[David Rosenthal]

There is something that can happen with ranked choice voting when there are three or more candidates and three or more voters. I learned it as the "voting paradox" or something.

Say there are three candidates A, B, C, and three voters, 1, 2, 3 and there preferences are as follows:

Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B

So A > B by two to one (Voters 1 & 3 vs. Voter 2).
And B > C by two to one (Voters 1 & 2 vs. Voter 3).
Therefore, A > C by majority preference.
But not so fast -- C > A by two to one (Voters 2 & 3 vs. Voter 1).

In other words, for the 3 voters as a whole A > B > C > A... It's a problem. Essentially, all three candidates have a similar claim.

This is not just an arcane philosophy problem. Imagine Voter 1 likes A & B about the same (a lot!) with a slight edge to A, but abhors C. And Voter 2 loves B, but really doesn't like either C or A, with A as a "lesser of two evils." And that Voter 3 doesn't really like any of them and considers it a coin toss. In that scenario, B would have the most popular support, but since the ranking system is flat and doesn't measure intensity, it isn't clear who would win, and if C won, the least popular candidate overall will have actually prevailed.

There is a calculation that can be done to predict the statistical probability of this actually happening depending on the number of choices and size of the electorate, and it is alarmingly high (something just shy of 10%, I seem to remember).

Just a thought to consider.

David R.

[Max Goodman]

Thanks for joining the discussion, David.

I'm not sure whether your argument is that, in the situation you describe, the result won't accurately reflect the voters' true desires, or that the election will not result in a winner.

Voters' second and third preferences exist whether they are allowed to name them or not. (And voters that really only like one candidate are not required to name other preferences in ranked choice. If they don't, their votes stop counting when their candidate is eliminated.) So it's impossible for the result to be less representative of the voters' true will because of ranked choice voting (though maybe it is possible for that fact to be clearer because those other preferences were expressed and recorded).

If the argument is that there might be no winner, maybe my math isn't up to the task, but it's hard to believe there's anywhere near a 10% chance of that happening if there are any reasonable number of voters. And I don't think ties are any more likely with ranked choice. Ranked choice, in fact, can break an initial tie that wouldn't be broken without it. In ranked choice, the tie would have to happen after all untied candidates had been eliminated. I don't see why that would be any more likely to happen in ranked choice than without it.

[David Rosenthal]

First, let me make clear that this is not my argument. It is a phenomenon some political scientists and social choice theorists have noted. Apparently, simply occurs some percent of the time where the conditions are met (3 or more voters and 3 or more options).

The point of it is that without mitigation of some sort (and perhaps still with it in some cases) there is some probability that the winner will not actually reflect to collective preference of the electorate.

I think there are ways of mitigating it, and they are often incorporated into models that are actually used -- allowing people to select fewer than the maximum number of preferences, scoring candidates to reflect intensity, "instant runoff" in which the selections are tiered into "rounds," requiring minimum percentages to move onto subsequent rounds, combinations of the or others, etc. I don't know to what extent these alterations reduce the probability of the paradoxical outcome.

Anyway, this leaves me at about my depth of understanding of this phenomenon. I am sure it would be easy to google about it and find out more. But, again, I am not making any kind of argument here, just sharing something I learned once.

David R.

[Matt Q]

Originally Posted by Max Goodman

If the argument is that there might be no winner, maybe my math isn't up to the task, but it's hard to believe there's anywhere near a 10% chance of that happening if there are any reasonable number of voters.

Lets assume that if we pick a voter at random, any order of preference is equally likely. There are three voters and three candidates.

It doesn't matter what order voter 1's preferences are. Just that voters 2&3 have placed each candidate differently from voter 1 and from each other.

