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Arrow's impossibility theorem

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Re: Arrow's impossibility theorem

#21

Post by rbkillea »

pixelcort wrote: post #15

How does Quadratic Voting fare in meeting these criteria? http://ericposner.com/quadratic-voting/

I'm pretty sure that Quadratic Voting is a way to assign prices to votes, whereas this theorem is about the features of a system in which the results are determined by vote. So the results of quadratic voting will meet this criteria because after the votes are bought they can be thought of as emanating from discrete voters who align their preferences with those of the buyer.

That is to say that the issues solved by Quadratic Voting and those presented in Arrow's theorem are orthogonal.

Re: Arrow's impossibility theorem

#22

Post by [deleted] »

chimeracoder wrote: post #12
schmit wrote: post #6

Earlier quoted context omitted.

In short: For a voting system (ranking of some candidates based on preferences of voters), it would be nice if: - A single voter cannot determine the ranking (as a dictator) - For every possible set of voter preferences, there is an outcome (not random) - If everyone likes candidate A over candidate B, then in the final ranking candidate A should be ranked higher than candidate B - If one prefers A over B when compar…

This is correct, with one addition: > Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case . Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satis…

[deleted]

Re: Arrow's impossibility theorem

#23

Post by mkehrt »

chimeracoder wrote: post #12
schmit wrote: post #6

Earlier quoted context omitted.

In short: For a voting system (ranking of some candidates based on preferences of voters), it would be nice if: - A single voter cannot determine the ranking (as a dictator) - For every possible set of voter preferences, there is an outcome (not random) - If everyone likes candidate A over candidate B, then in the final ranking candidate A should be ranked higher than candidate B - If one prefers A over B when compar…

This is correct, with one addition: > Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case . Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satis…

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!

Re: Arrow's impossibility theorem

#24

Post by verteu »

An amusing anecdote which illustrates why independence of irrelevant alternatives is desirable:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

Re: Arrow's impossibility theorem

#25

Post by SilasX »

kelvintran wrote: post #14
c-slice wrote: post #2

Can I get a simple wikipedia explanation? That was the most challenging wikipedia article I've ever read.

This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down. The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms. A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, s…

Yeah, but that takes the punch out of the theorem. It's saying, "hey, sometimes you have really screwy preferences, too bad."

Realistically, that kind of situation doesn't break a voting system. We can say "we don't care about that case -- just pick a random winner then", but it's no longer deterministic.

Is there a stronger version of the theorem that says there's no sane procedure even ignoring those cases?

Re: Arrow's impossibility theorem

#26

Post by VanillaCafe »

Maybe I don't quite appreciate the significance. The Informal Proof section seems to boil down to: If the vote is tied and there is one vote left, then that last vote determines the outcome. The existence of a swing vote in this circumstance doesn't seem very surprising.

Re: Arrow's impossibility theorem

#28

Post by asanagi »

chimeracoder wrote: post #20
ffn wrote: post #17

Earlier quoted context omitted.

>If one prefers A over B when comparing just A and B, then one should also prefer A over B when an additional option C is offered That assumption seems fishy to me if in a real life application, in particular because we assume offering new choices can't fundamentally change the agent/voter's preferences isn't exactly true to life. I'll give a real life, but slightly historically oversimplified example: Say I am a poo…

> Granted this doesn't invalidate the Arrow paradox in any way, I'm just saying the paradox isn't true to life because one of its core tenants doesn't quite hold. However, this plays rather fast and loose with the principles behind Arrow's theorem. First, you're violating the nondictatorial condition (you are the only person voting on your own choice), so it's kind of silly to apply the theorem here in the first plac…

>Secondly, you're saying that, as soon as Communism becomes an option (but only then), you'd rather die than serve the royal family.

Well, maybe. Supposing there's some information contained in option C, or the mere availability of option C, that reveals to you the futility of option A.

In other words, your preferences aren't necessarily transitive because the availability or unavailability of particular options are themselves a little information payload that could influence the decision.

Re: Arrow's impossibility theorem

#29

Post by kragen »

mkehrt wrote: post #23
chimeracoder wrote: post #12

Earlier quoted context omitted.

This is correct, with one addition: > Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case . Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satis…

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?

Re: Arrow's impossibility theorem

#30

Post by verteu »

VanillaCafe wrote: post #26

Maybe I don't quite appreciate the significance. The Informal Proof section seems to boil down to: If the vote is tied and there is one vote left, then that last vote determines the outcome. The existence of a swing vote in this circumstance doesn't seem very surprising.

"Part One" of the informal proof is trivial, for the reason you've described. But "Part Two" is not: It shows that if the "pivotal voter" from Part One votes B>C, then B will beat C in the election, even if everyone else votes C>B.
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