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Arrow's impossibility theorem

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Re: Arrow's impossibility theorem

#81

Post by logicchop »

reagency wrote: post #54
logicchop wrote: post #47

Earlier quoted context omitted.

I don't follow this "loophole" you mention. Afaik there are proofs of Gibbard-Satterthwaite that allow indifference in the rankings, and this "distinction" between rankings and scores is methodologically dubious: Arrow himself was sensitive about the meaningfulness of quantitative reports of preference, and I don't see any reason to believe that a "score" issued by a voter is any better or more meaningful than a simp…

Scoring lets you indicate ties, and strong preferences.

indifferent rankings already indicate ties, and what does a 'strong preference' value amount to here? I'd suggest that it only makes sense if you try to understand the "score" as being a relative ranking of N imaginary alternative candidates in addition to the actual candidates.. So a 1 to A and 99 to B only indicates 'strength' in that it suggests if there were 96 other candidates you'd put them above A and below B.. but now we are back to orderings over candidates. At any rate, you ought to be suspicious of quantitative assignments of preference..

Re: Arrow's impossibility theorem

#82

Post by logicchop »

kijin wrote: post #55
logicchop wrote: post #53

Earlier quoted context omitted.

(1) Not quite. The "deliberative democracy" camp is not interested in the measurement of group preference, and are instead interested in consensus building, political "rationality" (in hopefully some eventually-stabilizing sense), and so on. That is not a response to Arrow and his associates, it is just a different topic. (2) It is abundantly clear why a "procedure" should satisfy all of the requirements of the relat…

If we read early works (from the late 80s) in what is now called "deliberative democracy", we can see that it began as a response to certain models of democracy that emphasize preferences and procedures -- up to and including the participatory models of the 70s and early 80s. Although deliberative democracy is not a direct response to Arrow's impossibility theorem in particular, it was intended to sidestep its troubl…

I think you are a bit confused. What would Arrow's theorem explain? There is no empirical phenomena that we are puzzled about that Arrow's theorem solves (unless you are wondering: Why is it so hard to come up with a voting system that doesn't have the potential for goof-ball results? - Answer, because it is impossible..)

I don't know what you mean by the "trinity" in which "satisfaction of all pre-existing preference" is a part.. That's obviously not relevant; we aren't interested in a system that satisfies preferences. What we are interested in is a system that (1) isn't a dictatorship, (2) is fair [i.e., everyone counts equally], (3) allows us to decide any kind of potential matter _as a group_.

If you look at the conditions this way it should be glaringly obvious what this has to do with the (possibility) of democracy..

Re: Arrow's impossibility theorem

#83

Post by ClayShentrup »

gjm11 wrote: post #40

A few remarks: 1. Arrow's theorem concerns the situation where your election procedure needs to deliver (not just a single winner, but) a ranking of all the candidates. You might hope that relaxing this condition will help, but ... 2. There's a closely related theorem with the magnificent name of Gibbard-Satterthwaite, which says that if you have more than two candidates, any procedure that takes in ranked preference…

Some useful links.

http://ScoreVoting.net/ArrowThm.html http://ScoreVoting.net/GibbSat.html

Last year, I got some books signed by Arrow at his Palo Alto retirement home. It was pretty cool.

Re: Arrow's impossibility theorem

#84

Post by ClayShentrup »

gjm11 wrote: post #40

A few remarks: 1. Arrow's theorem concerns the situation where your election procedure needs to deliver (not just a single winner, but) a ranking of all the candidates. You might hope that relaxing this condition will help, but ... 2. There's a closely related theorem with the magnificent name of Gibbard-Satterthwaite, which says that if you have more than two candidates, any procedure that takes in ranked preference…

> Arrow's theorem concerns the situation where your election procedure needs to deliver (not just a single winner, but) a ranking of all the candidates.

Well, any social welfare function that can pick a single winner can also be used to form a complete ordering, by repeating the process with the previous winner eliminated. So any social welfare function is a social ordering function.

Re: Arrow's impossibility theorem

#85

Post by ClayShentrup »

logicchop wrote: post #81
reagency wrote: post #54

Earlier quoted context omitted.

Scoring lets you indicate ties, and strong preferences.

indifferent rankings already indicate ties, and what does a 'strong preference' value amount to here? I'd suggest that it only makes sense if you try to understand the "score" as being a relative ranking of N imaginary alternative candidates in addition to the actual candidates.. So a 1 to A and 99 to B only indicates 'strength' in that it suggests if there were 96 other candidates you'd put them above A and below B.…

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery.

You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaranteed item.

Re: Arrow's impossibility theorem

#86

Post by mkehrt »

kragen wrote: post #29
mkehrt wrote: post #23

Earlier quoted context omitted.

In college I was part of a club that had an elaborate election procedure for officers. I'm pretty sure it violates Arrow's Theorem, but it was also nonterminating!

