Arnold Kling  

What did Arrow Prove?

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Political Predictions... Technocratic Overconfidence...

Among those trying to popularize the Arrow Impossibility Theorem are Steven Landsburg, Tyler Cowen, and Alex Tabarrok. Alex writes,


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

Arrow showed that when a group chooses, there are no underlying preferences to uncover--not even in theory.

I think that most people who have had Arrow's theorem explained to them once remember it as saying something about voting. Many others would remember it as saying something like "You can't have a social welfare function," which means that you cannot get from individual preferences to group choices in a way that satisfies a particular set of axioms. Alex is saying that, conversely, you cannot get from group choices to "group preferences" that satisfy those axioms.

I think Alex comes the closest to dealing with my issue, which is that I would like to know the practical import of the theorem.

Suppose that one poses the problem as "Create an optimal decision-making algorithm for resolving group differences." For any individual, the algorithm is optimal only if he always gets what he wants. Obviously, no algorithm is going to seem optimal to everyone. However, the "nondictatorship principle" says to me that the algorithm must seem optimal to no one. Maybe I'm being dense, but I do not see why that is important.

To put this another way, suppose that this were a purely an internal problem, posed to your "inner economist." That is, you have multiple selves--a romantic self, a calculating self, a spiritual self, and perhaps others. These selves have different preference orderings. Does the Arrow theorem say something about how you can or cannot decide what to do in a situation where the selves disagree? Or does it simply say that one is going to be ruled by, say, one's heart?


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COMMENTS (17 to date)
Randy writes:

In a voluntary group, I can quit if there is no positive outcome. And in an involuntary group (e.g., a political organization), I have no expectation of a positive outcome as the objective of a political organization is always to benefit the inner party at my expense - that is, a declaration that I am a "member" doesn't make it so.

So, its cool, but what problem does it solve?

jsalvati writes:

It's critical to remember that Arrow's Theorem is ONLY about voting (and a specific kind of voting), and nothing else. People love to over interpret this, and while it's certainly suggestive that preference aggregation might be hard, it doesn't prove any more than it actually proves. For example, Arrow's theorem does not prove anything about scalar voting systems (where you specify how much you vote for an item with a scalar value rather than a Boolean value).

Rebecca Burlingame writes:

To me, Arrow's Theorem is just a simple way to show certain limitations of representative democracy. How much does it matter that the one elected does not actually represent our interests, even if we voted for him or her? Where the real problem lies is that we start over every so often with giant sized agendas that basically try to wipe out the other. Whereas if more resources were provided for agendas all along the spectrum at all times, a lot more economic productivity would actually occur. Oh, that's right, not enough money to make that happen...maybe there are other ways.

Doc Merlin writes:

@jsalvati

You are incorrect, it actually applies to any ordinal ranked preference system. Utility functions being an ordinal ranked preference system, then it applies to utility functions, this means that you cannot aggregate utility functions.

Robert Johnson writes:

Rebecca,

It seems like a continuous (I can change my vote at any time) voting system shouldn't be any more difficult or expensive to implement than, say, a system for choosing who is permitted to fly without additional security screening.

Oh.

MattW writes:

If you have multiple inner selves with differing preferences, you can measure how strongly each self holds those preferences. Multiple people with differing preferences can't really compare how strongly they hold certain preferences with a simple vote.
You can compare how strongly with some sort of market setup though.

Patrick R. Sullivan writes:

How about, 'political choices can never be optimal'.

Alan Crowe writes:

John Geannakoplos has three brief proofs of Arrow's impossibility theorem Here

I haven't mastered any of the three proofs yet. Nevertheless I think that I understand the significance of the non-dictatorship principle.

Individuals have transitive preferences. It would be pretty hopeless to try to aggregate their preferences otherwise. Indeed we are rather hoping that our method of aggregation will always produce a transitive aggregate preference. If there is any collection of transitive individual preferences such that our method gives a non-transitive aggregate preference we will have to admit that our method breaks down at that point.

"Unanimity" is the requirement that the method does the obvious thing when everybody agrees.

"Independence of irrelevant alternative" is the powerful and controversial axiom.

We want to design a voting method that respects Unanimity and Independence of Irrelevant Alternatives, but we also see preserving transitivity as important. How can we guarantee to preserve transitivity? Arrow's theorem (in Geannakoplos' presentation) says we have to cheat and choose one person as dictator.

This cheat clearly works. If we just let one person dictate the outcome then we get their preferences which are transitive by assumption. But we aren't actually aggregating the preferences any more. Every-one else's preferences as just discarded.

The non-dictatorship principle is there to outlaw this cheat.

ed writes:

Arrow's theorem is not "only" about voting. In fact it's not about voting at all! It's about preference aggregation.

Voting is a multi-player game, so we would expect people to vote strategically rather than to simply report their true preference ranking. And in fact we observe people voting strategically all the time. There are other theorems that deal with this issue, but Arrow's theorem does not.

Rather, Arrow's theorem simply shows that there is no way to aggregate preference rankings in a sensible way, even if there were a way to accurately find out what these preferences are. It destroys the fiction that there is a "will of the people" that can be discovered. The "will of the people" doesn't even exist.

