Why the Condorcet criterion is less important than it seems

by Alec Slatky // Published August 10, 2010

One of the criticisms made against instant runoff voting is that it doesn’t always elect the Condorcet winner – the candidate who theoretically would defeat all others in a one-on-one race. I will discuss this criticism through the lens of the instant runoff voting election for mayor in Burlington, Vermont.

Burlington’s 2009 Mayoral Election

Burlington (VT) is a three-party city, where the Progressive Party has established itself as an equal to the Democrats and Republicans in prominence and political power. In the 2009 mayoral election, there were three main candidates: Bob Kiss, the Progressive Party incumbent; Andy Montroll, a Democrat; and Kurt Wright, a Republican. There also were two additional candidates, including a strong independent candidate named Dan Smith who earned 15% of first choices.

The election utilized instant runoff voting. Once the two weakest candidates were eliminated, Kiss had 2,981 votes, Montroll had 2,554, and Wright led with 3,294. Montroll was thus eliminated. In the final pairing, Kiss defeated Wright by a margin of 4,313-4,061, keeping the incumbent Progressive in office. Although some Montroll supporters decided not to rank either Kiss (to Montroll’s left) or Wright (to Montroll’s right), a strong majority supported Kiss over Wright, vaulting him into the final round lead.

Supporters of instant runoff voting see this as a strong example of IRV at work. The frontrunner Wright had the most first choices and presumably would have won with plurality voting, but he lost when paired against his strongest opponent. Critics of instant runoff voting (IRV) focus on perceived problems with the election, however, and one of them is based on the fact that Montroll would have defeated both Kiss and Wright in a head-to-head election, yet was eliminated before the final round. More of Wright’s supporters backed the more centrist Montroll over the progressive Kiss and more of Kiss’s supporters backed Montroll over Wright. As a result, 4,067 voters overall preferred Montroll to Kiss, as opposed to 3,477 the other way around, and Montroll would have defeated Wright 4,597-3,668 in a two-candidate matchup.

The Condorcet criterion

Those are the basics in layman’s terms. Voting theorists would say that the Burlington mayoral race demonstrated that IRV fails the Condorcet criterion. It is named after the late 18th century French political scientist Marquis de Condorcet, who noticed a paradox in the aggregation of preferences through voting: even though any individual’s preferences are clearly defined, the collective ranking can be cyclic. In other words, in a 3-candidate race, it is possible for a majority of voters to prefer A to B, B to C, and C to A, making it hard to decide which candidate should be elected. Thus, there is not always a Condorcet winner – a candidate who defeats all others in pairwise elections.

The Marquis de Condorcet also noticed that even when there is a Condorcet winner, many voting systems do not necessarily elect that candidate. The basic plurality vote is one example, as was the method crafted by Condorcet’s rival, Jean-Charles de Borda – the Borda count, where voters rank candidates, and each ranking is worth a certain number of points. Other voting systems that similarly can fail to elect a Condorcet winner are instant runoff voting, approval voting, and range voting, and they are all said to fail the Condorcet criterion, which requires a voting system to always elect a Condorcet winner if there is one.

Condorcet proposed a new voting system to solve this problem, which today can be seen as a number of variants known as Condorcet methods. They differ in how they resolve ties in the absence of a Condorcet winner, but they all fulfill the Condorcet criterion. Such methods are not used by any government today, but nevertheless the Condorcet criterion continues to be one of the benchmarks by which a voting system is judged.

Why always electing the Condorcet Winner seems to make sense – but doesn’t

Nobody would dispute that in a two-person election, if more people like candidate A than candidate B, candidate A should win. It seems to follow, then, that in a three-person election, if more people like candidate A than candidate B, and more people like candidate A than candidate C, candidate A should win. The logic is simple enough: if you beat everybody, then you win. If the New Orleans Saints win every single football game on their schedule, they’re considered the best team, and the same standard should apply to elections. Indeed, if elections were held like football playoffs, then a Condorcet candidate would always win.

