What does a Condorcet cycle actually mean in the real world?

First of all, this is kind of a spin-off of a different thread, but seemed to make sense as its own thread. And this is actually pretty deep, at least to me. I’ve wondered about this for the over-15-years I’ve been obsessed with voting methods. So please indulge me as introduce the concept with a lighthearted little story, but which isn’t all that far from real life.

I have a daughter who is six, and she thinks being difficult is funny. (don’t worry, it’s all good. In exchange, I threaten to sell her for medical experiments. She thinks that’s funny too.)

So I ask her, “Would you like an apple or a banana for your snack?” She says “Definitely an apple!”

I then say “I’m sorry, I just remembered, we don’t have apples but I see we do have carrots. Would you like a banana or carrots for your snack?” She says “Banana! Duh!”

Then I say “Oh my gosh I’m so sorry I got confused, we actually do have apples, but we don’t have bananas. So I assume you want an apple, not carrots?” She says “No! Are you crazy, daddy? I want carrots!”

Now I realize we really have all three snack options. Which should I give her?

So basically we have a single person, who basically has a Condorcet cycle, so to speak. I asked her three questions, and she was able to answer each of them strongly and definitively. But if I asked her to rank the options in order, there is no way she could do so while remaining consistent with her pairwise choices.

So, we can write this off as her either a) changing her mind or b) just being difficult because she thinks it’s hilarious. In this case, I guess I can give her the carrots, and we go on with our day.

But what if we want to analyze whether it is actually reasonable for a single person to have the three simultaneous preferences of A > B, B > C, and C > A? Can you come up with any explanatory narrative that makes this a reasonable position to have? I’ll admit, I can’t.

Should we therefore simply conclude that I asked the question in a way that allowed for a nonsensical, self-contradictory answer?

Meanwhile, we all accept that in the real world with ranked ballots, with more than one person, we know that Condorcet cycles can actually happen. And when we do, it’s quite hard to say that any individual is contradicting themselves. Sort of like none of my daughter’s individual answers were irrational or illogical… it’s only when taken together they seem to contradict.

So, there are two possibilities. One is that, it is perfectly possible for a group of people to have the three simultaneous collective preferences of A > B, B > C, and C > A. Another is that this is a self-contradictory result, and is simply an artifact of the way we asked the question, but doesn’t really represent “truth” in a general sense.

I am personally leaning toward the latter. Especially if these preferences are strongly held, as opposed to being essentially a tie with a bit of roundoff error.

If you think the former is true (that it is actually valid and reasonable for a group of people to simultaneously and strongly prefer A to B, B to C, and C to A), can you come up with any sort of narrative which potentially explains, in a general sense beyond the specifics about how they filled out their ballots, the “truth” of that outcome?

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As for the individual person, the only way to solve this nonsense is to let the person solve it.
By asking the person to ranking (or score) the 3 foods at the same time, he will inevitably solve the contradiction in his own way (you cannot solve it in his place).

It depends on the cycle, consider the DV (or the IRV): adding the points at the beginning (with 3 candidates) you can get a result like this:
A[400] B[450] C[350]
In this context it can be valid for head-to-head that A>B>C>A but at the same time you also have total preferences.
Using total preferences, there are two ways to break the cycle:

  1. C is the worst candidate of all, with the least points (the one chosen the least), so the head-to-head where C>A is actually false. The winner is A.
  2. B is the candidate who has been chosen the most, with the most points, therefore, the head-to-head where A>B is actually false. The winner is B.

The problem becomes: in a context of maximum freedom (in which there are all the candidates to choose from) it’s more correct to say that:

  1. the one chosen least is the worst
  2. or the one chosen the most is the best.

It can be seen that the worst is C in both cases, while the best in case 1) is A, in case 2) it’s B.
The 1) is worth more often than the 2) so I think you are less wrong using 1).
Note that if there were only A and B, A would have won but with the addition of C (worst candidate) in 2), the victory would have gone to B. Methods that using 2) fail IWA, another reason why 1) is better in practice.

If the sum of the points were: A[400] B[400] C[400] with a Condorcet cycle, then we would find ourselves in a case of tie, and as such there is no way to solve it objectively correctly.
However, you can redo the election (hoping that some voters change their mind in the meantime), or choose randomly.

