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Author Topic: Question About Arrow's impossibility theorem
docmagik
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Came across a mention of Arrow's impossibility theorem today, but I don't think I'm quite bright enough to wrap my mind around the wikipedia entry.

http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem

Does anyone understand it enough to try to explain it more simply? I don't understand what it is or how it shows it.

[ November 05, 2008, 07:05 PM: Message edited by: docmagik ]

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docmagik
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Apparently, I am also not smart enough to wrap my mind around UBB code.
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Jhai
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As I recall, Wikipedia as a pretty poor explanation of Arrow's Impossibility Theorem. Try here.

If that doesn't answer your questions, let me know. I once wrote a term paper on the theorem (which is the central part of his thesis - a nice neat little book) for Advanced Logic.

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Jhai
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To expand a bit (just wanted to get that first post off):

Arrow's goal was to specify a few simple requirements that any logical person would think the rules for voting should include. He spelled these out very carefully, and then proved that there was no way you could make a set of voting rules that couldn't violate at least one of the rules.

The rules, simply put, are as follows:

1) No dictator - that is, you need to have more than one person voting. I think it's obvious why this ought to be in the set of rules - if only one person's vote counts, well, it's not a very good way to figure out what society, as a group wants, right? This one isn't controversial at all.

2) If everyone voting prefers Ms. X to Mr. Y, it shouldn't be possible for Mr. Y to win. It'd be quite the upset if Mr. Y won, despite the fact that everyone disliked him compared to Ms. X, hmmm? So our voting rules shouldn't allow that to ever happen. Again, not controversial.

3) People's preferences need to be transitive. You might have run across the idea of transitivity in one of your math classes, like geometry. Basically, this means that if you prefer Ms. X to Mr. Y, and you prefer Mr. Y to Sir Z, you should prefer Ms. X to Sir Z. It should also work if you don't have a preference between your choices. If you are indifferent between A & B, and indifferent between A & C, you ought to be indifferent between B & C. This one is a bit more controversial, especially if you put it into real world terms. For instance, suppose I'm choosing paint colors for my house. If you put a true black paint chip next to one that's almost, but not quite, black, I won't be able to see a difference between the two, so I'll be indifferent. Then if you put a slightly grayer chip next to the almost-black chip, I again won't be able to see a difference, and thus will be indifferent. Do this enough, though, and transitivity will eventually end up telling me I ought to be indifferent between black & white!

4) Voters need to know what their preferences are for all the voting options available to them. This means that I ask you if you prefer Mr. Y or Ms. X, you need to be able to give me an answer, even if your answer is "I'm indifferent". You can't not know what your preferences are. This one seems reasonable at first glance, but Arrow's Social Choice setup is used outside of just the voting box, which has a natural limit on the number of options you're given. For instance, a number of microeconomic theories require this point to hold true. But do I really know if I prefer 500 boxes of raisins and two jumbo jets over 3,000 camels or not? Well, I'd have to at least think about it some...

5) If you've decided you prefer Ms. X over Mr. Y, your preference between those two shouldn't change if I add on Sir Z on the ballot. This one is the most controversial of the bunch, mainly because we know enough about human psychology to know it's not true, even if we wish it were. By adding Sir Z the radical onto the ballot, I might make Ms. X, who you initially preferred, look fairly conservative - so you change your mind, and end up preferring Mr. Y, who leans liberal, over Ms. X. Note that I'm not referring to your preferences about Sir Z at all - it's just that adding him to your ballot made you change your mind about how you rank X & Y.

---

So what Arrow did was prove mathematically that all of these conditions cannot exist at the same time - which turned out to be fairly problematic for not just political scientists but economists as well. I hope that explanation is clear? Oh, and remember that these conditions are written in a more compact form mathematically, so they're normally not broken down into five parts - four or three is the norm (can't actually remember how Arrow writes it). However, when you translate it them into English it's easier to explain using five conditions.

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docmagik
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Jhai, thank you so much! That is a perfectly understandable explanation.

How important is the last one to the whole thing? Because like you said, the last one seems to be the one that explodes the whole thing.

If I'm trying to eat healthy, and I'm given the choice between a fun sized candy bar and an apple, I'll pick the apple. But if I'm given the choice between an apple, a fun sized candy bar, and a full sized candy bar, I might consider the fun sized candy bar a good compromise and grab it.

Or, it might go the other way. Given the choice of just the two, I might want the fun sized candy bar, since it looks so small next to the apple, but when you put a full sized candy bar next to it, it might remind me the fun sized candy bar is junk food and make me go for the apple.

