Popularizing Arrow’s Theorem II

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).

no-leviathanParadoxes 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.


Very interesting. Didn't Arrow question methodological individualism in economics? Thought that we should focus on institutions and collectives?

I use Larry, Mo and Curly and them deciding over breakfast. Hint: Mo tends to resolve the problem by insisting on his preference. To get more concrete I have used the example of a city council having to decide over a park, a school, or a parking garage for a use of land. The best book on the implications of Arrow that can be read by those not engaged in social choice is Riker's Liberalism Against Populism, and the best intro is in my opinion Dan Johnson's An Introduction to Political Economy.

Alex, I cannot comment on whether this makes Arrow's theorem more popular or not. A better candidate for popularity would Sen's Impossibility Theorem, in part because it doesn't rely on the IIA axiom - which is very tricky.

I don't follow your logic - are you saying that examples 1-3 are examples of individual irrationality but not necessarily of group irrationality?

I will wait for your reply.

(Technical note: example one is any of a failure of contraction consistency, property alpha, and not IIA.)

You can also discuss this as preferences for sets of things. We each have unique preferences for sets, and removing or adding something to the set changes the preference outcomes beyond the value of one item in the set.

In this very nice exposition of Arrow's theorem you present the result as demonstrating the impossibility of aggregating individual preferences. This assumes an atomistic point of view. Alternatively, the theorem may be taken as simply showing that the sui generous notion of the common good cannot be atomized; that it is a primary concept, ie unfactorable. We should not be lead, I would argue, from Arrow's theorem to political egoism, ie, banishing the notion of the common good from normative reasoning. We should just abandon the idea (which was already confusing prescriptive and descriptive inquiries) that we can define the common good by aggregation.

yes, well done.

In a contemporary economics that has lots of strategic behavior between various "selves" (e.g. my present oriented self vs my future oriented self), couldn't we also interpret Arrow's theorem to mean that we are all insane. Arrow's theorem looks less like failure of collective choice when we admit that individual behavior is not well described by stable, transitive preferences.(Maybe some French psychoanalysis could permeate economics).

This is indeed fascinating. Great post! Thanks, Alex!

anonymous: No type of voter would switch from Bush to Gore just because Nader dropped out. But the voting system, which is supposed to aggregate everyone's preferences into a single preferred outcome, probably would. It's not that people individually are crazy like Andy in the example, it's that any method of determining what people as a whole want must produce results which, if they were a single person's preferences, be crazy.

Sometimes I like chocolate, sometimes strawberry, and sometimes vanilla. The frequency I have just had my last selection will influence my next. Can one construct an ordering even for an individual that is stable?

Let's say nine people need to select one type of ice cream for everyone. Four like chocolate best, three like vanilla best, and two like strawberry best. The two that like strawberry best prefer vanilla to chocolate. Once strawberry is eliminated, the group would vote for vanilla five to four.

Andy walks into polling place, orders Bush. Before the vendor has a chance to take the ballet she says, "Sorry, we are out of Nader." "In that case," Andy says, "I'll have Gore." 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."

The groups choice isn't informative based on its choice: because there are actually TWO votes with TWO choices going on....

The less obvious vote is a choice of 1) process 2) politics.

To C voters, you either accept the dominance of bipolar option (bush or gore) and make a rational choice in Vote #1, or you reject bipolar hegemony and irrationally vote the C power in VOTE #2.

The fact that irrational acts sometimes become rational (Nader becomes viable) is interesting though.

*sigh* In all this blog discussion, has anyone yet mentioned the Arrow's Theorem is a Lie post on a LessWrong from a year ago? It points out that the result has much less practical application than you might thing.

Even more importantly, "Black Belt Bayesian" makes the point that so-called "irrelevant alternatives" aren't. That is, they give evidence about the relative _strengths_ of preferences and therefore SHOULD affect the aggregated preference ordering! So you don't even want the IIA axiom!

Worst of all, the usefulness of the entire Arrow Theorem becomes even *more* questionable* when you realize that even using a randomized tiebreaker violates the constraints. So really, who cares?

Thanks, Wonks_Anonymous, for that link about Sen's corresponding result -- just what I thought: it's crap as well, even by the standards of the "libertarianism sucks" crowd.

Great post, thank you Alex.

I have to say this explanation doesn't do much for me. The behavior described in those examples seems irrational, so why is it a surprise that a system can't translate irrational behavior into a rational result?