Personally, I like to think of a more “realistic” Condorcet cycle as three factions debating over two issues A and B, where one faction supports A but really hates B, another supports B but really hates A, and a third supports both. You get two cycles here, where AB > A > O > AB and AB > B > O > AB. (Or sometimes I do it the opposite way with passionate support and mild opposition.)

Another possible realization might be something like this:

  • Candidate A is an anti-establishment super-left candidate (e.g. Bernie Sanders)
  • Candidate B is a center-left establishment (e.g. Joe Biden, but maybe a bit more moderate)
  • Candidate C is an anti-establishment super-right candidate (e.g. Donald Trump)

If you position the voters in the right way and assume some voters care more about “blowing up the establishment” than left vs. right, you can probably get a Condorcet cycle.

I don’t see how that is inherently cyclic at all. And by saying “you get two cycles here” I’m not sure what you mean. Can we just talk about what real world situation would result in a single Condorcet cycle? Like A > B, B > C, C > A.

Trying to mix in multiple issues (A and B) in the way you are doing just seems oddly confusing. What does “AB > A > O > AB” mean? I can’t even tell what the choices presented to voters are.

I’m also a bit confused as to why there would be people who “support both” if everyone else is so polarized. Is there anyone who, for instance, supports A but is ambivalent toward B? It sounds like a very odd thing you are suggesting here, where there are two issues which are unnaturally correlated and that you aren’t quite sure if people are voting on them separately or combined into one.

Finally, are issues A and B both things that have zero middle ground? Say issue B is (drawing from current events) “should we remove public statues that honor racists and slavery proponents?” Aren’t there people who have nuanced views like “yes, we should remove some of them, maybe move them to museums, maybe add historical materials that place them in context, but don’t get carried away and remove statues of for instance George Washington just because he owned slaves but didn’t actively fight for slavery”?

Same goes for pretty much all issues. “should we defund/dismantle the police?” “should we reopen businesses?” “should we raise raise taxes on the wealthy?” “should we be tough on immigration?” All of these have middle ground positions. If you are voting for actual human candidates, it is possible for people with nuanced views to run for office.

But again, if you are trying to demonstrate things that are naturally cyclic, I don’t think you have done so, or at best it needs clarification as to why it makes a cycle. It seems more like you are conflating “confusing” with “cyclic.”

Regarding Sanders/Biden/Trump, I don’t see the cycle there either. You essentially give them two dimensions: anti-establishment vs establishment, and left vs. right. You can easily plot this in 2d space, and each voter would fall somewhere in that 2d space as well. Below is how I’d draw it, if A is Biden, B is Trump, and C is Sanders. Up is pro-establishment, down anti-establishment, and left and right are, well, left and right.

Everything else being equal (not that I think they are with those three choices! :slight_smile: ), the winner in my opinion should be the one closest to the median voter (represented by the black crosshair)

Regardless, I don’t see anything cyclic about that.

I should first ask you to be careful in your quoting, since you are making it look like I said the exact opposite of what I said.

You quote me as simply saying “it is perfectly possible for a group of people to have the three simultaneous collective preferences of A > B, B > C, and C > A.” However, I present that as one of two possible conclusions, and that I am leaning toward the opposite conclusion, which is:

So, onto the rest of what you say:

Of course. Or more straightforwardly, “we have apples carrots and bananas. pick one.” Easy to solve with an individual. Less easy to solve with multiple voters. (and not even 100% clear whether it is truly “nonsense” for a group of people to simultaneously think that A>B, B>C and C>A)

Beyond that, I’m not quite understanding how you are presenting something cyclic, other that simply being a 3 way tie. True ties aren’t as big a problem as cycles, because their likelihood drops as the number of voters increases. I think cycles probably also decrease in likelihood as number of voters increases, but not quite as dramatically as do actual ties.

Regardless, my hope is for something that is more of a narrative rather than just A, B and C and their numbers, or discussions of how things would resolve in particular methods such as DV or whatever.

As in, is there any real world situation you can imagine that could be seen as intrinsically cyclic? It’s ok if you can’t come up with one (I admit I can’t), but if such a thing exists, I’d love to hear about it.