So is the last one vital the whole thing, or not?

*

So how could, say, the second condition ever be violated?

*

Your explanation of the problems with the 3rd one make perfect sense as well, especially with regards to indiference.

I may not have a prefrence between 1-ply and 2-ply toilet paper, or between 2-ply and 3-ply toilet paper, but that doesn't mean I don't have a prefrence between 1-ply and 3-ply paper.

How vital is it that indifference be transitive?

*

And my last question:

On number 4, is it just vital that they know off the top of their heads, or would it be sufficient for the "rules" if we simply said, "the voter would have to be able to arrive at a prefrence for all options, once they gather sufficient information to make a choice," or something to that effect.

In other words, is the theorem that the guy has to know whether he wants the planes & fruit or the camels right now, or simply that he has to be capable of rendering a vertict eventually, be it A, B, or indifferent?

Because it seems to me like the concern for that one is partially resolved if you simplify it to, "The person has to be capable of making a decision eventually," but then you're stuck with the fact that there are simply some people who simply can't make up their minds.

*

Again, thank you for taking the time to talk about this. It's very gracious of you.

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Jhai
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Let's see.

They're all five required for the impossibility of Arrow's Impossibility Theorem to take hold and make it impossible for any set of rules to satisfy the requirements. There might be some voting schemes that could violate, say, 1 through 5 - and thus they would be "bad" by these rules - but if you take away number 5 it could be quite possible to come up with a social choice mechanism (SCM) that satisfies the first four. How viable that mechanism/voting rule would be for real life voting - with all the constraints on people's time, willingness, and so forth - I don't know.

However, even though eliminating number five makes a coherent SCM possible, it takes away a lot of flexibility we'd like to have in our SCM's from a theoretical perspective. Arrow was an economist, and his thesis wasn't just about creating an interesting problem for political scientists - it has some major implications for economists. Basically, keeping condition five in gives a lot of mathematical "wiggle-room" when we try to add up people's preferences to get a working model of an economy. You can still do things without condition five, but the (theoretical) results are a fair bit more difficult to get to. So Arrow burst a lot of microeconomic theorists' bubble with this problem of his. (I can look up an example for you when I get home, but I can't recall it off the top my head.)

Also, at least from my perspective, it'd be nice if we didn't let third alternatives change our mind about a choice between two things. I should be able to decide whether I prefer an apple or a "fun" size candy bar rationally, and the existence of a full-sized candy bar shouldn't sway me when I'm weighing my alternatives, darn it! But perhaps I just think that way because I've been indoctrinated into the ways of the professional economist.

---

Off the top of my head, two is really used just to set up the problem by requiring a Pareto efficient solution - basically, it says "let's at least get to the point where you can't make someone happier/better off without making someone unhappier/worse off." If condition two wasn't there, then if the SCM came up saying that the person everyone hated had won (Sir Z), we could make at least one person happier by choosing anyone else (say, Mr Y). Maybe 99 people like Ms. X the best & hate Y & Z, one person likes Mr. Y the best & hates X & Z - basically everyone hates Sir Z. Well, if we somehow ended up at the point where Sir Z was chosen (who knows how?), if we toss him out and get Mr. Y, one person is happier, and everyone else is no worse off. Condition two makes sure we'll always do that.

Arrow didn't originally have condition two in his formulation - I think because game theory and Pareto efficiency hadn't really been invented yet. He had conditions which were slightly stronger (i.e. required more assumptions), so it's considered an improvement to shift to condition two.

---

Again, to get the Impossibility, we do need transitivity to hold up to indifference, not just strong preferences (actual preferences of one thing over another). And, like with condition five, it's very helpful theoretically in lots of "applications" to have transitivity over weak preferences (i.e. including indifference), but you can do without it in some cases. (I'm not sure if this will make any sense outside my mind, but you can kind of think of it as the difference between a dotted line & a solid line - a solid line is just more "there" and you can work with easier than with a wishy-washy dotted line.) Anyways, even dropping indifference, you can often frame things so that in actual lab experiments people violate transitivity. Just showing a lot of options helps - if you get people to show a strong preference over various items on a table, after awhile they're likely to violate transitivity: they showed a preference for the mug over the pen, and for the pen over the marker, but like the marker more then the mug. Note that they can see all the options the whole time - so it's not like we're testing condition five.

