The Condorcet Criterion

[cfrank]

Since I started learning about voting theory I've struggled with accepting the Condorcet criterion. I think my position on it has been tempered a bit and softened, but I still have some criticisms I wanted to express.

In my opinion one of the strongest argument for the Condorcet criterion is very simple and as follows: It should not be that the winner of an election should have any other candidate preferred over them by a majority in a head-to-head election between the two. This is totally practical, since in reality a large majority could perhaps in principle simply overthrow the election results, for example. Unfortunately this is not possible to satisfy in general, as demonstrated by the Condorcet paradox. Hence, the condition is qualified to be upheld `"whenever possible," and one arrives at the Condorcet criterion. If one applies Condorcet's Jury Theorem, and assumes that in any head-to-head election there is an objectively ``better" choice, and simultaneously that each voter is more likely than not to indicate that choice as their preference in each head-to-head match, then we concoct some level of information-theoretic corroboration for the criterion.

There are issues with this argument, however, and also with the conditions for applying the Jury Theorem. Most substantially, and as recognized by Condorcet himself, the concept of a "better" choice is inherently subjective---that is, each voter has a different conception of what might constitute a good choice, hence the notion of a socially shared objective ideal is questionable at best. Even in that case, it does not seem theoretically appropriate to treat voters simply as independent, identically distributed Bernoulli random variables. More or less, this is equivalent to saying that each voter is already biased to make the ``correct" choice, or that already more voters than not will almost certainly make that choice. By that reasoning the result of the theorem is almost circular. At the same time there is no possible empirical support for the conditions supporting its conclusion in the context where individual preferences rather than the subjective identifications of a single socially shared objective is being indicated.

There are valid applications of the Jury Theorem: for example, in a jury. If one assumes that the jurors are competent, and that each has a higher chance than not of identifying the guilt of a defendant, then majority rule of the jury is likely to produce the correct verdict. The issue with extending the scope of the Jury Theorem is that, at least in theory, the guilt versus innocence of a defendant is an objective circumstance with the potential of being identified, whereas the ``best candidate" in an election simply is not, because it is by nature an opinion.

My second issue with the Condorcet criterion is the rather obvious observation that the majorities that elect the winners of each pairwise face-off can differ from each other substantially. This may not seem important on its face, but in fact, this is the root cause of the Condorcet paradox. A Condorcet cycle must have links that are supported by distinct majorities, one crowd shouting something here and another crowd shouting something else there. The social choice theorist listening to these different voices as though they come from one mouth will naturally be twisted into confused circles. It is true that there is necessarily an overlap between any two majorities, but there needn't be any overlap at all shared by any three of them. As such, while the premise of a Condorcet system seems positively-motivated, a Condorcet winner appears to be a figure that rises from the ashen remains of a chaotic competition, not necessarily a figure that represents a broad consensus of the electorate, although this may be the case.

The third issue I take with the Condorcet criterion is that the pairwise face-offs operate fundamentally within a majoritarian paradigm. May's Theorem only applies to situations where each voter can make only one of three indications---the first candidate, the second candidate, or neutral. This scope does not encompass general range voting systems, for example, so that it is not generally clear that a majority rules principle is the optimal method of social choice even between just two alternatives. The standard thought experiment might be called the ``pizza topping" problem, where three friends wish to contribute equally to purchase a single-topping pizza, but two friends prefer a topping that the third is allergic to. A majority rule decision is simply anti-social in this instance.

My final issue is that rank order systems put a large burden on voters, and it may just be impractical to expect each voter to indicate a full ranking of candidates unless the candidate pool is fairly small.

Anyway, I know that people like the Condorcet criterion, and I see its merits as well. I just wanted to know whether supporters (more or less) of it would be able to address those points in its defense, or raise any other criticisms that might point to better alternatives.

Thank you!

[Jack Waugh]

@cfrank said in The Condorcet Criterion:

assumes that in any head-to-head election there is an objectively ``better" choice

This is consistent with the fact that the pioneer of systems that we would today characterize as Condorcet-compliant was a Catholic apologist. Catholics and more broadly Christians and even more broadly Abrahamics believe in one god, whom they write about as though his name were God. Llull thought that when the leader of a monastery or a convent died, there would be one best candidate to replace that leader, and that God knew who that would be, and the monks or nuns would just have to ask each other for opinions and aggregate those properly to fix on the correct answer.

[rob]

@cfrank said in The Condorcet Criterion:

As such, while the premise of a Condorcet system seems positively-motivated, a Condorcet winner appears to be a figure that rises from the ashen remains of a chaotic competition, not necessarily a figure that represents a broad consensus of the electorate, although this may be the case.