The problem is that in football, each game is inherently a one-on-one contest – not the case with elections. Voters must consider a full range of candidates before casting their ballot, and candidates compete for votes with more than one opponent. If there is a Condorcet winner, it means that he or she is preferred to every other candidate – not necessarily liked more than other candidates and not necessarily ready to represent the constituents.

Let me explain. If candidate A beats both B and C, that candidate is preferred to both of them by a majority of voters. That doesn’t mean that he is liked more than them, or even at all – very conceivably, it could mean that he is simply disliked less. Condorcet winners are centrist by nature, regardless of the preferences of the electorate. In modern political terms, they are embodied by conservative Democrats, liberal Republicans, and centrist independents like Joe Lieberman.

Consider an election with three candidates: a strong liberal who commands between 40% to 50% of the vote, a moderate with about 10% to 15%, and a strong conservative between 40% and 50%. By being everyone’s second choice, the moderate will certainly be the Condorcet winner as long as neither of the two more extreme candidates earns a majority of the vote. If the electorate is moderate, then great – the Condorcet winner makes sense. But if the electorate mostly wants something to the left or right of the center, is it still the case that the moderate should always win? Wouldn’t the 80% to 90% of voters who lean clearly to one side prefer that their candidate have a nonzero chance of winning, as opposed to the impossibility of victory under Condorcet methods?

Agreeing that the Condorcet criterion is desirable is equivalent to saying that moderate candidates should always win. But despite the hand-wringing over increasing partisanship and polarization, there are cases where more off-center candidates are deserving of election, no matter how much one might hate their policies. Any election system that favors extremists would be considered unreasonable; the same rationale must be applied to moderates.

Note that moderates can win with instant runoff voting – for example, Joe Lieberman won a near-majority in 2006 in his re-election for U.S. Senator after losing the Democratic Party primary, and would have cruised under IRV. But having them always win would not reflect the fact that sometimes a city, state or country wants leaders who want to transform the center, to move voters their way, like a Franklin Roosevelt or a Ronald Reagan.

Just because a candidate is hated the least doesn’t mean he or she is liked the most. Put another way, quite often a Condorcet winner might actually never have a chance to be win a one-on-one race against other candidates because that Condorcet candidate lacks enough support to keep other candidates from running. So choosing the centrist candidate every time is just falling into the fallacy of the middle ground. It’s tempting to always compromise between two distinct positions, but the compromise might be worse than either of them. We shouldn’t always revert to the least common denominator when electing our representatives, and neither should our electoral system.

Turning to Burlington

Looking to Burlington, Democrat Andy Montroll had a clear chance to defeat a relatively unpopular incumbent mayor to his left. His party had not been elected mayor for three decades, but still ran well in many elections in the city. The mayor was vulnerable, but Montroll only secured 22% of first choices and only 29% when the field was reduced to three, basically failing to make the case for his candidacy to enough people. If Montroll had won due to Condorcet voting being in place, the resulting controversy in Burlington would likely have been far louder than the outcry against Kiss’s IRV victory. Having a candidate win after being in last place when the field was reduced to three would have taken a lot of explaining to voters. They might have accepted the results; more likely, they would have challenged them, particularly if they understood that Democrats would suddenly be the dominant party in mayor’s race even when failing to finish in the top two.

Conclusion

Though I framed failing the Condorcet criterion as a criticism of IRV, advocates of other voting systems have been faced with the same accusations. While the Condorcet winner usually deserves to get elected, there are cases to the contrary. Thus this particular voting system criterion is far less important in discussions of single-winner methods than one might assume.

Comments currently closed for Why the Condorcet criterion is less important than it seems