---

You don't have to know off the top of your head - but frankly, I still don't know if I prefer 500 boxes of raisins and two jumbo jets over 3,000 camels or not. I'm not sure if, given a lot of thought, I'd be able to make a decision. The idea I was getting at with that example is that people are pretty good at telling you their preferences about decisions they encounter in their daily lives, but have problems when it gets outside the realm of their normal lives. To take a more realistic example, very rich people can probably decide what configuration of aircraft (helicopters, big planes, little planes, etc) they'd like to have at their beck & call, but I don't know if I could properly weight the value of them well. I've never needed to, and it's quite difficult to imagine whether I'd prefer that third helicopter over two 4-seater planes or not.

---
I think that's all of 'em. You're welcome by the way, but don't worry - this is really fun for me. I don't get to talk about these sort of theoretical things very often, and I've spent a couple of years studying these things.

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rivka
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quote:
But do I really know if I prefer 500 boxes of raisins and two jumbo jets over 3,000 camels or not?
YES! Camels spit, you know. And bite.
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King of Men
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quote:
Originally posted by Jhai:
[QB]You don't have to know off the top of your head - but frankly, I still don't know if I prefer 500 boxes of raisins and two jumbo jets over 3,000 camels or not.

It seems to me that you could decide that after maybe an hour, if you can't make a choice, then you'll look up the market values and go for the more expensive choice. If you later change your mind, then you can sell them and get back the other, plus some cash.

Of course, that doesn't work in all cases, like choosing a politican to vote for.

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Jhai
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That assumes that I can easily sell the camels/raisins/jumbo jets. That's not necessarily true - in fact, in many economic models it's explicitly not true. Although, typically, free disposal is assumed (i.e. you have no problems getting rid of any stuff you don't want), but just because it's a hassle to deal with, rather than mathematically necessary. Models dealing with things with high disposal costs (like nuclear waste), don't, of course, assume free disposal.
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TomDavidson
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Actually, I've always had a question about Arrow's theorem: why does the "dictatorship" element matter? I mean, yes, you do hit a point at which hypothetical individual X's vote creates the "tipping point" for the final decision. But since it would be practically impossible for anyone to identify X in anything resembling a complicated scenario, there doesn't seem to be a lot of room for abuse. What's the perceived problem with the importance of X?
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Jhai
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Well, if you're okay with doing away with any of the conditions, then you're free to do so. The reason why most theorists reject that option for condition two is because it violates the intuitive idea of a social choice mechanism. Once you set up the system of rules, person X will be the only person who decides the winner every time, no matter the actual preferences of everyone. You're right that it seems unlikely to matter much in practical purposes - but, then, we're violating some condition every time we hold an election. Still, it seems pretty odd to say that we're trying to find out society's preferences, only to allow one person's opinion to rule over all.

Bear in mind that at this point I'm approaching Arrow from a very economic mindset, where we use these ideas all the time to do micro - a political scientist might give a very different answer to your question.

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El JT de Spang
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So, I was feeling smart because I understood the wiki article on first glance.

In fact, all of the conditions seemed self-evident. Oh, wait...I think that means I don't, in fact, understand it. Ah, well.

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TomDavidson
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quote:
Once you set up the system of rules, person X will be the only person who decides the winner every time, no matter the actual preferences of everyone.
But that's not the case in reality, is it? Because if people's actual preferences are really different in the aggregate, then the person who is X will change each time, too. Right?
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All4Nothing
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Some forums you visit and the more you read the dumber you feel for having wasted your time and poisoned your mind with reading nothing but nonsense. This forum I find myself learning and having the chance to experience knowledge I may never have come by. Thank you all for that!

BTW.....when I read the theorem what I took away at first glance was: That if you want to figure out a societies preference based on individual preferences you can only do so accurately if there are only two choices, cause when you add a third choice the whole thing breaks down.

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Jhai
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quote:
Originally posted by TomDavidson:
quote:
Once you set up the system of rules, person X will be the only person who decides the winner every time, no matter the actual preferences of everyone.
But that's not the case in reality, is it? Because if people's actual preferences are really different in the aggregate, then the person who is X will change each time, too. Right?
Maybe it'll help (at least for those of us mathematically inclined) if I give a good formal definition of the non-dictatorship condition:

quote:
Nondictatorship: No person i is decisive for every pair of outcomes in O. (where O is the space of all logically possible profiles of preference orderings and "every pair" refers to a comparison of two possible outcomes).

Decisiveness: An individual i, is said to be decisive for alternate x against alternate y if at every profile in the domain of the rule, if x Pi y for individual i, then for any agenda containing x, y will not be chosen - even if all non-i's prefer y to x. (Where x Pi y means that person i prefers x to y, profile refers to a possible set of preferences for everyone voting, "domain of the rule" means things that allowed in the SCM you've set up, and agenda refers to the voting choice being faced at that time).