Guys, this is social choice theory. The whole idea is that you want to design general voting rules where people with ordered preferences submit a vote, and something happens. That's it. Yeah, sure, if there was trade or side-payments, other things could be done. But we don't get rebates when we vote for congressmen, and they don't pay each other off to leave the race. Likewise, you need some structure on preferences to say anything at all; IIA is just the baseline, there's plenty of other assumptions you can bitch about.

Arrow's theorem is the starting point for perhaps thousands of papers. Acting like you're scandalized by the unrealistic assumptions or are going to overturn it with a simply thought experiment or whatever is pretty Philosophy 201.

Alex, I think you won this one. <a href"http://www.math.uci.edu/~dsaari/fourthgrade.pdf">Donald Saari also has a great paper on similar Arrow-related topics. Perhaps you should send this post to the Academy of Motion Picture Arts & Sciences (despite expanding their Best Picture field to 10, <a href"http://istanbulsunset.blogspot.com/2006/03/why-brokeback-lost.html">their simple majoritarian framework is just silly).

I'll try again.

We want a system to judge voting systems. I'll ignore the larger issue for now, and just look at voting systems. We invent examples where we know all the preferences of a voting population, and we look for pathological cases where a voting system gives the wrong result. We want a voting system which has no pathological cases.

Here are some criteria:

1. Unless the majority prefers A to B, A should not win. (Maybe if a voting system violates this, there might be a pathological case where *everybody* prefers B to A and A wins.)

2. Unless the majority prefers A to B, A should not win even when C is also running.

3. If a voting system does not satisfy 1 and 2, it mustn't satisfy them by throwing away votes until the problem goes away. We don't want any votes thrown away. (Maybe if it has to throw away votes to avoid a pathological case, we can find an example where it has to throw away all but one vote.)

Aren't these good criteria? We might want more criteria, but these are good ones.

What should happen when A > B > C > A ? No matter who is chosen as the winner, it fails. A can't win because C > A. Etc. So the voting system should not declare a winner in that case. We don't declare a winner when there's a tie. Why declare a winner this time?

If we really want these criteria, then one way to approach the matter is to look at voting systems where, for N candidates, voters make every pair-wise comparison. N(N+1)/2 pairwise votes or some equivalent. Because if you're going to judge by those preferences, a voting system has a better chance to get it right if it actually collects the data it's supposed to fit and not some other data.

@rojiani: Yes, maybe Arrow's Theorem has inspired useful, practical results by other people, in other papers. But the issue before us is whether Arrow's Theorem *itself* deserves popularization, and for that issue -- the one we're actually discussing -- it's perfectly relevant to render these "Philosophy 201" objections to its relevance, and ignore the authors who used it as a springboard.

If there's some other paper that does have a useful result, and relied on Arrow (and no, Amartya Sen's Liberal Paradox aint' it), then suggest *that result* as one being worthy of being popularized. But it still wouldn't substantiate the claim that Arrow's Theorem is insightful!

Rojiani, what I'm saying is that if you want a voting system that "works", it needs to explicitly fail the pathological cases.

Arrow says you can have pathological cases where any candidate you pick as the winner is wrong. So a winning voting system will announce that there was no winner -- just as a winning voting system will not pick a winner in a 2-candidate race that comes out a tie.

My further conclusion is that if you want to avoid picking winners when Arrow says not to, it will probably help if you collect all the information that Arrow uses to decide there is no winner. If you collect less information than Arrow uses, you might sometimes not have enough information to know there is no winner.

Those are my current conclusions.

@ Jeff R.

The Efron dice thing is a complete red herring. Scroll down and it explains that "Therefore the best overall die is C with a probability of winning of 0.5185."

So if you asked me to pick among K of these die, and then you roll them all and if mine wins, I get a prize, then I can always answer that question consistently. The problem is that my answer would depend on the "opponents", and there's nothing wrong with that. This is no different from betting on the world series, where my bet would change depending on what teams were in it, and it wasn't clear from their records which team was the "best", but we knew which teams were most likely to beat other teams. Just do the expected value of all the lotteries of the "prize" associated with winning.


If you want a vpting scheme that satisfies Arrow's criteria, then base it on Arrow's criteria.

He says there can't be any such thing? But Arrow's Theorem says which outputs are acceptable and which are not for each set of inputs.

And there *is* an acceptable outcome for each set of inputs. If none of the candidates is acceptable then "tie" is acceptable.

Did Arrow forget about "tie" as an output? I don't see how he could have reached his conclusion otherwise.

"1. For any finite collection of preferences, choose the first outcome from the list. Check whether it satisfies Arrow's Theorem. If it does, we are done."

