Arrow’s Theorem, Take Two

Tyler Cowen is of course one of the primary reasons to be grateful that you live in the age of the Internet. But none of us is infallible, and I believe Tyler has stumbled in his account of Arrow’s Theorem. His example:

Let’s say you had two people on a desert island, John and Tom, and John wants jazz music on the radio and Tom wants rap. Furthermore any decision procedure must be consistent, in the sense of applying the same algorithm to other decisions. In this set-up (with a further assumption), there is only dictatorship, namely the rule that either “Tom gets his way” or “John gets his way.”

Not true. A rule (or, in Arrow’s language, a social welfare function) has to prescribe a choice not just today, but every day, even as Tom’s and John’s preferences might change from one day to another. So there are in fact 16 possible rules. One is “Tom always gets his way.” Another is “John always gets his way.” Another is “Always turn the radio to jazz”, which seems pretty unreasonable since it prescribes jazz even on days when Tom and John both prefer rap. Yet another is:

  • If Tom and John agree, do whatever they agree on. If they disagree, turn the radio to jazz.

That last rule is particularly interesting because it satisfies every one of Arrow’s “reasonableness” criteria without anointing a dictator. What Arrow’s theorem says is that no non-dictatorial rule can meet all of those criteria.

Hold on a minute. I just gave you an example of a rule that meets all of Arrow’s criteria, and then told you that according to Arrow there is no such rule. What gives? What gives is the reason why Tyler’s example is irrelevant: Arrow’s theorem applies only when there are at least three options. With two voters and two options, the theorem fails and everything is copacetic.

In my own recent attempt to explain Arrow’s theorem, I assumed three voters and three options. It would have been simpler (and therefore better) to emulate Tyler by assuming only two voters (say Ann and Bob) arguing over three options (say Anchovies, Mushrooms and Pepperoni). Then you’re up against the fact that your rule must tell you what to do on days when their preferences run like this:

  Alice Bob
First Choice Anchovies Mushrooms
Second Choice Mushrooms Pepperoni
Third Choice Pepperoni Anchovies

On those days, you must either grant Alice a smidgen of dictatorial power by ranking Anchovies over Pepperoni even though she’s the only voter with that preference, or grant Bob a smidgen of dictatorial power by ranking Pepperoni over Anchovies. Once you’ve granted (say) Alice that smidgen of dictatorial power, Arrow’s argument demonstrates that — in order to satisfy his reasonableness criteria — you’ve got to grant her a bigger smidgen by ranking Anchovies over Pepperoni on any day when she has that preference. And then in order to continue satisfying his criteria, you’ve got to grant her yet another smidgen by ranking Anchovies over Mushrooms on any day when she has that preference. And then you’ve got to grant her another and another until finally you’ve made her an absolute dictator.

The details of this “argument by increasing smidgens” are in my earlier post, where you can just ignore the third voter (Charlie) to keep things a little simpler.

But Tom and John, living on Tyler’s island and facing only two choices, are exempt from all this and therefore an irrelevant diversion.


11 Responses to “Arrow’s Theorem, Take Two”

  1. 1 1 Ron

    Does Arrow’s theorem exclude what I consider the most fair algorithm?

    Minimize the choice number.

    In the example given:
      Anchovies would be 1 + 3 = 4.
      Mushrooms would be 2 + 1 = 3.
      Pepperoni would be 3 + 2 = 5.

    Therefore, the choice for this day would be Mushrooms.

    Granted, you must add to the algorithm what to do if there are two or
    more options with the lowest choice numbers.

    If this algorithm is, indeed, excluded, then how does this make Arrow’s
    theorem useful rather than simply needlessly prescriptive?

  2. 2 2 EricK

    Ron, One of Arrow’s axioms is that if choice A is preferred to choice B then adding new options can not mean that choice B is preferred to choice A (this is to prevent things like: “would you prefer chocolate or vanilla ice cream?” “chocolate” “We also have strawberry” “In that case “vanilla”)

    But minimzing the choice number can fall foul of this axiom in certain scenarios.

  3. 3 3 chris

    This is embarrassing (for me, not Steven) – first, let me say as a budding political scientist of a more theoretical and less mathematical bent that I have always had difficulty understanding Arrow’s Theorem. I’ve never had a teacher who could teach it either. Consequently, I get it in a big, vague way in theory, but that really means I don’t understand it.

    The idea that kick-started this attempt to explain it – that Arrow’s Theorem is both pretty important and almost completely…um… impossible to explain to people of even above-average intelligence – is absolutely true. I speak from experience. I have sought far and wide for an account that really brought it home to me. I read Riker’s book and it flew over my head. I read an account in Mueller’s Public Choice II, and that flew over my head and pooped on it to boot. Etc. I even tried to read Arrow’s little big book itself. Needless to say, it crushed my spirit.

    All of this is to say that, embarrassingly, even after reading Steven’s last post on the Theorem, I still don’t get it.

    But my vantage point can help to explain what it is that we dunderheads don’t get about the Theorem (everyone who reads this site seems to have no problem with it – it kind of makes me think this site must have the smartest readers on the net, by the by). Anyway, here goes:

    The first problem is the axioms. It is possible to explain what they mean relatively cogently, though things always get very blurry when people try to explain “independence of irrelevant alternatives” and one of the others I can’t recall. But what we fail to see is why just those axioms and only those axioms are the only ones there, and what exactly their logical connection is to each other. My sense is that if we got a really clear and forceful account of that, we would find the subsequent logic much easier to follow.

