In this post, I will try to describe the setting and the result as simply and as clearly as I can.

Suppose that there are more than two individuals in a given social grouping, and that there are at least three choices, denoted by A, B, C,...

The question is: how should a society order these alternatives?

Arrow starts by setting out what he claims are the minimal criteria that a social ordering should satisfy:

1) Completeness: It should always be possible to find a social ordering for any combination of individual preferences.
2) Monotonicity: If every individual prefers A over B, then society must prefer A over B.
3) Transitivity: If society prefers A to B, and if society prefers B to C, then society prefers A over C.
4) Independence of irrelevant alternatives: Society's ranking of A vs B should not be affected by the fact that choice C may or may not be available.
5) Non-dictatorship: One individual's preferences cannot completely define the social ordering.

Arrow goes on to prove that there is no social ordering that satisfies all of these criteria. The original proof of this theorem is probably not the easiest to understand; a google search on proof arrow impossibility theorem (no quotes) will yield others. And before you try developing a refutation, you should know that 50 years of determined attacks by very smart people have not put put so much as a dent in the this result.

It is difficult to overstate the importance of this theorem. It is not only one of the
most important results in economics to come out of the 20th century, I understand that
it's been cited (sorry, no reference to back this up...) as one of the most important result
in *mathematics* in the last century.

Those 5 conditions are all crucial; remove any one of them, and a coherent social ordering
is possible. But one of the strengths of the Impossibility Theorem is that no-one has
come up with a really convincing argument that one of these conditions should be set aside.
Again, before you try, remember that those conditions have survived 50 years of extremely
close scrutiny.

I'll close with a simple example. Suppose that there are 3 people, X, Y and Z, and that there are 3 choices, A, B and C.

Suppose also that:

X prefers A over B, and B over C (ordering is ABC)
Y prefers B over C, and C over A (ordering is BCA)
Z prefers C over A, and A over B (ordering is CAB)

If all three options are put on the table, there will be a 3-way tie for first place. Suppose instead that they agree to vote on each of the three possible pairings to get a social ordering.

If X sets the agenda, he'll ask for a vote to choose between A and B. Z will vote with him, so A wins over B. Next, he'll call a vote between B and C. Y will vote with him, so B wins over C. Since the order is now complete, X will declare that there's no reason to even vote for the A vs C contest (a vote he would lose).

Similarly, if Y or Z gets to set the agenda, they can engineer their preferred ordering by choosing the appropriate sequence of votes.

Since each of these outcomes will have been determined by majority vote, they are all 'democratic'. But the end result will be to simply ratify the preferences of the person who gets to set the agenda.

Arrow's theorem has also been presented as saying: Suppose that a social ordering must satisfy conditions 1)- 4) above. Then it must be a dictatorship.

Babblers who are interested in collective decision-making processes might want to spend some time thinking about this.

In another thread, I mentioned Arrow's Impossibility Theorem, the most famous result in the literature on social choice. The question he addressed was how individual preferences could be aggregated into a coherent social ordering.
In this post, I will try to describe the setting and the result as simply and as clearly as I can.

Suppose that there are more than two individuals in a given social grouping, and that there are at least three choices, denoted by A, B, C,...

...choices of what? Choosing what movies they want to see? Maybe that's not important.

The question is: how should a society order these alternatives?

...Society? Does that mean the three individuals you referred to above, or society at large?

Arrow starts by setting out what he claims are the minimal criteria that a social ordering should satisfy:

1) Completeness: It should always be possible to find a social ordering for any combination of individual preferences.

I guess I'm not understanding "social ordering". What is meant by this?

2) Monotonicity: If every individual prefers A over B, then society must prefer A over B.

Understood...ie, if every person in a society prefers chocolate over vanilla, then society prefers chocolate over vanilla

3) Transitivity: If society prefers A to B, and if society prefers B to C, then society prefers A over C.

...Ok, fairly simple logicd. If A>B and B>C then A>C, right?