  • nice blog
    Posted by cheap vps, 2010-11-02 22:41:49 (3 years ago)
  • For some voting tasks, a very central winner is usually appropriate. The chairperson for a diverse council is my favorite example. If the other members are elected by a proportional method, this chairperson will hold the swing vote. The others can form a majority without the chair, but that is not as likely. The Condorcet winner must try to please a wider constituency than an STV or party-list winner, and this too is a good motivation for the swing vote or chairperson to have. But even in this role, a Condorcet winner might be too inflexibly central. Perhaps a political system needs to weave left-right-left-right, to test policies, taxes and budgets. The system should not, I think, swerve left and right. And it should not be paralyzed by polarization. It should have a strong center as well as diverse voices. So as a rule, the Condorcet winner would be the best candidate to chair a diverse council.
    Posted by Robert Loring, 2010-09-03 13:11:02 (3 years ago)
  • There has not been much mention of the fact that Burlington reverted to an electoral system which would have elected the Condorcet loser among the three leading candidates. Hardly an impovement. Also, with a Condorcet system you can hurt the chances of your favorite candidate by giving a high rank to a consensual candidate, so there would be an impetus to vote for loonies and extremists as second and third choices. No system is perfect, but IRV seems to encourage more ``sincere voting'' in voters. Also note that Wright could have improved his chances by convincing enough centrist voters that he was a better second choice, so IRV also encourages conciliation among candidates.
    Posted by Claude Tardif, 2010-08-24 13:41:16 (3 years ago)

  • Actually this is one of the rare cases when I'll agree with FairVote. You can actually mathematically prove that the Condorcet winner is not necessarily the best candidate (no matter how you define "best", although I contend the uniquely right definition is "having the highest social utility").

    http://www.electology.org/criteria/majority
    Posted by Clay Shentrup, 2010-08-13 00:04:41 (3 years ago)
  • Paul Cuff,

    Score Voting has a better property: that even when there isn't a Condorcet candidate, removing an irrelevant alternative will never change the outcome.
    And it is simpler than Condorcet or IRV.
    And it seems to get as good or better Bayesian regret.
    Posted by Clay Shentrup, 2010-08-13 00:00:45 (3 years ago)
  • This is not the first time I've heard it argued that IRV's repeal was driven purely by Wright/Republican voters. But the data does not support that conclusion. Unfortunately, I cannot find the by-ward breakdown for the original vote, but when compared to the March 2nd 2010 election results, you can see that support for IRV dropped in all wards, and not disproportionately so in the Republican-favoring 4th and 7th. A great deal of Democratic voters also found something to vote against. (The best choice under plurality would perhaps have been for Kiss and Montroll to agree to a primary; hasn't Bernie Sanders done this when running for US congress?) Regardless; what you say is true: in a three-party city, elections are hard.)

    On single-peak utilities:

    Regardless of how you feel or what you believe a candidate "deserves", if the utility scores you provided reflect actual intrinsic utilities, then electing B is in-arguably the socially optimal choice. If you want to argue that the social optimum some how isn't the best choice (many argue utilitarianism is fundamentally flawed, but you seemed to speak in favor of it here), or that no election actually involves single-peak voters choosing among a uni-dimensional spectrum of choices (already granted), then you are welcome to do so. But given those stated assumptions which you seem to have agreed to for the purposes of this (sub-)argument, and the data that you have provided, your conclusion is wrong.

    That IRV fails to elect a Condorcet winner even under these idealized circumstances, I feel, is a problem; if you do not, I would love for you to explain how (perhaps by showing improved IRV performance in more "realistic" situations; although all the more realistic examples of which I am aware (naturally, all computer simulations, since intrinsic utility is unmeasurable in reality) show IRV's relative performance to scale at about the same rate as any voting method under such adverse environments; so you would have your work cut out for you.)

    That confusing sentence:

    Ah; so we're talking about difficulties of ballot access, and of an excellent, but little-known candidate attracting enough attention so that voters can become aware of what a good choice they would be? But I don't see how this is particular to Condorcet winners; ANY candidate under ANY electoral system could have these problems, could they not?

    Posted by Dale Sheldon-Hess, 2010-08-12 18:00:22 (3 years ago)

  • @ Dale:

    Thanks for commenting on another one of my posts. I’ll try to respond to your points one at a time.

    One:

    I don’t think voting would have changed much under plurality. One of two things would have had to happen for a changed result. Either the Wright voters would have had to switch to Montroll, which seems unlikely given the traditional gap between Democrats and Republicans, or the Montroll and Kiss voters would have had to coordinate on a candidate to support, another unlikely scenario given that Kiss was the incumbent but disliked by many Democrats. Plurality would almost certainly have elected Wright, and saying otherwise ignores what would be a massive collective action problem in a three-party city.