If individual i is decisive for every pair of alternatives in O, we say that the individual is a dictator.

That's from a well-written & explained proof that can be found here as a pdf. If you don't know a bit about mathematical proofs, you might get lost reading that, though. There's a constructive graphical proof aimed at teaching undergraduates here that might be helpful for the less mathematically-inclined among us.

So to answer your question, Tom, yes, it is the case in reality that the dictator, whoever he might be, decides the game once you've set up the rules for the game. Even if everyone prefers y over x, the dictator will force x to win over y (assuming you're not violating the other four conditions with your voting rules). The reason why we don't see this in real-life voting is because we're already violating one or more of the other conditions. You can read a bit more about dictatorship in Arrow from this recent paper (pdf).

All4Nothing - glad you're learning stuff. Your take-away is pretty good & succinct, btw. And note that run-offs if you have more than two contenders doesn't help solve the problem, because the way you pair the run-offs can completely determine the conclusion - anyone who's watched March Madness knows this.

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aspectre
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3) Sizzors? Paper? Rock?
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TomDavidson
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This reminds me of the time I spent back in high school geometry, arguing with my teacher that there was no reason I should accept that three points HAD to lie on a plane. [Wink]

--------

Seriously, though, I'm having trouble mapping the math to a real-world situation in which dictatorship is problematic.

I mean, yes, there will arise a situation in which person X casts the deciding vote between alternatives Z,Y,and W when 100 people are voting. But if 101 people vote, the person who happens to be "X" -- and the result of X's vote -- is different, right? And if the alternatives are actually Z, Y, and C, X also winds up being a different person, right? Or am I misunderstanding how the math maps to reality?

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fugu13
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It isn't that the person casts a deciding vote, it is that their preferences dominate in every vote.

Unpacking what Jhai quoted a bit, the key line (slightly translated) is, "If the dictator prefers x to y, then every time there's a decision where x could be chosen, y won't be, even if everyone else prefers y to x".

That is, it isn't someone deciding on one marginal vote, then someone else deciding on another. It is one person's preferences being the ones that dominate every decision.

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Jhai
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Tom, you're missing the fact that you've already violated another condition. The voting rule you're using is simple majority, right? I'm pretty sure that one violates Condition 5.

Consider the case where 7 voters have preferences A > B > C, 6 voters have preferences B > C > A, and 5 voters have preferences C > A > B. If we use a simple majority rule, then A wins. But if B were eliminated from the ballot, C wins. But by condition 5 (independence of irrelevant alternatives), our choice between C & A shouldn't change if we have B on the ballot or not. After all, society's belief about whether C or A is truly the better candidate doesn't depend on whatever B is doing. One of the two is the most preferred candidate (ignoring the possibility of indifference) - and shouldn't the way our social choice mechanism chooses reflect society's belief about the best candidate for the job?

So by violating condition 5 (I hope I've made a good case why it'd be preferable to not violate it), you've broken into the realm of a possible SCM that doesn't rely on a dictator. If you used a set of voting rules which didn't violate the other conditions, you'd be stuck with a dictator.

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TomDavidson
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quote:
The voting rule you're using is simple majority, right?
Doesn't have to be.

-------

I think what I'm hearing is that "dictatorship" is used here as what I like to call the "scary default setting" when I'm talking about software. In other words, since the other conditions are considered mutually exclusive, the only way to not have them apply is to limit the sample size to a single person.

Why not just leave out "dictatorship" altogether, then, and just say "you can't have these four things when there's more than one person voting?" By adding "dictatorship" as an option rather than making it the clear baseline, it seems to indicate that dictatorship itself is a problematic element. But it's not. It's that the point of the exercise is to find some way for all four of those things to coexist, and they can't.

I worry that by including non-dictatorship as one of the optional conditions to be satisfied, the theorem is applied inappropriately to all sorts of silly speculation on types of government (or economic choice).

It's like saying "Pick a fruit. Your fruit must be an orange, or else it must be simultaneously yellow, purple, cherry-flavored, and Fred."

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fugu13
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You can have all four of the other things co-exist, you just can't have them co-exist without having only one person's preferences ending up mattering.

And that isn't necessarily through them just issuing edicts. It could be with fairly complicated voting rules that look like they're democratic.

Arrow's Theorem just has a variety of conditions that are generally considered favorable (for a wide variety of reasons) to have in a democracy (and no one person being the one whose opinions are ultimately followed is one of them), and shows that those conditions cannot all be met at once.