4) Independence of irrelevant alternatives: Society's ranking of A vs B should not be affected by the fact that choice C may or may not be available.

Huh?

5) Non-dictatorship: One individual's preferences cannot completely define the social ordering.

Yes

Arrow goes on to prove that there is no social ordering that satisfies all of these criteria.

These were criteria? I thought they were propositions in a theorem.

The original proof of this theorem is probably not the easiest to understand; a google search on proof arrow impossibility theorem (no quotes) will yield others. And before you try developing a refutation, you should know that 50 years of determined attacks by very smart people have not put put so much as a dent in the this result.

I wouldn't dream of it...because I don't have the slightest idea of what you just described.

...I've stopped here. Perhaps I'll take it up tomorrow.

...*sigh*

[ 29 February 2004: Message edited by: Hinterland ]

Like Hinterland, I am somewhat confused by the oblique wording of question 4. It doesn't establish how C is irrelevant. Define irrelevant. This will determine whether it is a desirable criterion.

[ 29 February 2004: Message edited by: Mandos ]

You seem to be missing a lot of functions in logic there, especially when they cross more than one issue.

How about if X then A, or if B then not Y, or an if and only if A, then Z (introducing a new, connected issue, X,Y,Z).

What happens to the theorem then? There are paradoxes in logic, which would then manifest no?

And where's predicate logic in all this?

And can't we make the same claim as some economists, that people do not alway arrange (all) preferences?

[ 29 February 2004: Message edited by: Tackaberry ]

quote:

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Since each of these outcomes will have been determined by majority vote, they are all 'democratic'. But the end result will be to simply ratify the preferences of the person who gets to set the agenda.

---

Good that 'democratic' is put in quotes as clearly semantics are a big part of the issue. A head to head knock out competition which possibly can produce 3 results obviously is not the best choice and stretches the meaning of democratic. If I remember the method of providing the most 'democratic' choice is to give each player 1 vote which they can divide however they wish between the three choices. For example:

Person 1's vote:

A = .9
B = .1
C = .0

Person 2's vote:

A = .2
B = .5
C = .3

Person 3's vote:

A = .2
B = .1
C = .7

Totals

A = 1.3
B = 0.8
C = 0.9

"A" being the choice.

So Arrow (with your agreement) says rational society is impossible.

Big Friggin' Deal.

quote:

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We've seen society work, and we've seen it become disfunctional. and strangely enough, the points at which we've seen it break down have been the junctures at which your pseudoscience had significant influence.

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I think what IIA does is reduce the possibilities for manipulation. An agenda-setter just has to make sure the right combination of items are on the list in order to get the outcome he wants.

My own opinion is that as far as social sciences are concerned, it's difficult to imagine a more important issue than the question of how best to incorporate individual preferences into a collective decision-making process. And it's not just economists who are interested in this question.

What I find fascinating with the result is that Arrow tackles the question head-on, in a rigourous way. His approach is reverse-engineering: set out some specifications for a social choice rule, and see what they imply.

quote:

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Many babblers have mentioned their unhappiness with existing arrangements for social decision-making (a mixture of the market and representative democracy). I simply wanted them to be aware of what some of the issues are in social choice theory and to let them know that wanting "more democracy" is not the end of the story. The goal of the simple example was to show that there are any number of decision rules that can all claim to be 'democratic', and all will have their weaknesses. The question is what sort of weaknesses we're willing to live with.

---

To me, 1-3 are just standard characteristics of various kinds of mathematical systems. All sorts of things should be able to satisfy them. The only one with "moral content" are 4 and 5. It's not surprising that they are the tricky ones to satisfy. Since we agree that the necessity of 4 is a little doubtful, and that 5 is rather desirable, then it seems to me that this discussion is predicated entirely on a strawman---that someone was planning to propose something that attempted to satisfy all 1-5. But no one is.

To me, as others have noted, number 4 seems to be the key weakness. I've been trying to think of an example of introducing an option that reverses the previous ordering, but I can't think of one.