Sorry, I didn’t understand you wanted to deal with the problem more generally.
The cycle is obtained (in reality) because of “relativity”, specifically, if I ask a person to vote for A,B,C his vote will be:
A[0] B[1] C[10]
while if I ask him to vote only between A and B, his vote will be:
A[0] B[10]
In logic (theory) we would like to have absolute values, but the “relativity” of people generates (in practice) this apparent contradiction, which must be managed in some way.

However, “relativity” is often linked to intelligent beings, who therefore find ways to solve this cycle (the ways are the “numbers” I told you about, in the previous comment).

The only real context in which this cycle would remain unresolved is a context of “stupid relativity”, and the only one that comes to mind is this:
An algorithm that is set to compare advertising in pairs, and having to choose which advertising is the best to show between A,B,C, could find itself in an infinite cycle of evaluations (A>B>C>A) in which it will never know which advertising is the best to show.

If I want to exaggerate I could also give you this example:

  1. For the seed it is better in order: to grow, bear fruit, die.
  2. For the already grown plant it’s better in order: to bear fruit, die, grow (it cannot grow more than that).
  3. For the plant that has produced fruit, it’s better in order: to die, grow, bear fruit (it can no longer grow or bear other fruits, therefore dying is the only thing that it “choose”). Dying, the fruits fall and release the seeds that make the cycle start again.

This cycle could be understood as a Condorcet cycle, I think :sweat_smile: .
I know that the plant can grow and bear fruit several times, mine is just an exaggerated example in which the plant, at different times (relativity), “wants” different things.

Hmmm, I’m not getting what you are saying here.

I mean I would understand how if you remove C, you now normalize so B gets increased to 10. But I’m not seeing a cycle.

So my hypothesis is that Condorcet cycles are not a reflection of any sort of general reality about the voters and their preferences, but are just a sampling glitch that can potentially happen in near-tie situations, especially with a small numbers of voters.

Another way of saying it is this. Say you have 10,000 voters, who use ranked ballots (*). You look at a randomly selected subset of the ballots, say 100 of them, and see that pairwise, A beats B, B beats C, and C beats A – i.e. a classic cycle.

That is a good indication that if you were to look at the other 9,900 ballots, A B and C will be in a tight race. Statistically that is true based on your sample of 100.

But it is NOT any indication that there will be a Condorcet cycle within those 9,900 ballots, and especially not the same Condorcet cycle. If there is one, it is simply because of the near three way tie, and it is just as likely to be, for instance, B beats A, C beats B and A beats C.

But also the greater number of ballots reduces the chances of a cycle. Because, again, the cycle isn’t representative of anything general about the voters’ preferences, so cycles should tend to be eliminated by the “Law of Large Numbers”.

Do you see any reason this wouldn’t be true?

  • let’s assume – just to keep things simple – that they aren’t going to any particular effort to be strategic, since they don’t know how others will vote, and aren’t savvy enough to know how to game a condorcet election.

The cycle arises from this problem of “relativity”; if instead of normalizing A[0] B[10], the vote remains A[0] B[1], then the cycle could not be born (but excluding “relativity” is wrong because it exists among the voters). Maybe I have dealt with the problem too broadly.

I think you are right but not for the reason indicated by you.
I think that the more the votes increase, the more the possible results increase and among these possible results, cycle cases are less and less frequent. But this is something I think, I don’t have a demonstration to support it.
One criticism could be: “candidates tend to want to support the interests of the voters, and this makes the candidates more similar to each other, increasing the possibility of having cycles”.
However, even by reducing the cycles, when they arise they must be resolved in some way.

I think we are kind of talking past each other or something.

With large numbers, things tend to converge on a target. Like, the more times you roll a die, the more the it tends to converge on 1/6th for each of the possible outcomes. 1/6th is the convergence target.

My hypothesis is that a cycle is never going to be a convergence target. A cycle is something that only happens because of slop due to low numbers of voters. To me this is a very significant theoretical insight if true, and has wide implications, even for approval voting, STAR voting, IRV etc.