    The second problem, then, is this “smidgen” notion. Why does a smidgen of dictatorial power become absolute necessarily? Forgive me, but I simply look around me and I don’t see many ABSOLUTE dictators. But I’m probably just not getting the idea here. Can the dictator be a group? A party? An institution of some sort? And anyway, how is “being a dictator” different from “being a representative” or someone who is delegated decision-making authority?

    The point is just that these are the questions that occur to folks in my IQ range, and the people who try to teach the theorem seem to take it for granted that the answers are obvious. Maybe they are – to really smart logicians.

    I may be in a minority, I may simply be really bad at the kind of formal-logical thinking the Arrow Theorem requires, but with all of that noted, I still don’t get it. Not really. In theory, as I said, I do, in the way I get most public choice theory without being able to see it all Riker-style. But if I was trying to tell my grandfather what cool stuff I learn in political science, and I mentioned Arrow’s Theorem, and he asked, “Huh? How’s that?” I couldn’t begin to explain it to him.

    I also read, fwiw, Alex’s post about the cycling problem and the wacky “preferences” of groups – that was pretty good, and I think he was onto something in attempting to explain it that way. I still couldn’t put the pieces together, but it felt like a start. Just my humble two cents.

    PS – I’ve noticed that we who have trouble with Arrow also have major trouble with Bayes. I’ve tried to read “simple” accounts of Bayesianism, and I never get past the third or fourth paragraph without crumbling to pieces. It’s kind of similar to Arrow. I get it in a big vague way. But not “really.”

    Thanks, though, for making the effort (I mean that).

  4. 4 4 Steve Landsburg

    Ron: EricK has this exactly right.

  5. 5 5 John Faben

    re. the two-player, two candidate version. What does it even mean to satisfy the hypotheses of Arrow’s Theorem in this set-up? I can’t quite figure out what the relevant version of IIA says, or does that axiom just become vacuous?

  6. 6 6 Steve Landsburg

    John Faben: IIA becomes vacuous in that case.

  7. 7 7 Harold

    “Argument by increasing smidgens” -I love it! Is this a formal logic term?

    In this example, it is clear (using Ron’s reasoning) that Mushrooms should be the choice. This seems intuitive. In the 3 person example there was no clear winner. It was arbitrary that Alice got her way on Tuesday, but here we have a valid reason for giving Bob his first choice.

    I think if you follow the logic through using Bob as the dictator, it still works.

    However, on some days you will probably have to ignore the “minimize the choice number” rule which gave Bob his way at the beginning.

    You could avoid this by having M/A/P, P/A/M as the choices – each then has 4 points in Ron’s scheme.

    Perhaps here is a simpler contradiction – if you follow Arrow’s rules, you must contradict the intuitive “minimise choice number” rule?

  8. 8 8 kev

    Wow! Are you guys for real? You’re talking about voting in a 2 person scenario? Do you not see how absurd this is? I disagree with my son as to what pizza topping to order. He says anchovy, I say mushroom. I suggest we vote on it- because I just read Prof. Landsburg post and that’s what 2 reasonable people do- don’t they? I vote for mushroom, he votes for peperoni OMG! What just happened?! Voting didn’t solve anything! Does this explain Arrow’s theorem? No. My son, all of four years old, explains why- ‘Daddy you big stoopid, voting is only good when Mummy is there.’
    And then the tears start and the tantrums and the drunk dialling once he’s safe in bed.

  9. 9 9 Steve Landsburg

    Kev: But in the case of two people and two toppings, there *is* a voting system that meets all Arrow’s criteria — we get whatever we both agree on, or mushrooms if we disagree.

    So if your claim is that you can already see the problem in a two-person, two-topping scenario, you have misunderstood what the problem is.

  10. 10 10 kev

    My objection is to the use of the word ‘voting’ rather than ‘bargaining’ or ‘comparing preferences’ in a 2 person scenario. Why? Well, we’re talking about popularizing Arrow- i.e. helping ordinary people see that this theorem encapsulates something empirical or provides a hueristic.
    The notion of 2 people voting rather than bargaining or discussing conjures up an absurd image. Nothing to do with Econ, everything to do with ordinary language- the latter being the binding constraint on ‘poupularization’.
    I wasn’t saying there was a way to collapse three options into 2 and still keep Arrow. ‘So if your claim is that you can already see the problem in a two-person, two-topping scenario, you have misunderstood what the problem is.’I agree, if that were my claim I’d have misunderstood the problem.
    The other point about a 2 person bargaining situation is that the common-sense approach is to think of the 2 parties getting into the mechanism design business for themselves- i.e. Alice and Bob may start a dialogue on what’s a good way to resolve this, perhaps a coin toss? At this point intensity of preference and contingent dynamic considerations can express themselves- Alice may stipulate that pepperoni gives her gas so if Bob is calculating the odds on some nookie tonight that option should be handicapped appropriately.
    The problem here is that once Game theory and mechanism design get entangled then, my intuition is, you have a complexity problem with rules out impossibility results in advance.
    Which is not to say that this area of study shouldn’t be popularized or that it isn’t highly relevant to the ordinary blokes. Here in England, Ken Binmore became a folk hero for the recent 3G auction.

  11. 11 11 kev

    there *is* a voting system that meets all Arrow’s criteria — we get whatever we both agree on, or mushrooms if we disagree.

    Actually, if one revives non-imposition there is an ordinary language argument for Arrow as follows- If the only information fed into a S.W.F calculator are preference rankings with no further information about consequences for the ‘voter’ then you can get very bad results.
    Alice likes anchovies, mushrooms not to so much and pepperoni not at all. Her preferences remain the same but today she gets a piece of advice from her Doctor- there is a sound medical reason for her preferences. This changes things. It would probably affect how Bob’s vote if he has this information. Can a deontic voting rule capture this change?

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