This seems less of a complaint about Condorcet methods, than a complaint about reality.

Reality can be rather messy. If anyone is advertising Condorcet as being able to magically fix that, they are overselling it.

"Broad consensus" doesn't mean everyone agrees with the result or is even ok with it. Or, if it does, it simply isn't achievable by a voting system. Condorcet might reasonably claim to find the "best compromise." You might even claim it tends to elect the "first choice of the median voter." There are other ways of wording it, but "broad consensus" is either overly vague, or simply inachievable.

My second issue with the Condorcet criterion is the rather obvious observation that the majorities that elect the winners of each pairwise face-off can differ from each other substantially.

This is one reason I don't the term "majoritarian" as applying to Condorcet methods. There is a majority from each pairwise comparison, but the final result is not a majority of anything, it is something else entirely.

So I guess I don't get this complaint. You have suggested that you don't like things that are majoritarian, but here you see that it "differs substantially" from having a singular majority that wins. I would expect you to see that as a positive.

The standard thought experiment might be called the ``pizza topping" problem, where three friends wish to contribute equally to purchase a single-topping pizza, but two friends prefer a topping that the third is allergic to. A majority rule decision is simply anti-social in this instance.

To me, this is outside the scope of what a voting system should do (although maybe something like the quadratic voting might attempt to deal with things like that). The solution to that sort of thing, to me, is representative democracy, not a better voting system.

Say you've got an organization where one elected office is in charge of the menu, including choosing pizza toppings. Some candidates' platforms might include "don't choose foods that some people are allergic to", while other candidates may be have a "sucks to be you" attitude toward the food-allergic. But this isn't the only issue that people care about. With a reasonable voting system, people can account for all of their concerns when they choose who they vote for. So a person with allergies might prioritize that issue highly (and presumably at the expense of other food preferences they may have) and those without won't care so much, so they won't necessarily vote against the candidates that account for food allergies.

So, a reasonable solution is still arrived at that may well mean "no foods that a significant subset are allergic to." Even though the people who care about that issue are the minority, since they care very strongly, they can still get their way.

Any voting system that otherwise tries to capture strength of preference (which is basically what you are asking for) is going to, by nature, not be "one person one vote." I can't get behind a system that says "everyone's vote is equal, unless you can demonstrate that your particular issue (e.g. food allergy) must override everyone else's preferences." Or that you expect voters to "be nice" rather than vote their interests....that tends to breed resentment and doesn't work well beyond very small groups, such as where people who understand that their own needs might impose on others will try to make it up by being flexible on the next thing. (i.e. "since they were cool about not making me order a separate pizza without mushrooms, I'm just going to go with what everyone else wants when it comes to choosing which beer to buy a pitcher of")

My final issue is that rank order systems put a large burden on voters, and it may just be impractical to expect each voter to indicate a full ranking of candidates unless the candidate pool is fairly small.

I can agree to an extent, but I don't think we should assume that any Condorcet system requires voters to rank all candidates. You could even present a ballot interface identical to STAR voting. (which is actually my preferred type of ballot for Condorcet elections) Even on a ranked ballot, I think it is fine if voters just don't bother ranking candidates that they think have no chance of being a front runner (or maybe I should say, would have no chance even if voters were forced to rank all candidates).

I can think of one Condorcet system which accounts for this even more, which I have called "Delegated Condorcet." I described it here 10 years ago back when I was rather gung-ho on it:
https://www.quora.com/If-America-moved-away-from-the-electoral-college-and-the-first-past-the-post-voting-system-what-system-should-it-move-to-and-why/answer/Rob-Brown-13
and I mention it briefly here but also address some of the stuff above regarding how representative democracy addresses tyranny of the majority:
https://www.quora.com/What-would-the-best-form-of-democracy-be-like/answer/Rob-Brown-13

I could say more on how I look at Condorcet methods and why I think they are superior to others (or, why the Condorcet criterion is important, or why methods that tend to elect Condorcet candidates are better than those that don't), but I'll leave it at that for now.

[cfrank]

@rob I revisited this and I think Delegated Condorcet is a really interesting idea, what are your latest thoughts about that concept?

I am also reading this paper to try to get a balanced view of the Condorcet criterion:
https://www.journals.uchicago.edu/doi/full/10.1086/682019#

[rob]

@cfrank said in The Condorcet Criterion:

@rob I revisited this and I think Delegated Condorcet is a really interesting idea, what are your latest thoughts about that concept?

I haven't given it a lot of thought in the years since. It seemed like a good idea, and probably would work pretty well if adopted, but I doubt people would trust it enough to adopt it in the first place.

It might be best for local elections where there isn't a huge amount of publicity and the public tends to only know one or two candidates out of a larger field.