    Two:

    That’s not at all what I did. As I said, I used Burlington as a “lens” because this criticism is so common when discussing that race. To illustrate my point, I could have made up an example election, but instead I used a real one, and there would be no difference in how I approached the issue.

    Three:

    Yes, I am essentially using a one-dimensional spectrum with single-peaked preferences, which is not perfect – about 15% of Kiss voters preferred Wright to Montroll, for example – but it’s sufficient to get my point across. The point is that beats-all winners can be nobody’s first choice, and so they shouldn’t be elected.

    I disagree with your claim that under these assumptions, the moderate and thus Condorcet winner always deserves to win. Consider a situation where candidate A and his 47 supporters are at -1 on a unidimensional spectrum, candidate B is at 0, with 10 supporters, and candidate C is at 1, with 43 supporters. If we use distance from the winning candidate as a metric for negative utility, candidate A would earn 96 in negative utility, B would have 90, and C would have 104. Based on your claim, candidate B should win. I would say candidate A should win (assuming the B voters are split 50-50 between A and C as second choices). Candidate B is essentially the least common denominator candidate: liked by few, hated by few. That’s not someone I want in office.

    Four:

    You’re right, this sentence was confusing. What I meant to say is that someone who would be a Condorcet winner if he ran, might not even get the chance to run because other candidates are more popular and can get ballot access, whereas the would-be Condorcet winner might not be popular enough to get on the ballot.

    Five:

    Admittedly, my conclusion that an election of Montroll would cause more controversy than Kiss’s victory did is an educated guess. As Peter said, a large part of the movement to repeal IRV was grounded not in any theoretical concerns, but the fact that Wright lost. The existence of a possible strategy that could have benefited some voters played some role, but my hunch is that any other alternative voting system would have been repealed for exactly the same reasons.

    @ Peter

    Thanks for commenting. And I agree that the main reason for IRV’s repeal was that the plurality winner, the winner that people are used to, lost. I can’t say for sure, but my gut tells me that Democratic voters, though perhaps not Democratic leaders, were also upset by the election outcome. If you have any insight into this assumption, I’d be very interested in hearing it. Because my hunch is that if Montroll were elected under a Condorcet method, Kiss supporters would be equally upset and the election system would go back to plurality.

    @ Charles

    Thanks for commenting. First, I admit I was unclear, but I absolutely understand what the Condorcet criterion is. What I meant to say is that a candidate who would be a Condorcet winner if he ran might not even get a chance to run due to lack of support. Essentially, potential Condorcet winners could have such weak support that they decide not to enter the race, or cannot due to ballot restrictions.

    Again, you’re right, in that “median” could be substituted for “moderate” in this post, and it would mean the same thing. I used the latter term because it’s relevant to American politics today – Condorcet winners in elections where the two other candidates are polarized would generally be moderate. Moderate doesn’t imply complete centrism, but positions that are more towards the middle than those of their peers. We’re really just splitting hairs here.

    I take umbrage with the accusation that I don’t understand the social choice theory I’m discussing. This isn’t really a justification for election of extreme candidates, but rather a justification for not electing more moderate candidates in every single election. My personal political views (I’m not registered with either the Democrats or Republicans) would generally align with more moderate representatives, but I think that there are cases where extreme candidates (no matter how much I dislike them) deserve to win. If that to you is a “justification for extreme candidates winning elections,” then so be it. All I’m saying is that sometimes, moderates don’t deserve to win; no more, no less.

    @ Prof. Cuff

    Thanks for commenting. I go to Princeton now, which is why I addressed you as “Professor” rather than “Paul” – no matter how many times some professors insist that I call them by their first name, it’s feels awkward to me.

    You make a compelling point. Only the Condorcet methods truly satisfy your criterion (range does also, technically, but if a non-winner is removed from the pool then voter strategy will surely differ). But I think a more serious flaw is a proclivity to elect the least common denominator candidate, as I discussed. And in cases where A beats B only if C is in the race, I don’t find that characteristic abhorrent (though it is unfortunate) if C reflects a significant portion of the population – candidates should ideally reflect the electorate, and election systems should be based off of that fact. If nobody really likes B, then he doesn’t deserve to be elected.