Voter 2's first-pick preference will be different from voter 1's 2/3 of the time (because there are 3 candidates). Given that 2's first preference is different, his second & third choices will both also be different 1/2 of the time. (There are only two possibilities with voter 2's two remaining candidates)

So the probability that two voters will have preferences which place every candidate differently is: 1/3

Voter 3's first preference will be different from voters 1 and 2, 1/3 of the time. Given that 3's first preference is different, his second & third choices will also be different 1/2 the time.

So, the probability the third voter will have preferences which place each candidate differently (from the first two, who are also different from each other) is: 1/6

And the overall probability is 1/6 * 1/3 = 1/18 = 5.56%

But that's with three voters only. Do the odds decrease with more voters? I can't see how. Does it increase? My brain's too tired to work that out.

OK, so I cheated and googled: it increases, it tends towards a limit of 8.77%

[Max Goodman]

Thanks, David and Matt.

David, ranked choice voting can give a result that doesn't exactly reflect the voters' actual weighted preferences. The same is true of the current system (only voters don't leave proof of this when they vote).

Matt, I don't see how these probabilities of voters having different preferences correlate with probabilities of their being no clear winner. As far as I'm understanding the calculations, they show the probability that ranked choice voting will give a different result than the current system would. That's not something to be avoided; that's the whole point.

[Matt Q]

Hi Max,

What this shows is the probability of a paradoxical result. A result in which the majority preferences are:
candidate A over candidate B,
candidate B over canditate C, and
candidate C over candidate A.

Not that anyone individually necessarily has those preferences, but collectively, that's the result.

So who's won?

-Matt

[Max Goodman]

Ranked choice voting cannot create such a paradox of preferences; it can only reveal a paradox that would have existed anyway.

It is certainly true that ranked choice voting does not in all cases yield a result that is clearly the most representative of the voters' will. (In cases like the one you and David outline, no such result would be possible.)

I think I understand now that no one is offering this scenario as a reason to shun ranked-choice voting. That's where I was going wrong.

[David Rosenthal]

Originally Posted by Max Goodman

Thanks, David and Matt.
David, ranked choice voting can give a result that doesn't exactly reflect the voters' actual weighted preferences. The same is true of the current system (only voters don't leave proof of this when they vote).

Good point. I suppose the paradox holds with 3+ voters and 3+ choices whether it's ranked choice or good old fashioned pick one, there is just no evidence of ranked preferences in the latter beyond the single vote.

Meanwhile, for my part I can confirm that I did not bring this up to shun ranked choice. I just remembered it from when I was a philosophy student a long, looooooong time ago.

I think in general I am in favor of ranked choice to models. I think the "instant run-off" model makes most sense to me, though I admit I am not super well-versed in the various preferential voting models.

David R.

[Matt Q]

Max,

Yes, I definitely see significant problems with a first-past-the-post system. Because of this I'm also interested in understanding the strengths and weaknesses of various alternative systems. It's definitely a weakness of a system that it sometimes doesn't produce a winner. (There are ways to tweak it so it does, each of which have pros and cons). Plus I do also just find this situation interesting on a purely intellectual level (I like paradoxes and probability!).

-Matt

[Max Goodman]

Originally Posted by Matt Q

I'm also interested in understanding the strengths and weaknesses of various alternative systems. It's definitely a weakness of a system that it sometimes doesn't produce a winner.

Hi, Matt. We're on the same page regarding our goal here: understanding. Regarding the strengths and weakness of ranked choice, I don't think you and I are understanding each other. I may be misunderstanding one of your terms, but as far as I do understand:

A paradoxical result (posts 36 and 37) is not the same as failure to produce a winner (39), and neither of those outcomes is any more likely under ranked choice than under a vote-only-for-your-favorite-candidate system. (A paradoxical result may feel more likely under ranked choice, but only because that system can reveal a paradox that would otherwise have remained hidden. Ranked choice may therefore be more interesting to those intrigued by paradoxes and probability, but that's not a weakness.)

I hope the bolding doesn't feel like shouting. My post feels overlong, and I want to help those who don't care to read the whole thing.