Straight Arrow

arrowTyler Cowen asks which economic ideas are hardest to popularize. Arnold Kling nominates the Arrow Impossibility Theorem. Tyler responds with an attempt to popularize it. Alex Tabarrok weighs in with another. Here’s my own attempt:

Every day, Alice, Bob and Charlie split a pizza with one topping — anchovies, mushrooms or pepperoni. Their preference orderings change from day to day — some days Alice is in the mood for mushrooms, other days the very thought of mushrooms makes her queasy. Every day, they have to call in their first, second and third choice pizza orders. (The pizza delivery place insists that you specify your second and third choices in case they run out of something.) So Alice, Bob and Charlie need a method for translating their individual preferences to a pizza order.

Now as it happens, this past Tuesday, their preferences ran as follows:

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

I’m not going to tell you in any detail what system these three were using to determine their order, but I will tell you that on Tuesday they reported Anchovies as their first choice.

On Wednesday, Alice ranked Anchovies over Pepperoni again. (I don’t remember whether she ranked them first and second, or first and third or second and third — but that doesn’t matter anyhow.) I don’t remember anything about Bob’s or Charlie’s rankings. But I don’t have to remember anything to know this: On Tuesday, only Alice ranked Anchovies over Pepperoni, whereas on Wednesday, Alice plus possibly some others ranked Anchovies over Pepperoni. So if the pizza order ranked Anchovies above Pepperoni on Tuesday, then it should certainly have ranked Anchovies over Pepperoni on Wednesday, no?

Any other outcome would have seemed unreasonable to Alice, Bob and Charlie, so when they designed their system, they designed it with the following feature:

  • If we list Anchovies over Pepperoni on our order on one day, and if none of the people who prefer Anchovies over Pepperoni change their minds about that the next day, then we should list Anchovies over Pepperoni the next day as well.

Because this was built into their system (and because they’d listed Anchovies over Pepperoni on Tuesday when only Alice had that preference), they always listed Anchovies over Pepperoni on any day when Alice preferred Anchovies to Pepperoni. Alice, in other words, was sort of an “Anchovy/Pepperoni dicatator”.

Now on Thursday, Alice was in an Anchovies/Pepperoni/Mushroom sort of mood. I don’t remember much about Bob’s or Charlie’s moods except that they both favored Pepperoni over Mushrooms. Since everyone preferred Pepperoni to Mushrooms, of course Pepperoni was listed higher than Mushrooms in the pizza order. Once again, any other outcome would have seemed unreasonable to Alice, Bob and Charlie, so they’d designed their system with this feature also:

  • Whenever we unanimously prefer Topping X to Topping Y, Topping X should rank higher than Topping Y on our order.

In fact, though I don’t remember everything about Thursday’s pizza order, I can figure it out. First, Alice (the Anchovy/Pepperoni dictator) preferred Anchovies to Pepperoni, so of course Anchovies ranked higher than Pepperoni. Second, everyone preferred Pepperoni to Mushrooms, so Pepperoni ranked higher than Mushrooms. And using my IQ test skills, I can figure out that Anchovies must have ranked higher than Mushrooms.

By the same logic — On any day when Alice prefers Anchovies/Pepperoni/Mushrooms in that order, and everyone else prefers Pepperoni to Mushrooms, Anchovies must rank higher than Mushrooms.

Now here’s another design feature these three agreed on:

  • Our preferences about Pepperoni should not affect the relative ranking of Anchovies and Mushrooms.

Therefore, the green boldfaced statement above should remain true if we drop all the Pepperoni-related assumptions. In other words, On any day when Alice prefers Anchovies to Mushrooms, Anchovies must rank higher than Mushrooms. That is, Alice was not just an Anchovy-Pepperoni dictator. She was an Anchovy-Mushroom dicatator as well.

On Friday, Alice’s preferences ran Mushrooms/Anchovy/Pepperoni, while the other two both preferred Mushrooms to Anchovies. Since they all preferred Mushrooms to Anchovies, Mushrooms came out higher than Anchovies on the pizza order. Since Alice was an Anchovy/Pepperoni dictator, Anchovies came out higher than Pepperoni. Our IQ test skills tell us that Mushrooms came out higher than Pepperoni. And the same would be true on any day when Alice preferred Mushrooms/Anchovies/Pepperoni and everyone else preferred Mushrooms to Anchovies. But the ranking of Mushrooms vs. Pepperoni was designed to be unaffected by how anyone cared about Anchovies, so the Anchovy-related information can’t be relevant. This tells us that on any day when Alice prefers Mushrooms to Pepperoni, Mushrooms rank higher than Pepperoni. She’s not just an Anchovy/Pepperoni dictator and an Anchovy/Mushroom dictator; she’s a Mushroom/Pepperoni dictator also.

