Arrow’s Impossible-to-Understand Theorem

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Tyler Cowen writes,

Can one economist in forty properly define the “independence of irrelevant alternatives” axiom behind the Arrow Impossibility Theorem

That one I think I understand. What I do not understand is the “nondictatorship” axiom. Does it mean that ex ante no one is given the privilege of selecting for the group? Or does it mean ex post that no one’s preferences are perfectly satisfied? My impression, which may be wrong, is that it is the latter, in which case I think that nondictatorship is a misleading term for it. [UPDATE: commenters point out that I am wrong, and that it is an ex ante assumption]

As an undergraduate, I spent a lot of time trying to understand the Arrow Impossibility Theorem. I was much more fluent with it then than I am now, and yet I never thought that I had it. I think that every attempt to explain it in simple terms is wrong.

I agree with Tyler that much of modern finance is not well understood. The Portfolio Separation Theorem is very important, and yet few can explain it. My attempt is here.

One of my long-standing complaints is that nobody is sure whether aggregate demand slopes up or down. In undergraduate macro, it slopes down. In graduate macro, with inflation on the vertical axis, it slopes up (although in graduate macro nobody draws it–they just write down equations.)

12 thoughts on “Arrow’s Impossible-to-Understand Theorem”

  1. One of the helpful mantras I learned from puzzling for hours over Thermodynamics is “this just lets the math work.”

  2. The non-dictatorship requirement is against an ex-ante dictator.

    If we declare an Emperor, we do have a decision process which is Pareto efficient and independent to irrelevant alternatives.

  3. The best explanation I know of “independence of irrelevant alternatives” is a joke:

    A guy is ordering dessert in a resturant and asks what kinds of ice cream they have.

    Waiter:“Vanilla and choclate.”

    Customer“OK. I’ll have some vanilla.”

    Waiter:“Wait, we have strawberry, too.”

    Customer“Oh. In that case I’ll have chocolate instead.”

  4. No, it’s the former. Ex ante there’s nothing odd about someone getting their ideal outcome (e.g. the median voter). The axiom just blocks one cheat. Namely,. we can’t escape the impossibility theorem by substituting one individual’s preference which does satisfy all of Arrow’s conditions and call that the social preference.

  5. Another illustrative example of the irrelevance of independent alternatives.

    Poll worker: would you like Gore or Bush?

    Electorate: Gore

    Poll worker: would you like Gore, Bush or Nader?

    Electorate: Bush

  6. A dictator is an individual who has his way regardless of his/her preference ordering and regardless of every other person’s preference ordering. Non-dictatorship means that there is not a dictator.

  7. I’d always understood it to be ex ante: there can be some individual whose preferences match the SDF output, but if we rejigged other people’s preferences, the SDF shouldn’t still give that guy’s preferences as output unless the re jigging were trivial.

  8. Yes, it’s ex-ante, as wd40 and others explained very well above.

    Although calling it “ex-ante” is maybe confusing. This isn’t a random variable, its a function that is required to generate a complete, transitive social ordering for *any* valid combination of individual preferences, i.e. “unrestricted domain.” (The individual preferences are also required to be complete and transitive.)

  9. I read comments only to see if someone explained the final Arnold’s point, about the demand curve. No luck.

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