What are you talking about? What is "the first outcome"? Arrow's theorem applies to functions, not lists of preferences or alternative, so I don't understand what "satisfies Arrow's theorem" even means. Are you suggesting we use some voting rule, but apply it iteratively?

"2. If not, choose the second outcome from the list. Check whether it satisfies Arrow's Theorem."

Again, what?

"By finite induction, there will be an outcome which satisfies Arrow's Theorem, or there will be none. If no outcome satisfies Arrow's Theorem, call it a tie."

I mean... what?? Outcomes and preferences do not 'satisfy' Arrow's Theorem, voting systems are the subject of Arrow's Theorem.

If you want ties, OK, add ties. We get a tie, now what? Ok, let's use randomization among the set of Pareto undominated elements of the set of outcomes.

Suppose there are two elements of the Pareto undominated set, one you like more than the other. If the voting system is responsive to your report, there was a tie, and you can block it by lying. This is a serious problem that follows in the wake of trying to patch up the idea of voting systems. Note that this issue isn't as severe if Arrow's is false, since we could pick a voting rule that chooses the unique Pareto-optimal choice, since no individual would be a dictator and wouldn't be able to block an outcome on his own.

Rojiani, I presented an approach toward building precisely that sort of machine. You choose outcomes and test whether they satisfy Arrow's Theorem and accept one that does. If no outcome satisfies Arrow's Theorem then you call it a tie.

How can that be wrong?

@ J Thomas

Ignore ties for a moment.

I suppose what you are saying is to create a voting system that assigns a voting system to every possible list of preferences that might be submitted, so that at each preference list submitted, the voting rule "satisfies Arrow's" (which, of course, I still don't know what that means). This is a composition of functions, which is itself a function, and is just another social choice rule, so it will fail Arrow's.

Now we add ties.

OK... what is an acceptable tie? You are essentially weakening the Pareto condition, since if we drop that from Arrow's criteria, the rule "Choose an outcome randomly for every possible vote "satisfies" the other criteria. So when we get a profile like:
1: A>B>C
2: B>A>C
3: C> {A=B}
We randomize over A and B? That's nice, since A and B are pareto undominated, OK.

What about
1: A>B>C
2: B>C>A
3: C>A>B
the Condorcet paradox list? So you want to randomize over A, B, and C?

Notice that "no dictator" and "pareto optimal" mean that no single agent can block a pareto optimal allocation by manipulating his vote --- if a social choice rule existed satisfying Arrow's criteria, strategic manipulation would not be possible by a single agent, since that would make him a dictator. But without these conditions, the serious issue of manipulation comes into view. For example, agent 3 can lie, vote according to ACB, and A is chosen for sure. With Arrow's specific criteria, these strategic issues are not an issue, but with them, we run into new kinds of troubles. You CAN add randomization to relax the "domain restrictions", but there is an important implication you are ignoring.

Rojiani, when I replace Arrow's criteria with criteria which look weaker and which look reasonable, it appears to be false. But I notice that you reason that I must be wrong because I get a result that violates Arrow's Theorem.

I enjoy arguing with you about this but I think I should study Arrow's Theorem in greater detail instead, since I believe I have found a self-contradiction in Arrow's Theorem unless I change the meaning around a little bit, while you believe in Arrow's Theorem. So arguing about it would be useless.

Perhaps at some point you might humor me by explaining Arrow's proof or some simpler proof of the theorem? Neither you nor anybody else has any obligation to do that, but if you want to....

I looked at the informal proof in Wikipedia.

First they argue about a specific situation where they are wrong as shown by the following example:

Choice of A B C. Everybody chooses B to be last. So the outcome must have B last. If everybody chose B to be first then the outcome must have B first. Suppose one voter switches from B last to B first. And then another and another. When one particular voter makes that switch, the outcome immediately switches from B last to B first, in one move. They argue why B cannot spend one turn at an intermediate spot.

But they do not consider that for one turn B could have a tie. Consider the obvious case where you count the votes for first place and the votes for last place. When there are an even number of voters, when voter N/2 switches then B is half first and half last. This is not the same thing at all as being in the middle, but it is very much a tie.

In the second part they discuss the situation when the one vote that would switch B from bottom to top is instead put into the middle. Then if that vote changes to put B first, B will be first, and if it changes to put B last then B will be last. That voter is the dictator who decides the outcome which is not allowed.

However, their definition of dictator was that the dictator decides the order for every collection of preferences, not just for the ones where every other voter has chosen B first or B last. So that voter is not actually a dictator at all, he is only a dictator for one particular (R1...RN) in L(A)^N, not for every (R1...RN) in L(A)^N.

So the informal proof is bogus.

I should look at a real proof.

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