What we’ve discovered is that any Anchovy/Pepperoni dictator is also an Anchovy/Mushroom dictator and a Mushroom/Pepperoni dictator. Interchanging the names of the toppings, we could as easily have discovered that any Anchovy/Mushroom dictator (e.g. Alice!) is also an Anchovy/Pepperoni and a Pepperoni/Mushroom dictator — and so on until every pair of toppings appears. In other words, Alice is an absolute dictator. All of her preferences are fully reflected in the pizza order, every single day.

Now to get this ball rolling, I had to assume that Anchovies came out on top on Tuesday. But if Mushrooms had come out on top, I could have proved that Bob is an absolute dictator, and if Pepperoni had come out on top, I could have proved that Charlie is an absolute dictator. Regardless of what happened Tuesday, someone must be an absolute dictator.

And what if there had been more than three voters or more than three toppings? Then the argument gets more complicated to keep track of, but not more complicated in spirit. The conclusion remains the same — if you have a system that translates a collection of individual preference orderings into a single “social preference ordering”, and if that system is designed to have certain features that strike many people as reasonable, then the system anoints a dictator.

Arnold? Tyler? How did I do?

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16 Responses to “Straight Arrow”


  1. 1 1 Bennett Haselton

    When I first heard of this problem many years ago and the apparent paradoxes that come with it, I thought: You can do away with the paradoxes if you allow for gradations of preference, where one person liking Pepperoni MUCH more than Mushrooms, would count more than someone else liking Mushrooms slightly more than Pepperoni.

    That would cause you to reject one of the rules listed above:

    “If we list Anchovies over Pepperoni on our order on one day, and if none of the people who prefer Anchovies over Pepperoni change their minds about that the next day, then we should list Anchovies over Pepperoni the next day as well.”

    Suppose on Monday, Alice would have paid $1,000 to have Pepperoni instead of Mushrooms, and Bob and Charlie were nearly indifferent. By logic of economic efficiency, they should have Pepperoni. But then suppose on Tuesday, Alice wanted Pepperoni only slightly more than Mushrooms, while Bob and Charlie both wanted Mushrooms a lot more than Pepperoni. Even though the conditions for your rule above are satisfied, they should still not order Pepperoni on Tuesday.

    This also seems to resolve the simplest version of the paradox, which is that if you start with the preferences chart at the top of the blog post, then it’s impossible to satisfy “If 2/3 of voters want A instead of B, we should have A instead of B”, because no matter *what* pizza topping you order, you order something where 2/3 of voters would have preferred another one! However, even though 2/3 of voters prefer Anchovies over Mushrooms, the remaining 1/3 like Mushrooms MUCH more than Anchovies. And if you assign more weight to strong preferences, then all three toppings are in a tie.

  2. 2 2 Nick

    Somewhere my hons. micro lecturer is jumping for joy (and preparing handouts of this post)

    Excellent summation!

  3. 3 3 Harold

    Good way to put it. I have one niggle. The Arrow Impossibilty Theorem has 3 axioms, this approach is putting the “dictator” one in a special position. This is OK to illustrate the point, but can lead one into a small error. I remember someone stating something like the only fair voting system ends up appointing a dictator. This is not the case, as it states there is no fair voting system that can satisfy all three axioms. There is no particular reason to pick the dictator one as “special”.

  4. 4 4 Scott F

    Why is it reasonable to say that our Pepperoni preferences should never affect the relative ranking of Mushrooms and Anchovies? In the real world when both Charlie and Bob prefer Pepperoni over Mushrooms and Anchovies, but Alice does not why should we expect them not to take that into account with their ranking. I think I’m missing something (there’s a lot to keep track of).

  5. 5 5 Ken B

    Oh good lord. I think you have made Kling’s point.

    My own suggestion is to imagine some being like Maxwell’s Demon, and show him thwarting the imagined ‘best choice’. Lots of logic problems become simpler for most people when they can track some purpose or intent, nad that might be a useful stepping stone. (Like using selective breeding as a stepping stone to natural selection.)

  6. 6 6 wheninrome15

    @Bennett.

    Any attempt to overturn Arrow’s impossibility theorem would have to work in EVERY case, because that’s the content of the theorem. Arrow isn’t saying you can never get the optimal outcome (for instance there is no problem in arrow’s system when everyone happens to agree on the leading candidate); the point is that no matter what system you design, it will go wrong for SOME preferences. So in particular, your approach would have to account for the special case where in fact everything happens to have the same weight. “No gradation” is a special case of “gradation of preferences,” so it seems the latter will not patch up the problem.