And if you're looking for a pertinent example, consider the 2000 US presidential election. If Nader had not been a candidate, it seems fairly likely that Gore would have beaten Bush. With Nader on the ballot, the Gore-Bush ordering was reversed.

It would appear that the US electoral process does not satisfy IIA. The implications for manipulation are clear: anyone willing to wager that the GOP did everything it could to keep Nader's candidacy viable?

I hasten to add that Nader was irrelevant only in the sense that he was not directly involved in the comparison of Gore vs Bush. In this context, Bush would have been irrelevant in comparing Nader vs Gore, and Gore was irrelevant in comparing Nader and Bush.

And I would also add that this has nothing to do with whether or not Nader had a right to run, or whether or not should have run; it's simply a comment on the properties of the decision rule used to elect presidents in the US.

[ 02 March 2004: Message edited by: Oliver Cromwell ]

quote:

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Originally posted by Oliver Cromwell:
It's not for me to defend the theorem; it's a mathematical proposition whose validity has been demonstrated to everone's satisfaction. The only question is whether or not it's interesting.

---

Very well put.
Many people here are essentially taking the practician's stance that theory is irrelevant, and so while perhaps true it is *not* interesting. Which is sometimes true (and more often in human-related sciences than in other ones), but it's usually worth taking a look at the theory before concluding it. In this particular case I'd tend to agree with them--we're way too far from perfection to have to worry about limit cases at this juncture, and in any case the messiness of real people lends itself poorly to that degree of formality.
It's noticeable for instance that game theory has often been very poor at predicting how real people will behave in games, partly because real people are not perfect calculators, partly because real people are optimized for certain game assumptions and tend to apply them even when the formal game they're in does not. F'rinstance, people do revenge, figuring consciously or just because of evolutionary heritage, that it will deter the next person who wants to pull a fast one even if it's unprofitable in itself. Game theory figures people shouldn't do revenge--except if there's repeated games. But people operate as if there'll be repeated games unless they make a strong conscious effort to think otherwise. But just the fact that some game theorists have figured some of this stuff out does suggest it's not hopeless.

I'd tend to agree that the theorem is presumably true on its own terms. And even if it has little relevance to actual decision-making, it's still fun to think about.
It suggests that no matter how close we get to true social justice, there would always be need for tinkering with the system.
But I wonder if it would be theoretically possible to reach a system such that its self-modification could oscillate between two states, each of which failed to deal properly with a different small subset of problems?

It would be if people who were going to vote for Gore, once presented with the possibility of voting Nader, then went and voted for Bush.

Look at it in terms of ordering: If you're a Democrat, you've got Gore>Bush. Now, add Nader, and different Gore-preferrers will fit Nader into that equation differently, either as Nader>Gore>Bush, or Gore>Nader>Bush. But I doubt anyone saw it as a case of Nader>GoreBush.

I've become 'sold' on the validity/necessity of IIA. What I'd argue now is that axiom 5 can be minimized to such a degree that decision-making *is* collective by any standard you care to apply. 'Setting the agenda' could be minimized to the point of having a rotating meeting chair, or a randomly selected one, minimizing the degree of agenda setting to the point (almost) of nonexistence.

That may be some form of reductio argument, my philosophy days are far behind me.

Your point of 'having to accept some degree' of failing to completely satisfy all five axioms is a good one, but I think there's a case for reducing one or another of them to virtually nothing.

quote:

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One answer may be found in a series of psychology experiments conducted at Princeton University in the 1950s. Princeton social psychologist Solomon Asch showed a room of participants a series of slides displaying sets of vertical lines. Two of these lines were clearly the same length, while the others were obviously very different. The subjects were then given the seemingly trivial task of identifying which pair of lines were the same. But there was a trick: Everyone in the room except for one person had been instructed beforehand to give the same incorrect answer. The real subject of the experiment was the lone unwitting participant, and the real test was of an individual's ability to disagree with his or her peers.

