Don’t Be So Arrow-Minded

Arrow’s Theorem states that no voting procedure can satisfy a certain set of “fair” axioms. The rhetoric goes as follows:

The procedure should deterministically translate an input consisting of everyone’s preference orderings, to an output, which we take as a collective preference ordering. Pretty reasonable, right?

The procedure should not just copy a single voter’s preference ordering. Pretty reasonable, right?

If everyone prefers A to B, the procedure should prefer A to B. Pretty reasonable, right?

If we add a new candidate, specifying for each voter where that candidate stands in the preference ordering, then this should not affect the relative order of the other candidates. Pretty reasonable, right?

Not right! Not right!

This last axiom is called “independence of irrelevant alternatives” (IIA). And indeed, the alternatives themselves are irrelevant. If you change me from a water-lover to someone indifferent about water, but keep my attitudes toward tea and coffee the same, that shouldn’t change whether the collective prefers tea to coffee.

But if, knowing that I prefer tea to coffee, you learn that I prefer tea to water and water to coffee, this still informs you about the strength of my preference for tea. And it’s only reasonable for this preference strength to make the collective prefer tea in close situations.

Whether Fred has the same DNA as the serial killer the police is looking for is not relevant in the sense that if you changed Fred’s DNA, you’d change Fred’s guilt. But it is relevant in the sense that learning about Fred’s DNA should change your opinion of his guilt. The second “I” in “IIA” denies only the first kind of relevance.

If IIA doesn’t matter, then it’s not clear Arrow’s theorem says anything practically interesting. It’s true that a perfect voting system is impossible. But that’s because there are always incentives to lie about preferences, leading to strategic voting. The Gibbard-Satterthwaite theorem says this applies to all non-dictatorial deterministic systems.

But now it turns out the two theorems are actually the same thing! Huh. I don’t understand this on an intuitive level. It undermines the whole reason for me writing this, but I’ll post anyway.

(PS: Some of you may claim all talk of the “strength” of preferences is nonsense. This is nuts. Arrow allows you to communicate only the ordering, but in the real world preferences clearly do have different strengths; if necessary you can give them an operational definition by looking at preferences over lotteries, or by comparing to other questions that you’re not voting on. This viewpoint doesn’t even allow you to say that if you prefer A > B > C, your preference A > C is stronger than A > B. And if preferences don’t have strengths, then it’s not the case that adding irrelevant alternatives leaves preference strengths unchanged.)

(PPS: To the extent that people’s true interests aren’t what they think, the problem becomes one of finding the right answer and not so much one of fairness.)

5 thoughts on “Don’t Be So Arrow-Minded

  1. And if preferences don’t have strengths, then it’s not the case that adding irrelevant alternatives leaves preference strengths unchanged.

    That doesn’t make sense. If preferences didn’t have strengths, it would be meaningless to ask whether preference strengths were changed or not after adding additional information.

    Preferences do have strengths, and offering irrelevant choices can elicit information about those strengths. I think that’s what you meant to say.

  2. What I meant was, I think people intuitively think that preference strengths remain unchanged when irrelevant alternatives are added, but if there are no preference strengths then it’s not the case that this is meaningful and true.

  3. Pick a voter at random and have their vote be the final choice
    From the fairness point of view, this is perfect in a way. However, it’s bad from the “finding the right answer” point of view, as it’s maximally vulnerable to random noise. Maybe there’s some sort of compromise.

  4. Pick a voter at random and have their vote be the final choice

    That is part of what Hay Voting does. Hay Voting also uses a quadratic voting-mass transfer scheme to elicit preferences. It is a truth-revealing protocol, on the assumption that the voter wants particular candidates to win.
    It’s by Peter de Blanc and Marcello Herreshoff.

    The compromise Steven seeks may be realized in Multistage Hay Voting. I invented it after seeing Peter de Blanc’s writeup of Hay Voting. Basically, you repeat Hay Voting as a series of successive approximations. More details are in my write-up.

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