  7. 7 7 SB7

    @Bennett, Arrow specifically says in his presentation of this that we are considering only what the pairwise preferences are, not the relative strength of those preferences.

    I believe, but I am not sure, that for any discreet system of expressing preferences strength, such as listing a preference as extreme/strong/moderate/slight, you can reduce it to a situation in which there is only a yes/no preference by introducing extra, “phantom” voters. So if Alice strongly prefers X to Y, you introduce three Alices into your transformed, simple-preference voting system who each prefer X to Y, and if she extremely prefers X to Y you introduce four Alices, etc. Again, not sure about this, but I think you can do this.

  8. 8 8 Justa Commenter

    This was a lot better explanation than Tyler and others attempted yesterday, at least the examples, the thrust of the theorem and the impossibility makes more sense. On the other hand, it’s not clear why anybody would care that these assumptions lead to impossibility. Sure, they are reasonable sounding, but that hardly holds water; plenty of logically inconsistent assumptions sound reasonable.

  9. 9 9 Harold

    Just found the quote I was thinking of earlier. “The only way to satisfy all the requirements is to select one voter and give him all the votes. The only democratic solution that meets the minimal requirements for democracy is to annoint a dictator!” (Extra marks for identifying the quote)

    This is surely wrong, as appointing a dictator does not satisfy the requirements, one of which is that there is no dictator.

    You can just as easily satisfy the “no dictator” rule and abandon one of the others, for instance the independance of irrelevent alternatives. Surely the correct way to express it is that there is no way to satisfy all the requirements?

  10. 10 10 Noah Yetter

    Sadly, you lost me. And I already understood the theorem! Needs to be shorter and less contrived.

  11. 11 11 Doc Merlin

    @SB7
    @Bennett Haselton

    Yes, you need interpersonal utility comparisons in order to avoid the problem with the other two conditions. This means you need to be able to say who’s utility function is more important and when. The problem is, once you do that, you have in effect anointed a dictator.

  12. 12 12 Steve Landsburg

    Harold: You are right that there’s no important reason to pick out the dictator axiom as “special”, but having taught this material several times, I find that it’s a little cleaner when it’s presented this way.

    Arrow’s theorem says “No social welfare function satisfies Axioms 1, 2 and 3.” My preferred statement is “Any social welfare function satisfying Axioms 1 and 2 must violate Axiom 3.” Of course, one could permute the 1,2, and 3 in any way one wanted to get other equivalent statements. But I’ve found that it’s a little easier — for me at least — to get the ideas across when I state it my preferred way.

  13. 13 13 Harold

    And a very good way to get the idea across it is too. However, there is a difference between ” Any function satisfying 1 and 2 must violate axiom 3″ and “The only way to satisfy all the axioms is to select one voter and give him all the votes”.

    I have no problem with the post above, but perhaps added on should be that you could do the same for the other axioms also (in the same way as you make clear that Bob or Charlie could have been the dictator).

  14. 14 14 David Grayson

    Wow, that was a great proof! Thanks for phrasing it so well. I wonder how difficult the proof would be for an arbitrary number of voters and an arbitrary number of candidates.

  15. 15 15 dave

    ‘On Tuesday, only Alice ranked Anchovies over Pepperoni, whereas on Wednesday, Alice plus possibly some others ranked Anchovies over Pepperoni. So if the pizza order ranked Anchovies above Pepperoni on Tuesday, then it should certainly have ranked Anchovies over Pepperoni on Wednesday, no?’

    im sure i am missing something, as always..but it seems to me that you go on to explain how wednesdays ‘system’ is established with an invalid assumption.

    what if tuesdays system is that alice’s picks will be the order, and wednesdays is that bob’s picks will?

    i only got more confuzzled as i read further. =]

  16. 16 16 sheila

    What if Alice Bob and Charlie are asked to come up with a voting rule which is non-dictatorial? I suppose you’d then get a violation of ‘independence of irrelevant alternatives’- at least if their tastes were sufficiently different- which begs the question of its ‘naturalness’.
    There is another more general problem. I guess I prefer mushroom to anchovy but it isn’t a life or death thing. What if Bob is allergic to anchovies? Or Alice can’t have peperoni for religious reasons?
    Cardinal utility could capture that, at least in theory. I don’t see how ordinal measures can. If a voting mechanism is loosing information, so to speak, it’s bound to be sub-optimal.
    The Myerson-Satterthwaite theorem also comes to mind as one which could not arise if the agents decide to do the mechanism design themselves, perhaps carving up ‘the gains from trade’ in a stochastic manner.
    The reason for my guess is that you’d be getting a sort of impredicativity- utility is based on two things viz. the game about the game as well the actual payoff. Perhaps, if this line of enquiry were pursued you might get Godel type impossibility results showing the limits of this approach.

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