That sounds very interesting! How did your nonterminating election procedure work?

Oh, god. I don't remember the details, but it involved crossing off the bottom third of candidates every round until there was only one left. But somehow, this didn't always get rid of everyone, and you could get stuck in a state where there was no bottom third.

Re: Arrow's impossibility theorem

#87

Post by dllthomas »

logicchop wrote: post #79
dllthomas wrote: post #64

Earlier quoted context omitted.

I don't see this. Could you elaborate?

I recall that Bordes and Tideman were a good source on this issue; I believe the (relevant) paper is "Independence of Irrelevant Alternatives in the Theory of Voting." -- The upshot is that the condition known as IIA (I think it is sometimes known as Sen's condition-alpha, and the condition that many people harp on, including Michael Dummett) is in fact a stronger condition than is needed for the result; roughly, tha…

Ah, I think your phrasing was confusing. "IIA" is considering an attribute of the ballots, but including IIA is stronger than necessary for the theorem to hold - it can be replaced with the much weaker "nothing but the ballots".

Re: Arrow's impossibility theorem

#88

Post by logicchop »

ClayShentrup wrote: post #85
logicchop wrote: post #81

Earlier quoted context omitted.

indifferent rankings already indicate ties, and what does a 'strong preference' value amount to here? I'd suggest that it only makes sense if you try to understand the "score" as being a relative ranking of N imaginary alternative candidates in addition to the actual candidates.. So a 1 to A and 99 to B only indicates 'strength' in that it suggests if there were 96 other candidates you'd put them above A and below B.…

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery. You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaran…

Yes, we know that preference involves more than ordering and that "strengths" are relevant to decision making. We know this because it explains an obvious empirical matter, that there are scenarios like the one you cite where we prefer X over Y, but end up choosing Y over X. Not a puzzle; not a paradox; how it is. And of course the way we tease out the "strength" of your preference involves, e.g., looking at how your choice varies between counterfactual scenarios involving different probabilities.

But the question is whether a "score" given by a voter on a ballot indicates anything psychologically interesting, and, more importantly, whether it indicates anything relevant to the "judicious" selection of a candidate. It seems to me that, empirically, the only meaning we can _seriously_ assign to a "score" on a ballot is a relative-scoring-as-indicative-of-relative-preference. I don't know how "10 to A, 20 to B" could indicate anything beyond the simple fact that the voter prefers B to A. And surely my "10 to A, 20 to B" needn't have the same meaning as your "10 to A, 20 to B" -- I think all we can really say is that we both prefer B to A. What else?

Now, you might want to try to interpret these "scores" as something like a hypothetical question put to the voter about how much they might pay to have this-or-that person elected; but if you know anything about self-reports of this kind about hypothetical scenarios, you know that people are horribly inaccurate (either intentionally or not), and that the setup is contrived. Serious measures of preferences involve actual stakes, markets, etc. Good luck with that here.

Re: Arrow's impossibility theorem

#89

Post by _dps »

ClayShentrup wrote: post #85
logicchop wrote: post #81

Earlier quoted context omitted.

indifferent rankings already indicate ties, and what does a 'strong preference' value amount to here? I'd suggest that it only makes sense if you try to understand the "score" as being a relative ranking of N imaginary alternative candidates in addition to the actual candidates.. So a 1 to A and 99 to B only indicates 'strength' in that it suggests if there were 96 other candidates you'd put them above A and below B.…

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery. You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaran…

You are implicitly presuming an expected utility framework (more generally, that people rank outcome distributions based solely on their long run central tendencies).

This isn't the case (or even uniquely formalizable) in base (non-e.u.) utility theory which requires choice-order preservation under arbitrary monotonic transformations of the scoring function. Expectation ordering is not necessarily preserved under such transformations.

E.U. is a common paradigm because it is a useful approximation and allows numerical calculations but it is not all (or even, the core) of utility theory except in very limited circumstances (e.g. purely monetary payoffs and linear utility of money).

Re: Arrow's impossibility theorem

#90

Post by dragonwriter »

reagency wrote: post #54
logicchop wrote: post #47

Earlier quoted context omitted.

I don't follow this "loophole" you mention. Afaik there are proofs of Gibbard-Satterthwaite that allow indifference in the rankings, and this "distinction" between rankings and scores is methodologically dubious: Arrow himself was sensitive about the meaningfulness of quantitative reports of preference, and I don't see any reason to believe that a "score" issued by a voter is any better or more meaningful than a simp…

Scoring lets you indicate ties, and strong preferences.

> Scoring lets you indicate ties, and strong preferences.

Non-forced rankings allow you to indicate ties, and quantitative indications of strength of preference are unlikely to have consistent meanings among voters, so treating them has having the same meaning (as, presumably, any voting system using them must) is inherently problematic.

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