Asch demonstrated a stunning effect: Faced with a decision that, in isolation, no one would ever get wrong, the unwitting subjects went against the evidence of their own eyes about one-third of the time. In psychology, Asch's result is famous, yet its implications for what we might call "social decision-making" (decisions that are influenced by the previous decisions of others) are largely unappreciated by the general public, or even researchers who study decision-making. And social decisions are everywhere. From the everyday (choosing a movie or a restaurant) to the profound (choosing a religion or a career), each one of us is influenced consciously and unconsciously by our friends, families, colleagues, and role models in ways that make the boundary between what we decide for ourselves and what others decide for us almost impossible to distinguish.

In many situations, social decision-making isn't a bad idea at all. After all, the world is a complicated place, and other people often do have information that we lack. So, we can often do reasonably well, or at least no worse than the people we are copying, by letting them do the hard work for us.

But sometimes the people we are copying aren't working either, and that's where the problems come in. When everyone is looking to someone else for an opinion—trying, for example, to pick the Democratic candidate they think everyone else will pick—it's possible that whatever information other people might have gets lost, and instead we get a cascade of imitation that, like a stampeding herd, can start for no apparent reason and subsequently go in any direction with equal likelihood. Stock market bubbles and cultural fads are the examples that most people associate with cascades, because they are generally accepted to represent "irrational" behavior (although, curiously, not to the people who are participating in them—just ask a teenager why she wants to get her navel pierced; she won't say "because it's a fad"), but the same dynamics can show up even in the serious business of Democratic primaries.

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On the reference to fractals, I would suggest looking at non-linear dynamics (aka chaos theory or complexity theory), it has a lot of application to how economic systems behave in the real world

However, it should be noted that this area of economic thought is heavily conditioned by ideology - and I'd argue that it's ultimate purpose is ideological.

Specifically, its function is to promote a "libertarian" view of current democratic institutions as irrational, inefficient, and consistently perverted by "special interests" (used in the same sense that Mike Harris used the term). It's tied back to Hayek and the whole "capitalism = democracy" school.

If you want an example of where this kind of thought leads - which is basically to the conclusion that "government sucks" - here's a gem from the Fraser Institoot - "Many still assume that governments act in the public interest ..."

One of the questionable/debatable assumptions is that "collective choice" is simply the sum of individual preferences. From a different viewpoint, you can criticize this "axiom" as ill-founded: it ignores the possibility that, whatever "collective choice" means, it is a much more complex pheonomenon than a sum of individual preferences; and it seems to have very little explanatory and predictive power in the real world (as opposed to in the pages of some economics journals.)

Economists who indulge in this kind of theorizing are trying to extend their assumptions about "economic man" (a rational, utility-maximimzing individual) to include the realm of political theory. (To some extent, this reflects the views many economists have that theirs is the only social science worth speaking about, and all the historians, poltiical scientists, sociologists, etc. are really just dunces.)

The median voter will understand ...

[ 09 March 2004: Message edited by: Wellington ]

The claim that "it's ultimate purpose is ideological" is going to require some backup. You may not like the result, but you cannot dismiss it out of hand without a better argument than "it has no substance." On what ideological grounds are you willing to challenge any of the five criteria upon which the theorem is based?

It should also be noted that the theorem has nothing to say about the optimality of market outcomes. As I noted in an earlier post, the market is simply another way of making choices about resource allocations, and that it is also subject to the Arrow impossibility theorem. Unless you are willing to make extra assumptions such as diminishing returns to scale and no externalities - assumptions that Arrow does not make - markets will not be optimal.

There is nothing in the theorem that claims that collective choice is the sum of indivividual preferences. In fact, you look at the theorem more closely, you'll find that the term 'sum of individual preferences' is a meaningless expression in this context. I mean that literally: how does one add the orderings ABC + BCA + CAB?

It may be that you didn't mean that. But do you really mean to say that individual preferences should be ignored in collective decision-making?