    I have two issues with your claim that Condorcet winners would be elected if voters had perfect information. For situations where that is true, it still doesn’t mean that the Condorcet winner is liked, only that voters are trying to avoid electing their least favorite option; it doesn’t change which candidate really deserves to win, because those votes are strategic, and voting systems should encourage honest voting. Secondly, there are certainly cases where your statement is untrue, with the stipulation that the same candidates are running. For example, in Burlington, do you think that enough Wright voters would switch to voting for Montroll, even though Wright garnered much more initial support? It’s possible but by no means certain – voters are not always rational. Under range or approval voting, tactical voting could likewise cause voters to only approve of their top choice, even after the preferences of society are revealed. I agree that your contention is generally correct, but as the vote totals for 3 candidates get closer, the likelihood that the Condorcet winner will not be elected – even with perfect information – increases.

    Posted by Alec Slatky, 2010-08-12 14:54:48 (3 years ago)
  • A property of the Condorcet criterion not often mentioned is that the Condorcet winner (when one exists) is the unique winner if you demand that removing non-winners from the pool doesn't change the outcome. This is very compelling. How can we make a convincing argument for a system where A wins the election over B only if C is also in the race? Also, pretty much any voting system would conform to the Condorcet criterion if you allowed voters to recast their votes in rounds with full disclosure of the preferences of society.
    Posted by Paul Cuff, 2010-08-12 10:04:42 (3 years ago)
  • Two fundamental problems with the essay. The clearest is that the third-to-last paragraph demonstrates that Alec doesn't understand what a Condorcet winner is. Because the Condorcet winner is defined as someone who beats every other candidate on a one-to-one pairing, it makes no sense to say that a Condorcet winner might never have a chance to win a one-one-one race. (That's either a logical contradiction, or an indication that Alec is mixing up theory and practice in voting systems.) The other problem is that this essay confuses the "moderate" position and the "median" position. The example he uses is a unidimensional, left-right case, in which the median is always the Condorcet winner. The MEDIAN might espouse liberal, moderate, or conservative views, but the median is always the median. At the end of the day, it seems that Alec wants to build a justification for extreme candidates winning elections. If that's what you want, make that argument; don't confuse it with social choice theory you don't understand.
    Posted by Charles, 2010-08-12 07:06:37 (3 years ago)
  • Dale has a lot to say here. But he sure doesn't know much about Burlington'.: The issue of Montroll being a Condorcet winner did not have the slightest impact on the repeal debate - in fact Montroll and nearly all Democratic elected officials in the city supported keeping IRV. The repeal was about Wright not having won despite leading in first choices and the mayor getting really unpopular after the election due to a scandal. No one thought of Wright as a spoiler. Heck, he almost won! Put it another way. If Wright had won, with Montroll still being the Condorcet winner, there wouldn't have been a peep about repeal.
    Posted by Peter, 2010-08-11 17:41:58 (3 years ago)
  • Right; html formatting. Sorry...

    One:

    Wright would (almost) certainly not had won if the election had been held under plurality. It is likely that voters would have realized the potential of a spoiled election, and voted differently.

    The problem is, the Wright voters--due perhaps to unfamiliarity with IRV and the false-claims by IRV proponents that such a thing was impossible--failed to notice that Wright could still be a spoiler under IRV.

    Two:

    Citing Kiss' victory as an example of why the Condorcet criterion isn't important, and therefore, IRV's failure to support it is irrelevant, is circular logic: you begin by asserting electing Kiss was right because that's what IRV did, and then use it to prove that IRV is good because it did the right thing.

    Three:

    If the moderate choice truly is a bad choice (which I acknowledge it often is) than voters, if they are rational, will rank it below both of the extreme choices (or at least not rank it higher than their less-preferred of two extremes).