I'd be interested in seeing one example of the theorem being applied inappropriately to silly speculation on types of government that relies on calling not having a dictatorship one of the five conditions that can't all be met together.

(which is equivalent to saying, if we don't have a dictatorship, we can't have all of the other four things, so I'm still not certain where your objection is).

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Jhai
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I feel like this is a "translation of a proof to the English language" issue. The other four are only mutually exclusive if you want to have more than one person's vote matter. The way the proof runs in Arrow's original formulation is that you try to get the other four to work together, and then you end up having only the dictator's vote mattering - which you've already said is a bad a thing at the start. You don't have to say that it's a bad thing at the start, but why not? It's pretty obvious that ending up with a dictatorship fails the whole social part of the social choice mechanism.
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TomDavidson
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quote:
You can have all four of the other things co-exist, you just can't have them co-exist without having only one person's preferences ending up mattering.
But my point in that case is that the "dictatorship" in those situations is not a form of dictatorship that should be considered particularly problematic, since the identity of the dictator and the result of the vote changes with the number of people voting and the options available.

----------

quote:
I'd be interested in seeing one example of the theorem being applied inappropriately to silly speculation on types of government that relies on calling not having a dictatorship one of the five conditions that can't all be met together.
I've seen people summarize Arrow as "it's been proven that the only fair form of election is a dictatorship." That's obviously incorrect, but I think it's encouraged by the selection of terminology.
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Jhai
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quote:
Originally posted by TomDavidson:
quote:
You can have all four of the other things co-exist, you just can't have them co-exist without having only one person's preferences ending up mattering.
But my point in that case is that the "dictatorship" in those situations is not a form of dictatorship that should be considered particularly problematic, since the identity of the dictator and the result of the vote changes with the number of people voting and the options available.
The person doesn't change once you've set up the rules of the game - the options available don't matter. Rules, of course, include who is voting. And I, at least, think it's incredibly problematic to have a voting system that actually doesn't depend on what society - or everyone besides that dictator - wants. It's sort of like saying that that cabal of evil white men (they are white, right?) that control the media and the economy don't matter because no one knows who they are, and they have a rotating membership.


quote:
Originally posted by TomDavidson:

quote:
I'd be interested in seeing one example of the theorem being applied inappropriately to silly speculation on types of government that relies on calling not having a dictatorship one of the five conditions that can't all be met together.
I've seen people summarize Arrow as "it's been proven that the only fair form of election is a dictatorship." That's obviously incorrect, but I think it's encouraged by the selection of terminology.
I've seen it used that way - but only by people who are examining the issue in a very basic way, like undergraduates who run across it in Game Theory 101 - and only if their professor is relatively incompetent (which means that I'm more worried about what they're getting out of the Prisoner's Dilemma, frankly). Anyone who actually uses it in their professional life - i.e. social choice theorists, economists, biologist, etc - understands what the theorem does & does not give you.
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fugu13
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Of course the outcome of a vote changes with the options available. And no, the dictator would not change with the options available.

I'd have to think more about the number of people voting question, but in most voting systems not voting would be effectively voting for one of the options (or perhaps several of them in some proportion), in which case the theorem would still hold, the outcome would not change (holding the options constant), and the dictator would remain the same person.

And the question of options available is mostly just because that's a flexible formulation of the theorem (which can still be proven). The theorem doesn't deal with how options are picked to be voted upon.

Not having a dictator is one of the five unfair conditions shown not to be possible to avoid in Arrow's Theorem, so I suspect it is less terminology, and more the idea that a dictator would be cool (at least if the person making the silly statement were a dictator).

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TomDavidson
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quote:
And no, the dictator would not change with the options available.
Assuming that each person involved might change their preferences based on the different options (or number of options), why doesn't that produce a different individual dictator?

quote:
And I, at least, think it's incredibly problematic to have a voting system that actually doesn't depend on what society - or everyone besides that dictator - wants.
But that's what I'm not seeing in the system; it's precisely that conclusion that I'm not seeing as something inevitable. I mean, as I understand it, the "dictator" is actually produced by a combination of the number of people voting, the things there are to vote about, and how they'd vote for them. In this case, it turns out that what they're really voting for is the dictator, who -- once chosen -- then casts the single actual vote. That doesn't seem like an exploitable system to me.
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fugu13
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The theorem assumes (reasonably) people have preferences over all the possible options -- that is, if A and B are up against each other, the person will prefer A to B. Things like that. If A and B aren't up against each other, the person still has that preference, just not the option.