You do, but the less and less likely they are, the less you have to worry about those ways of resolving them being perfect. (I mean, how much do people worry about ties in a political election? They are possible, but if you have large numbers of voters, they are very little concern)

The possibilities of cycles is also a big part of people’s concern about gameability of elections that meet the Condorcet criterion. Since there don’t seem to have been many large scale Condorcet elections in the real world, we don’t know for sure, but I’m saying this is way overstated.

I don’t know if you are missing why this is so important in a theoretical sense. I’ll admit I’m struggling to put where I’m going with this into words.

It may be true, but if I’m not mistaken, there have been some elections with IRV in which the cycle occurred, so you have to manage it.
It must also be considered that maybe the Condorcet cycles were also in elections that do not use IRV but no one noticed (in FPTP, AV, SV, etc there could be cycles without being able to see them).
Furthermore, if a voting method were optimal then it could also be used in “smaller” surveys (such as surveys on the internet, surveys in company departments, etc.) and in those cases the cycles can present themselves more easily.
The general problem is “2 or more people have to choose 1 alternative among many”, not necessarily thousands of people.

Choosing how to break the cycle generates different winners, that’s what makes it important (albeit rare).
However, you are right in saying that there are bigger problems to manage.

Yes you have to manage them, but I am suggesting they be managed more like ties, which we don’t worry too much about. You need to have some language in the law as to how to resolve them, but they are a practical issue rather than a theoretical issue.

Yes, this is getting at where I’m going with this. When you acknowledge that there can be things that are there but invisible, you are with me in looking at this in the abstract.

Approval voting, for instance, can sort of “paper over” such things because it has a certain randomness introduced by people having to arbitrarily establish thresholds. The hope is that these are essentially random decisions and, with enough voters, they tend to cancel each other out and it converges on the “correct” result. So, for instance, a Condorcet winner can actually exist in an approval election, but you can’t see it directly by looking at the ballots. It’s an abstract concept.

And my current thinking is that the cycles don’t exist when looking at it in the abstract. Even in a ranked “Condorcet type” election, there would still always be a true Condorcet winner, but it is hidden when there is a cycle.

This hope seems to me very unfounded. People (although many) aren’t random in the mathematical sense of the term.
A group of people who support a certain ideology (perhaps populist), may have overall different thresholds from a group that supports another ideology (non-populist).
To say that “tend to cancel each other” you need a fairly rigorous demonstration because if (and when) it were not true, you would get a very wrong result.

I don’t understand what “looking at it in the abstract” means.
The cycle can also exists in methods such as AV and SV but these methods use the process 2), which doesn’t consider the “relativity” of the voters. Hiding (not considering) “relativity” also hides the cycle, but this is wrong because it’s a fact that “relativity” exists.

Maybe I’m being charitable because we’re on an approval voting web site, and approval voting seems to be gaining traction, etc.

I honestly have my concerns about it, and really don’t like that to set your approval threshold meaningfully, you need to know how others are likely to vote.

Regardless, you can’t argue that approval provides less information than Score or ranked methods, right? When you approve A and B but not C, the system isn’t able to tell which of A and B you prefer – something that might end up being important. But luckily, some subset of voters will hopefully express that preference by picking A and not B (or vice versa). It may be that some of that subset has the exact same internal feelings about the candidates as you do, but they simply chose the approval threshold differently from you.

If I have three or four kids I am serving snacks to (apple/banana/carrots), and we do an approval vote, approval has a high likelihood of not collecting enough information, either resulting in a tie or simply getting a less than ideal result.

But if I do the approval vote with a whole classroom of kids, the chance of this being the case goes down. Approval’s coarse granularity can be compensated for, by having a larger number of voters.

Ok, well this is important so indulge me here. No point moving on until you understand that.

When you say “in FPTP, AV, SV, etc there could be cycles without being able to see them” I would say you are looking at it in the abstract. In other words, a cycle is a hypothetical concept, beyond just being something directly measurable by looking at ballots. As is the concept of a Condorcet winner (and various other concepts). We can talk about each in the abstract.

