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?