So, to reiterate, the dictator is not influenced by the options available or the number of people voting. The particular dictator will depend on people's preferences and the voting rules in question, of course.

And no, people aren't voting for the dictator who casts the actual vote. There could be a complicated series of voting rules involving run-offs between options which are too close together or somesuch; the net effect is that only one person's opinions matter: there is no direct election (unless that's the 'voting rules' chosen).

There are other ways the person isn't a literal dictator, too. For instance, changes in preferences change everything.

But there's no worry about that part of Arrow's Theorem applying in most situations -- our voting rules rarely come close to dealing with situations where most people prefer either A or B to C, yet C still wins (because A and B split their part of the vote), much less every aspect of the four conditions that must be met before an effective dictator becomes impossible to avoid.

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Jhai
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Fugu, it's not true that the dictator changes based on people's preferences over the options to vote on. To quote a bit from the paper on dictatorships I linked earlier:
quote:
In the standard multi-profile world, where all preference profiles are allowed (the so-called “universality,” or “full domain” assumption) a dictator is a very bad thing indeed. A dictator in such a world forces his (strict) preference for x over y even if everyone else prefers y over x.
The paper continues to say that if you have a single-profile world - i.e. the preferences are taken as given, then you can end up with "innocuous dictators", such as the guy who is indifferent to everything. But the innocuous dictators are only innocuous with very specific preference profiles - given another profile & the same rules, you end up with a non-innocuous dictator.
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TomDavidson
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quote:
And no, people aren't voting for the dictator who casts the actual vote. There could be a complicated series of voting rules involving run-offs between options which are too close together or somesuch; the net effect is that only one person's opinions matter...
But that's my point. In effect, by choosing to vote, and by voting a certain way (i.e. those changes in preferences which change everything), they are actually selecting -- blindly -- which person gets the real vote. And the specific person who gets the vote arises organically and naturally from the number of people voting, the preferences they're voting for, and the number of preferences available.

What I don't understand is why that's an undesirable circumstance. If the system can fairly produce one person with a vote based on the expressed preferences of everyone else who voted, are the latter people really being marginalized? Aren't their expressed desires actually reflected -- albeit unknowingly -- in the person who gets to matter?

-----------

quote:
it's not true that the dictator changes based on people's preferences over the options to vote on.
Jhai, this is the part that confuses me. I can't make the math here work out when I try to translate it to reality. Doesn't the identity of the dictator specifically depend upon placement within the matrix of choices? How is it possible, for example, for a complete "fringe" option to wind up the dictator under a model that meets the other four requirements?
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Jhai
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Since there seems to be a severe lack of mathematical reading ability in this thread ( [Smile] ), let's go over that definition I posted earlier.

Here's the most important bit:
quote:
Decisiveness: An individual i, is said to be decisive for alternate x against alternate y if at every profile in the domain of the rule, if x Pi y for individual i, then for any agenda containing x, y will not be chosen - even if all non-i's prefer y to x.
To translate that entirely into English:

A) There's some person, who we'll call i, who will be described below. We're going to describe him in such a way that he can be called "decisive".

B) We've set up the rules of the voting system so that we know who is voting, and how they will vote (pick one option, rank them all, assign points, etc), and how we will combine the votes (add together all the points, give points based on rankings, guy with the least votes wins - whatever) - this is what is meant by "the domain of the rule". Note that this doesn't tell you what is being voted on or how many options there are - we've laid out the rules to roulette, not said how many times we're going to spin the wheel. Also note that the number of people voting only affects who will be dictator, and doesn't affect whether there will be a dictator - i.e. we don't know who will do the best at the roulette game unless we know who is around the table, but we do know that there will be someone who does the best (or the least worse - we're not getting into probability theory here).

C) Our decisive voter i prefers x over y - if he were deciding which of the two to vote on (at this point it doesn't even matter if they're on the ballot), he'd go with x.

D) Now let's suppose x and y are both on the ballot - that is, they are options that we can vote on (since otherwise we're just talking about i's preferences, and that's not what we're here to do today). To use the correct terminology, that means that we have an agenda containing x and y. The word "profile" is sometimes used rather than agenda.

E) Because i prefers x over y, on our ballot which contains x and y, y can never win - no matter what the other people voting think about x and y.

E is really the key point here - this is what makes us call voter i decisive. A dictator is someone who is decisive for every possible x and y pairing contained in the universe - whether we're voting on them or not. Now, if we aren't voting on, say, a and b (if they aren't on the ballot), then i's decisiveness doesn't matter for a and b. But if we did happen to vote on them, it would - and i's preferences would rule over everyone else's.