I had a similar conversation a while back (not about voting), and here is something I used to get this across. Look at the below two images. They both represent a figure that is black and white (no shades of gray) and has smooth curves. Both of them are imperfect representations, but they are imperfect in different ways. The one on the left is truly black and white (there are no gray shaded pixels), but the curves are not smooth because of the low pixel resolution. The one on the right has even lower pixel resolution, but it compensates with gray-shaded pixels to provide additional information as to the shape of the figure. There are advantages to each, and each contain about the same amount of information.

But we can speak abstractly about the actual, hypothetical figure (that we understand to have perfectly smooth curves and that every point is either black and white) independently from their practical representations.

If that makes sense to you, i can tie that back into what I was trying to get across.

you can’t argue that approval provides less information than Score or ranked methods, right?

Of course I can, if you ask me which food you want to eat between:
pizza, meat, pasta, broccoli, insects, shit
my threshold would make me choose at least “pizza, meat, pasta” but pizza is still better than meat or pasta and this important information is lost.
Even if I only support Pizza, I’m still saying that meat sucks me as much as shit, which is absolutely not true.
Approval in fact provides less information; the only thing that can save the AV is to show that:

Approval’s coarse granularity can be compensated for, by having a larger number of voters

but it does not seem to me that there is a demonstration to this.

It seems to me that you assume a random and uniform distribution of the voters in terms of interests, but in reality the distribution is far from uniform.
If in a state there are: 600 people who support A and 400 people who support B, even by increasing the number of people (children, future voters), the distribution would remain on average 60% A and 40% B.
If there are 3 groups of voters in a cycle, even if the number of voters increases, the % don’t change on average. To change the % you need some big changes in society (not so much in the amount of people).

But we can speak abstractly about the actual, hypothetical figure (that we understand to have perfectly smooth curves and that every point is either black and white) independently from their practical representations.

The problem in this context is that different voting methods in some cases generate very different forms (images), therefore there is no “actual, hypothetical figure” to refer to (it is not known even if it exists as unique).
What can be done is to find rules that are evident enough to be used to exclude figure, so that it is easier to understand which is the right hypothetical figure.

You misinterpreted me. I guess I should have said “you can’t argue against the fact that approval provides less information, right?” I am agreeing that approval provides less information, and the stuff I say following that should make that clear.

Well, then we aren’t going to have a productive conversation about it if we can’t agree on that basic point. When you say “different voting methods generate different forms”, that is true of the graphic representation of the letter S as well. (and you could add a third one that represents it as a polygon made of ~20 straight lines, for instance, which also has advantages and disadvantages). Each shows different aspects of the ideal figure, but they all ultimately converge on the same ideal figure.

Remember, the ideal figure in a voting context is not necessarily a “winner”. It is just the full preferences of the population. We try to capture enough of that to derive a winner, and different approaches can produce different winners. But there is still an abstract concept of the combined preferences of the electorate, independent of ballot type, independent of whether the voters are strategic, etc.

Again, I think you seem to be understanding this when you say that there can be cycles in approval but they are not visible. If there are cycles, they must exist somewhere. And so that we can have a reference point and this conversation doesn’t just go all over the place, I think we need to give them a home. That’s exactly what this “ideal figure” of the collective preferences of the electorate is.

Can we at least agree that Approval ballots, especially with a small number of voters (such as my example with 4 kids choosing a snack), is insufficiently capturing the full picture of what the voters actually prefer?

Can we agree that plurality voting is even less sufficient?

Can we agree that if you hold a vote in a club, even with very expressive ballots (such as Score with a range of 0 - 100), but due to practical considerations only 50% if the members are available to vote, that you are also insufficiently capturing the full picture of preferences?

Each shows different aspects of the ideal figure, but they all ultimately converge on the same ideal figure.

You are assuming that there is a perfect (ideal) result to which all methods converge (or try to converge).
What I am saying is that there are different categories of methods that point to different ideal results.
The fact that all the criteria cannot be met means that the ideal figure cannot be just one.

But there is still an abstract concept of the combined preferences of the electorate, independent of ballot type, independent of whether the voters are strategic, etc.

For me there is an abstract concept of absolute honest preference (range) of the individual voter, which if it were possible to know, would make SV the best method among all.
In concrete reality, however:

  1. There is no absolute range that voters can use to cast an absolute vote.
  2. Even if it existed, the voters would use tactical votes.

