better to say that the set of weak Condorcet winners should have some, but not total priority to win. Example (parentheses are used to indicate implied rankings):<blockquote>3 A(>B1=B2=B3)

Weak CWs have also been called Condorcet non-losers, with the requirement that they always win when they exist being called Exclusive-Condorcet. <ref>{{Cite web|url=http://www.votingmatters.org.uk/ISSUE3/P5.HTM#r2|title=Voting matters, Issue 3: pp 8-15|last=|first=|date=|website=www.votingmatters.org.uk|url-status=live|archive-url=|archive-date=|access-date=2020-05-09|quote=Exclusive-Condorcet (see Fishburn[2]). If there is a Condorcet non-loser, then at least one Condorcet non-loser should be elected.}}</ref>

 

Schulze has proposed a generalization of the Condorcet criterion for multi-winner methods:<ref name="Schulze 2018">{{cite web | last=Schulze | first=Markus | title=The Schulze Method of Voting | website=arXiv.org | date=2018-03-15 | url=https://arxiv.org/abs/1804.02973v6 | access-date=2020-02-11|page=351}}</ref> Suppose all but M+1 candidates are eliminated from the ballots, and the remaining candidates include candidate ''b''. If ''b'' is always a winner when electing M winners from the M+1 remaining candidates, no matter who the other M remaining candidates are, then ''b'' is an M-seat Condorcet winner.

 

See also https://arxiv.org/abs/1701.08023.

 

See [[Self-referential Smith-efficient Condorcet method]].

 

Approval Voting (and thus Score Voting when all voters use only the minimum or maximum score) is equivalent to a traditional Condorcet method where a voter must rank all candidates 1st or last. Score Voting where some voters give some candidates intermediate scores can be treated as Approval Voting using the [[KP transform]], and thus treated as a traditional Condorcet method in the same way as Approval Voting.

 

 

Every Condorcet method is susceptible to burial in at least some elections, and Condorcet is also incompatible with [[later-no-harm]] and [[later-no-help]]. However, if a method has the property that a coordinated majority can always force its outcome, then prefixing a Condorcet step to that method can never increase the proportion of elections where strategy pays off.<ref name="Green-Armytage Tideman Cosman pp. 183–212">{{cite journal | last=Green-Armytage | first=James | last2=Tideman | first2=T. Nicolaus | last3=Cosman | first3=Rafael | title=Statistical evaluation of voting rules | journal=Social Choice and Welfare | publisher=Springer Science and Business Media LLC | volume=46 | issue=1 | date=2015-08-11 | issn=0176-1714 | doi=10.1007/s00355-015-0909-0 | pages=183–212|url=http://jamesgreenarmytage.com/strategy-utility.pdf}}</ref><ref name="Durand Mathieu Noirie 2014">{{cite web | last=Durand | first=François | last2=Mathieu | first2=Fabien | last3=Noirie | first3=Ludovic | title=Making most voting systems meet the Condorcet criterion reduces their manipulability | website=Inria | date=2014-06-17 | url=https://hal.inria.fr/hal-01009134 | access-date=2022-01-12}}</ref>

 

The Condorcet criterion and methods that pass it have been criticized for certain reasons. Some common arguments are:

 

 

 

 

 

 

 

 

 

 

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This criticism can easily be averted if the voters on either side of the Democrats simply refuse to rank the Democrats above the other side..

 

 

It is rather common for pollsters to do head-to-head matchup polling to see who is likely to win in an [[FPTP]] general election. Condorcet polling can be done in a similar way, except more efficiently, by allowing polled voters to rank or rate the candidates.<ref>{{Cite journal|last=Potthoff|first=Richard F.|date=2011-07-01|title=Condorcet Polling|url=https://doi.org/10.1007/s11127-010-9646-1|journal=Public Choice|language=en|volume=148|issue=1|pages=67–86|doi=10.1007/s11127-010-9646-1|issn=1573-7101}}</ref>

One strategy common to most Condorcet methods is to prevent a candidate from being a Condorcet winner by [[burying]] them (giving them a pairwise defeat against another candidate).

 

'''Alternative definition of "beat" that is claimed to be more consistent with the preferences, intent and wishes of equal-top-ranking voters:'''

 

[[Category:Voting system criteria]]

 

Analogues to the Condorcet criterion have been proposed in non-voting contexts; it appears in many places when discussing how to aggregate ranked information. It has been used to discuss search engine rankings <ref>{{Cite web|url=http://www10.org/cdrom/papers/577/|title=Rank Aggregation Methods for the Web|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref> and metasearch <ref>{{Cite web|url=http://www.ccs.neu.edu/home/jaa/IS4200.10X1/resources/condorcet.pdf|title=Condorcet Fusion for Improved Retrieval|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref>. The concept of a [[Smith set ranking]] (which is sometimes referred to as the "extended/generalised Condorcet criterion") helps structure how the ranking of all options should be, rather than only the winner.