Edit: and in case it isn't clear, since the dictator is required to have an opinion on every option (that's one of the other four conditions), and since for every pair we compare, the dictator's preference decides who will not win, once he compares all the pairs, he'll have decided the winner by making everyone else a not-winner. It's like March Madness, except that there's one dude who decides who doesn't get to advance - once you've decided that enough, there's a winner.

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fugu13
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Jhai: I think you're misunderstanding. I'm saying that if various people suddenly start preferring B to A where they used to prefer A to B (in the preferences that back any particular voting decision), while there were still necessarily be a dictator (since the voting rules haven't changed, and they're what insure the other four conditions hold), but who that dictator is might be different. After all, otherwise we would always be able to name who the dictator would be given any set of voting rules that met the other four conditions, without knowing anyone's preferences, and I'm pretty sure that's not true (and could be easily proven by showing there are some symmetric voting rules that meet the other four conditions).

Again, Tom, no, the number of people voting does not matter, and the number of preferences available does not matter. And changes in individual preference only matter if they're really changes in preference, not just changes in voting because of changes in the options available.

As for why this is bad, it is because the preference of the dictator wins out even if everyone else holds an opposing preference.

But yes, in many ways the theoretical dictator is less offensive than people assume is meant by dictator.

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Jhai
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quote:
Originally posted by TomDavidson:
quote:
it's not true that the dictator changes based on people's preferences over the options to vote on.
Jhai, this is the part that confuses me. I can't make the math here work out when I try to translate it to reality. Doesn't the identity of the dictator specifically depend upon placement within the matrix of choices? How is it possible, for example, for a complete "fringe" option to wind up the dictator under a model that meets the other four requirements?
You've got two options here, Tom - one, you could work through one of the proofs I linked to above to see the math work itself out, or two, you can come up with a voting system that meets the other four requirements - and then I'll show you how it's really a dictatorship. I've never tried that hard to find a system that works for other four conditions, but then, I've followed the math, and understand the proof, so I don't feel the need for real world examples. If you can come up with a voting rule that fulfills the other conditions, I'd like to see it.

I think the constructive proof I linked to above for undergrad math basically shows the dictatorshipness for a simplified set of options as an example.

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fugu13
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Jhai: I'm pretty sure differing numbers of voters can be figured in under the voting rules/framing of the options, since non-voting is just equivalent to a special sort of vote.
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Jhai
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quote:
Originally posted by fugu13:
Jhai: I'm pretty sure differing numbers of voters can be figured in under the voting rules/framing of the options, since non-voting is just equivalent to a special sort of vote.

You probably could - I haven't thought about the math, but it seems like you could declare not voting as an indifference over all the options for the purpose of the mechanism. However, you still need to have a total number of voters possible - otherwise you run into problems with infinity. So you don't need to know who's at the roulette table, but you do need to know who's at the casino, and it does need to be less than infinite.

And okay on the other one - I thought you were saying that as what the options are changes, then the dictator changes.

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fugu13
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Yeah, but you could set the total number of voters to any arbitrary finite number, and account for 'voters' that didn't exist in a way similar to how 'voters' that didn't actually vote were accounted for (not quite the same way, as some voting mechanisms will have quorum rules and the like that would not be impacted by the imaginary voters).

And whether or not not voting was indifference would depend on the mechanism. In some mechanisms, not voting would be equivalent to a no vote or a yes vote (on a yes/no to a particular options scenario), for instance. So it would be pretty complicated, but it would work.

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TomDavidson
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quote:
I thought you were saying that as what the options are changes, then the dictator changes.
As a practical matter, isn't this true? As the options change, the person who fits the criteria "prefers x to y for all cases" changes -- since the actual, specific options x and y are changing, and thus the people who prefer x to y are a different set of individuals.

As a result, the person who actually winds up dictator might be someone who prefers x to y or someone who prefers y to x. And who that person is is determined by the preferences of the other people voting.

Ergo, their preferences have "selected" for a dictator, who in turn makes the decisive vote.

-------

For my part, I can't come up with a way to entirely satisfy the latter four conditions. Presumably, however, it's possible. If it were not, the non-dictatorship condition would be unnecessary.

As for working through the math myself: heh.