Practical reality makes it impossible to know the ideal result, which therefore is as if it did not exist.
Eventually, this concept can be used in simulations where first honest absolute votes are created, then they are converted into practical votes and one looks at which method achieves the ideal result several times (which is what I would like to do in a future project).

Can we at least agree that Approval ballots, especially with a small number of voters, is insufficiently capturing the full picture of what the voters actually prefer?
Can we agree that plurality voting is even less sufficient?
We can agree that if you hold a vote in a club, even with very expressive ballots (such as Score with a range of 0 - 100), but due to practical considerations only 50% if the members are available to vote, that you are also insufficiently capturing the full picture of preferences?

Yes.

The candidates take positions on the two issues and each combination of issues is represented by one candidate. So one candidate supports A but not B, one supports neither (O), etc.

You could think of these as a city trying to start 2 independent construction projects (e.g. build a freeway, build a light rail transit system) that will provide a greater public benefit but seriously inconvenience a small portion of the population.

A lot of times with real issues, there is one side or the other whose position is hard-core NO on some issue. E.g. Democrats have varying views on defunding/abolishing police but Republicans are almost entirely against all of it.

It is cyclic because a majority prefers Both to A Only, another majority prefers A Only to Nothing, and another majority prefers Nothing to Both.

There is no rule saying we have to be distributed normally around one “center”. I think today’s politics have several clusters. When you have multiple clusters it is possible for Condorcet cycles to be more common.
For example, take your voter crowd, scale it down by 80%, and put three copies: one to the southwest of A, one between B and C but closer to C, and one north of B (so they on average think B > A > C). Then you can in theory get a Condorcet cycle.

But all that really means is that there is no “majority winner”. If a majority of voters prefers A over B, a majority prefers B over C, and a majority prefers C over A, then those must all be different majorities. None of them actually comes out on top. That seems entirely self-consistent to me.

I know this diagram was related to a specific example, but generally, I don’t think it’s as simple as politicians and voters all fitting on a point on an n-dimensional graph where you can just find the median voter point.

Things like how much you trust a politician’s honesty and integrity etc. wouldn’t work like this. Every voter wants politicians who are honest, so you can’t map the voters like this. It’s down to their opinions on which of the politicians are honest.

Also, even in the n-dimensional graph scenario, voters have different priorities and would give different weights to different issues so even though they would have a position on the graph, their vote wouldn’t necessarily reflect that because their position on one or more of the axes might not affect how they vote.

I haven’t modelled any of this, but I’m guessing that it could mess up the median voter thing and potentially allow for the creation of cycles.

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I think it’s worth pointing out that some Condorcet cycles can be resolved if you allow voters to weaken their pairwise votes. If you’re using some form of

, then you might see some voters putting less than their full vote in some pairwise matchups. Think of it like this: if pairwise matchups are thought of as Score voting matchups (i.e. giving one vote to one of the candidates is equivalent to giving that candidate the max score and the other one a 0 in that matchup), and every voter scores the candidates “consistently” (i.e. a voter scores each candidate the same in each matchup), then there can never be a cycle, because the Score voting winner will always win. And if all but one of the voters scores consistently, then the Score voting winner can only suffer a pairwise defeat if that single voter alone can overturn one of their narrower pairwise victories by putting more pairwise power in the opposing direction in that matchup. I’ve written more on this at

Weakening pairwise votes can be done, to some extent, using equal-ranking. In fact, if a voter uses a probability to decide whether or not to express their pairwise preference between two candidates, and sets the probability equal to their “utility” in the matchup (i.e. they use a 30% probability of voting A>B if they would score A>B with a 30% score margin, and a 70% probability of equal-ranking otherwise), then the rated pairwise preference ballot can be emulated in that manner to some extent.

This issue has been discussed briefly on Wikipedia, with references to longer discussions:

and, with relevance to runoff-elections,

It’s interesting to me that both IRV (RCV) and STAR have an implicit two or multi-stage process, and are also implicitly resolving the paradox by adding another metric.

That’s why I like Approval Sorted Margins as a composite method – it satisifies IIA when a Condorcet winner exists, and when it doesn’t, the approval ranking is used and modified to the minimum extent possible to determine a pairwise consistent ordering.