Category: Political Science

Dirk writes:

I vote that this post deserves a follow up post with more clarification. If anyone is against this please express your vote with inaction. Us laymen would like to understand this a little better. For the record, the reason I got on a tangent about the law of large numbers was that I watched Boudreaux's lecture and understood it in terms of 3 parties but kept thinking if there were 3000 parties it was unlikely that exactly 1000 would have preference A, 1000 preference B, and 1000 preference C. I guess I'm used to thinking in terms of run-off elections and not the sort in the example. That is why I couldn't grasp why things should "collapse" back to an island situation where n = 2 or 3.

Arnold Kling comments as well and not everyone is happy.  

Return to the oft-neglected difference between intra-profile and inter-profile versions of the theorem.  Most commentators and expositors have in mind an intra-profile version of the theorem.  They set up an example of people and preferences and show how cycling or some other paradox of choice or voting is possible.  Observers then wonder whether this cycling is likely as the number of people increases, or as preferences change, and indeed sometimes it is not, as Gordon Tullock pointed out long ago and as Dirk above wonders.

That's interesting stuff, but those fun and practical-sounding expositions are not Arrow's theorem as Arrow wrote it up.  Think of Arrow's theorem as modal in nature: "Maybe there is no paradox with current preferences, but there exist possible preferences where everything goes screwy, under any decision rule satisfying a few criteria."  Arrow showed that claim is related to something like: "if we apply a specified decision-making procedure across all possible preference configurations, consistent application means the same person gets her way each time."

That's called Arrovian dicatorship, but it does not have to be either harmful or unjust or not even necessarily undemocratic.  It just means that one person — the same person — is always getting her first choice, across these modal worlds with differing preference configurations.

This more metaphysical and more originally Arrovian version of the theorem is perhaps why Arnold Kling finds it difficult to apply the theorem to practical problems.  It is not about the likelihood or relevance of cycling (though it is a jumping-off point for those analyses).  It is instead a deep result about the implications of consistency, combined with limited information about the value of ordinally ranked outcomes.

The intra-profile versions are still important.  For intra-profile versions of Arrow, start with Kemp and Ng (1976).  Here is a good summary article on that literature.  Samuelson, by the way, remained somewhat recalcitrant when it came to the theorem.

Allowing in even limited amounts of interpersonal comparability defuses the paradox, as shown by Kevin Roberts (ReStud, 1980) and Amartya Sen (see the essays in Choice, Measurement, and Welfare).  That said, interpersonability can lead to other paradoxes, as shown by Derek Parfit and his Repugnant Conclusion.  Paradoxes everywhere, and you must choose which ones to live with. 

I take the practical upshot of Arrow's interprofile theorem to be this: when you make a judgment, it is our assessment of the interpersonal comparisons (or intersport importance comparisons, for scoring a decathlon) which is doing all the work.  Be very careful with those. 

Neither Tullock nor Samuelson was happy with Arrow's theorem, especially when it came to practical implications, so it is fine if you wish to add your name to that list.  But I also think they each missed Arrow's point a bit and that of the major economists of his time he was probably the deepest thinker, albeit not the best practical thinker.

Arnold Kling is not satisfied with Tyler’s popularization of Arrow’s impossibility theorem asking (in the comments) “what exactly did Arrow show was impossible?”  Here is my attempt.

• Andy walks into an ice-cream shop and seeing that they have chocolate, strawberry and vanilla, orders chocolate.  Before the vendor has a chance to scoop the ice cream she says, “Sorry, we are out of strawberry.”  “In that case,” Andy says, “I’ll have vanilla.”  Strange right? Yet however strange we might think this behavior is for Andy it is routine when groups make choices (just substitute Bush, Nader, and Gore for chocolate, strawberry and vanilla.)  (Failure of IIA.)

• Andy likes sprinkles on his ice cream.  Andy walks into the ice-cream shop and seeing that they have chocolate, strawberry and vanilla chooses chocolate.  Before the vendor has a chance to scoop the ice cream she says “Today, chocolate comes with free sprinkles.”  “In that case,” Andy says, “I’ll have vanilla.”  Strange right?  Yet once again it is easy to show that improving an option can result in a group rejecting that option (n.b. no psychology is involved in this result). (Failure of positive association.)

• Andy walks into a supermarket and is offered apples or bananas and chooses apples.  Andy is then offered bananas or coconuts and chooses bananas.  Andy is then offered apples or coconuts and Andy chooses coconuts. (A>B, B>C but C>A).  Strange right? Not only is this strange, if Andy has preferences like this the supermarket can take all his money with a money pump. Once again, however, it is quite possible for groups to cycle in just this way even if every individual in the group has perfectly normal (transitive) preferences. (Failure of transitivity).

Paradoxes such as the above have been known for centuries.  What Arrow showed is that no decision mechanism can eliminate all of these types of paradoxes. (n.b. Arrow’s theorem actually applies to any mechanism for aggregating any rankings not just voting and not just preferences.) We can tamp down some paradoxes but only at the expense of creating others (or eliminating democracy altogether.)

More generally, what Arrow showed is that group choice (aggregation) is not like individual choice.

Suppose that a person is rational and that we observe their choices. After some time we will come to understand their choices in terms of their underlying preferences (assume stability–this is a thought experiment).  We will be able to say, “Ah, I see what this person wants. I understand now why they are choosing in the way that they do.  If I were them, I would choose in the same way.”

Arrow showed that when a group chooses, there are no underlying preferences to uncover–not even in theory. In one sense, the theorem is trivial. We know or should always have known that a group doesn’t have preferences anymore than a group smiles. What Arrow showed, however, is that without invoking special cases we can’t even rationalize group choices as if leviathan had preferences. Put differently, the only leviathan that rationalizes group choice has the preferences of a madman.

Arnold Kling presents it as an under-popularized idea, so here is my attempt to popularize it:

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."

Arrow's axioms, by invoking "the independence of irrelevant alternatives" (read: pairwise comparisons) require that all information in the problem be ordinal.  A decision procedure such as: "Turn the radio (and allocate other resources) to which station makes the winner happier" is ruled out per se.  Which person is happier cannot be expressed in the language of pairwise comparisons, which allow you to claim only that "John prefers jazz to rap" and so on.  "Let Tom and John trade" also won't do, whether or not transactions costs are high.  That's not a social welfare function as Arrow defined it.

I sense the above paragraph isn't a very charming popularization (it wouldn't make the Freakonomics movie), but let's continue.

That's a two-person example.  With 17 persons, Arrow's clever method of partitioning shows that conflicted decisions ultimately must break down to one-on-one comparisons of different individual preferences, returning us to the island/radio example.

For a truly popular audience, you could go on about the difference between inter-profile and intra-profile social welfare functions.

I believe that a popularized version of Arrow's theorem, which explains where the main result actually comes from (strong IIA), is not very impressive to most people.  That may be one reason why the theorem doesn't get popularized so much.

Here is one very good transparent version of a proof, which requires virtually zero mathematics.  If you work through this paper, you will understand the theorem and where it comes from.  Here is a related journal article.