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fugu13
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No, you're not getting it. Preferences are over all possible options. The dictator is only determined by the preferences of all voters (in the loose sense Jhai and I have been discussing) and the voting rules. The particular set of options available for any one round of voting only change the outcome, not who the dictator is (and it is the dictator who determines the outcome, assuming all four of the other requirements are met).
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Jhai
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No, it's certainly not true Tom. I'm not sure what or who you're referring to when you say "the person who fits the criteria 'prefers x to y for all cases'". There need not be someone like that as dictator or anything else. First off, given the last condition, saying that someone prefers x to y, stop, is the same as saying someone prefers x to y for all cases, if by cases you mean the different ways that the ballot could be drawn up - i.e. the agenda.

The criteria for the dictator is someone who is decisive in all parings. Whether those parings actually happen to be on the ballot doesn't matter. Or, to be more explict:

Case 1: A, B, C, D, E and F are the possible options that could be put on the ballot - let's say that together they cover all logically possible options on the issue we're voting on. All of them are put on the ballot, and the dictator, who we'll call Mr. X, prefers them in the letter order - i.e. A is his first choice and B the second and so forth. Everyone else prefers them in the reverse order - i.e. F > E > D, etc. In this case, A will win the election, even if everyone else prefers F.

Case 2: Exactly the same universe as in case 1 - the voters, voting rules, and preferences stay the same, except that A is left off the ballot. In that case, Mr. X likes B best out of all the options available to him, and therefore B wins, 'cause he's still the dictator.

It's logically possible to come up with a set of rules where this exact outcome would come true AND the other four conditions are satisfied. Difficult, yes, but quite possible.

Edit: note that it's also possible to come up with a set of rule that satisfy the other four conditions and sets up one of the other people voting to be the dictator. So then F would win in both cases - but it's not because he has a majority. It's because now the dictator is someone who likes F the best. The point is that the way the voting rule is setup let's the dictator chose, and that may not be what society wants by any reasonable definition we'd like to use for "what society wants".

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TomDavidson
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quote:
The particular set of options available for any one round of voting only change the outcome, not who the dictator is (and it is the dictator who determines the outcome, assuming all four of the other requirements are met).
Yeah, like I've said, this is specifically the part I don't understand. If a dictator is defined as the person whose preferences are authoritative over all possible options, and arises not because of a deliberate mechanism but rather a mathematical inevitability, how can the identity of that person not change based on the selections of the other voters?

---------

quote:
Case 1: A, B, C, D, E and F are the possible options that could be put on the ballot - let's say that together they cover all logically possible options on the issue we're voting on. All of them are put on the ballot, and the dictator, who we'll call Mr. X, prefers them in the letter order - i.e. A is his first choice and B the second and so forth. Everyone else prefers them in the reverse order - i.e. F > E > D, etc. In this case, A will win the election, even if everyone else prefers F.
This is exactly the kind of example I was imagining and not understanding. In what kind of bizarre system would this be true, that the person whose choice is authoritative favors the least-favorite choice of everyone else, provided the other four requirements are kept intact? I can't think of a scenario.

[ November 06, 2008, 01:52 PM: Message edited by: TomDavidson ]

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Jhai
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To answer the first bit:

Because the dictator is determined by the voting rules and the preferences in the universe, not the options on the ballot. Thus it doesn't matter what people do and do not select - since selection is partially determined by what is on the ballot. People can't select an option that's not on the ballot.

To answer the second bit:

(I think) Because you're thinking of all the real-life voting schemes we have, and in all of those we're violating at least one of the other conditions.

Here's a simple voting mechanism that gives you that outcome: Everyone on Earth votes. In the case of everyone besides Mr. X, their first choice gets 1 point. Mr. X's vote gets two hundred billion. We add up all the points, and the option with the most points win. (Note: I'm not sure if this violates the other conditions, since we haven't talked about preferences. But theoretically, it could not)

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TomDavidson
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Well, yes. I thought of those scenarios. I was trying to think of more likely ones, though. [Smile]

It seems to me, like I said earlier, that the practical effect of the "no dictator" requirement is just to drive home the fact that there's no useful way to reconcile the other four.

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Jhai
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You can certainly think of it that way. The way the proof flows certainly leads one to essentially come to that conclusion - there's no way to get a social choice mechanism that contains all of the other four conditions. But mathematicians and theorists are fussy, and like everything spelled out precisely, which means that non-dictatorship is a condition.
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TomDavidson
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Thanks, Jhai. I think I understand. [Smile]
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Jhai
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Good. I was getting desperate, and horrible analogies were starting to suggest themselves to me. [Smile]
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jebus202
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"I know all these words, but I just can't parse this."

I'm pretty confident you guys are talking gibberish and just pretending to understand each other to confuse everyone else.

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