    The difficulty comes in that, under some ranked-order voting systems, it may seem tactically advantageous for a voter to dis-ingenuously rank a known-to-be-terrible choice below a personally-disliked-but-popular choice, as a way of defeating what they perceive as their true-favorite's greatest competition. Such naive tactical exaggeration could cause a known-bad, but "moderate", choice to win the vote. Discouraging such tactics would be a good design goal for any voting system.

    But it is wrong to assume that the moderate choice is always the Condorcet winner; this is true only if all choices can be placed along a spectrum, and all voters have a single-peaked preference function. And in that case, the moderate choice is not only the Condorcet winner, it is the socially-optimum choice; so electing it would certainly be good! (Note: this case is one of the few places where utilitarianism has actually been proven to work as a valid social-choice metric.)

    Four:

    "Put another way, quite often a Condorcet winner might actually never have a chance to be win a one-on-one race against other candidates because that Condorcet candidate lacks enough support to keep other candidates from running."

    I have no idea what you're trying to say with this sentence.

    Finally:

    I disagree with your conclusion that a Montroll victory would have inspired significantly more public gripping. There simple isn't going to be a lot of difference between having 71% of the voters wanting someone else to have won, and 77% of voters wanting someone else to have won. With three strong candidates, there will always be a lot of pissed off voters in the end; the question is, will they be upset because they lost, or because they felt cheated, lied to, and betrayed by the system?

    What we can know for certain is that the voters of Burlington found it difficult to believe that, if only more of them had voted for Kiss, Kiss would not have won. And I think THAT was harder to explain than electing a strong compromise candidate would have been. Which is probably why they voted to reject IRV.

    Posted by Dale Sheldon-Hess, 2010-08-11 15:30:49 (3 years ago)
  • One: Wright would (almost) certainly not had won if the election had been held under plurality. It is likely that voters would have realized the potential of a spoiled election, and voted differently. The problem is, the Wright voters--due perhaps to unfamiliarity with IRV and the false-claims by IRV proponents that such a thing was impossible--failed to notice that Wright could still be a spoiler under IRV. Two: Citing Kiss' victory as an example of why the Condorcet criterion isn't important, and therefore, IRV's failure to support it is irrelevant, is circular logic: you begin by asserting electing Kiss was right because that's what IRV did, and then use it to prove that IRV is good because it did the right thing. Three: If the moderate choice truly is a bad choice (which I acknowledge it often is) than voters, if they are rational, will rank it below both of the extreme choices (or at least not rank it higher than their less-preferred of two extremes). The difficulty comes in that, under some ranked-order voting systems, it may seem tactically advantageous for a voter to dis-ingenuously rank a known-to-be-terrible choice below a personally-disliked-but-popular choice, as a way of defeating what they perceive as their true-favorite's greatest competition. Such naive tactical exaggeration could cause a known-bad, but "moderate", choice to win the vote. Discouraging such tactics would be a good design goal for any voting system. But it is wrong to assume that the moderate choice is always the Condorcet winner; this is true only if all choices can be placed along a spectrum, and all voters have a single-peaked preference function. And in that case, the moderate choice is not only the Condorcet winner, it is the socially-optimum choice; so electing it would certainly be good! (Note: this case is one of the few places where utilitarianism has actually been proven to work as a valid social-choice metric.) Four: "Put another way, quite often a Condorcet winner might actually never have a chance to be win a one-on-one race against other candidates because that Condorcet candidate lacks enough support to keep other candidates from running." I have no idea what you're trying to say with this sentence. Finally: I disagree with your conclusion that a Montroll victory would have inspired significantly more public gripping. There simple isn't going to be a lot of difference between having 71% of the voters wanting someone else to have won, and 77% of voters wanting someone else to have won. With three strong candidates, there will always be a lot of pissed off voters in the end; the question is, will they be upset because they lost, or because they felt cheated, lied to, and betrayed by the system? What we can know for certain is that the voters of Burlington found it difficult to believe that, if only more of them had voted for Kiss, Kiss would not have won. And I think THAT was harder to explain than electing a strong compromise candidate would have been. Which is probably why they voted to reject IRV.
    Posted by Dale Sheldon-Hess, 2010-08-11 15:29:45 (3 years ago)