Doc Merlin writes:

"You are incorrect, it actually applies to any ordinal ranked preference system. Utility functions being an ordinal ranked preference system, then it applies to utility functions, this means that you cannot aggregate utility functions."

This is much like saying that there is no sense in which even numbers (as a class) are either larger or smaller than odd numbers (as a class); therefore it makes no sense to ask whether 8 is bigger than 5.

If we throw away everything we know about 8 and 5 other than their parity, then we can't compare their size. If we throw away everything we know about our utility functions other than the ordinal rankings they induce, then we can't aggregate them in a way that satisfies Arrow's conditions. But if you don't insist on throwing the extra information away, then of course there's more you can do.

Have you noticed, by the way, that you *can* aggregate utility functions? There are several ways to do it. One is called "addition".

Kurbla writes:

I think independence of irrelevant alternatives is wrong.


    101 voters: A C B1 ... Bn
    100 voters: C A B1 ... Bn

A is social preference over C. OK, I buy it. But, according to IIA,


    101 voters: B1 ... Bn A C
    100 voters: C B1 ... Bn A

A should be the preference over C. I do not accept it. In most of situations, I think C should be social preference over A, and that's how voting systems actually work.

Why? I think this is the answer: not only order of the possible choices matter, but relative values attributed to these choices. Irrelevant alternatives do not influence order of possible choices, but estimate the values associated with choices.

Also, non-dictatorship have no sense to me as well. Naming that pivotal voter "dictator" is misleading.

R. Richard Schweitzer writes:

Perhaps this is all a bit esoteric.

Group "choice" does not necessarily involve the concept of "preference."

A useful reference point might be found in the studies of Public Choice Theory on how group decisions are formed.

Radford Neal writes:

the "nondictatorship principle" says to me that the algorithm must seem optimal to no one. Maybe I'm being dense, but I do not see why that is important.


That a particular decision seemed optimal to someone, or even everyone, would be unremarkable. My understanding was that the principle concerned the process, not a particular result - that there wasn't some fixed person who always sees the decision as optimal. That might also be unremarkable if the set of possible decision problems to be addressed were limited, but assuming that there are many varied decisions to be made, one fixed person always getting their way doesn't seem much like democracy.

Radford Neal writes:

Steven Landsburg writes:

Have you noticed, by the way, that you *can* aggregate utility functions? There are several ways to do it. One is called "addition".

I don't think so. Utility functions aren't really functions. They're equivalence classes of functions, that are the same up to adding a constant and multiplying by a constant. Because of this, the addition operation is not well-defined.

andy writes:

You typically have a 'pivot voter' in situations when the society is divided on a point. Or to put it in other terms, there is not much difference if A or B is selected, both options are supported by many people; both options are probably quite reasonable choices, the 'dictator' isn't dictating too much.

WillJ writes:

@Kurbla: "Also, non-dictatorship have no sense to me as well. Naming that pivotal voter "dictator" is misleading."

No, that's not what the non-dictatorship principle is about. It's not calling that pivotal voter a dictator. He's not a dictator, because if others had voted differently, he wouldn't have gotten his way.

It seems like a lot of the comments on all of these blogs are misinterpreting the non-dictatorship principle. I think it's best to think of it as follows:

Don't think of it as an axiom that you start with out of some abstract concern. (If you do that, it seems completely unmotivated.) Start by just thinking of all the other axioms. Can you have a system that satisfies all those axioms? Well, certainly they work with individual preferences. Heck, most of economics is grounded on individual preferences being rational, so of course individual preferences can satisfy those axioms.

So that gives us one way of making social preferences satisfy those axioms. Just pick a person from the group, and have the group decision be whatever he wants, regardless of whatever the hell anyone else in the group wants. That effectively turns the social choice problem into an individual choice problem. And, due to what I said above, it can thereby satisfy those axioms.

But of course that's cheap. That doesn't "count." Hopefully there's some other way of satisfying those axioms. What Arrow's theorem says is that, no, there is not.

J Thomas writes:

Alan Crowe and WillJ are right about the dictator issue.

To be a dictator, they throw away everybody else's votes and do what you want *every time*. Not just sometimes. And the procedure is to throw away the other votes and do what you want. If you have N voters and their outcome each time happens to make sense, and then you are voter N+1 who happens to vote along with the majority on every issue, that doesn't make you a dictator.

Try out this explanation of Arrow's theorem and see hows it fits for you:


In a two-person election it's possible to get a tie whenever there are an even number of votes. An election system which declares that somebody wins when there is a tie, is a bad system. When the voters are evenly divided, you should have a tie.

When there are more than two candidates then there are new ways to get ties. Voting systems which do not declare a tie when it ought to happen are not ideal. Likewise voting systems which declare a tie when somebody ought to win, are not ideal.

Arrow's Theorem proves that voting systems which do not declare ties will sometimes give bad results. It's possible for them to give good results every time there ought to be a winner. But the times they declare a winner when it's really a tie count against them.

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