Arrow's impossibility theorem • Hacker News Board

Re: Arrow's impossibility theorem

#41

Post by justifier » Thu, Jun 04, 2015, 2:23 AM UTC

gjm11 wrote: post #38

SomeStupidPoint wrote: post #34

Earlier quoted context omitted.

I don't see why that anecdote makes it desirable -- can you explain?

The anecdote illustrates why it's desirable: Morgenbesser's change of mind doesn't make any sense, and the reason why it doesn't make any sense is that it violates IIA: whether they have cherry pie shouldn't make any difference to his preference between apple pie and blueberry pie. (Except that it might -- e.g., imagine that making cherry pie is incredibly difficult and most kitchens can't manage it, and that the spe…

in that case.. i thought about it and, if i can still change my order, i think i want the blueberry

Re: Arrow's impossibility theorem

#42

Post by SilasX » Thu, Jun 04, 2015, 2:28 AM UTC

gweinberg wrote: post #39

SilasX wrote: post #25

Earlier quoted context omitted.

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?

No, of course not. It's really easy to come up with a system that always comes up with "good" results if you rule out "screwy" voter preferences, with a sufficiently restrictive value of "screwy".

"Screwy" in this context means the sort of intransitive situations referred to in the parent of my last comment.

Re: Arrow's impossibility theorem

#43

Post by belovedeagle » Thu, Jun 04, 2015, 2:33 AM UTC

verteu wrote: post #24

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 p…

I hate to be that guy but there are plenty of reasons why this might be logical behavior. (I say "that guy" because I'm not prepared to argue for their relevance to voting.)

The most trivial example is that he simply changed his mind—but obviously this has nothing to do with IIA.

Another reason is that he might be dining with someone (who is, perhaps, away from the table). Say that he knows his partner's preferences are C>B>A. Say his own preferences are B>A>C, but that he also has a preference for being able to sample two distinct choices. Initially, he believes that his partner will order B, so he orders A to maximize his range of desserts. Upon learning that the "irrelevant" choice C is available, he will have to order his top choice, B, himself.

The same scenario could play out with himself as the second diner, in the sense that he knows he will have two opportunities to visit this restaurant (and that their dessert choices will remain constant). If his preferences are C>B>A, and for whatever reason he decides to order his second-most-preferred dessert first and his most-preferred dessert on the next visit, then his choice is logical.

Perhaps he's of the opinion that a kitchen which prepares cherry pie cannot prepare an adequate apple pie.

The point is, alternatives are rarely "irrelevant". In fact, including the word "irrelevant" in IIA is begging the question: we are tasked with determining whether dominated alternatives (for example) really are irrelevant, and our answer might be "no".

Re: Arrow's impossibility theorem

#44

Post by wildmusings » Thu, Jun 04, 2015, 2:37 AM UTC

The criteria considered by the theorem concern how democratic a voting system is. I'd argue that 'democraticness' is secondary to two other criteria: accountability and government-effectiveness. Accountability means that bad rulers can get voted out of office when enough people are displeased. Government-effectiveness means that the resulting government can practice good governance. When considering these criteria, I think plurality/first-past-the-post voting systems are superior. Unpopular leaders are voted out and elections usually result in single-party majorities that can govern effectively.

Re: Arrow's impossibility theorem

#45

Post by logicchop » Thu, Jun 04, 2015, 2:56 AM UTC

verteu wrote: post #24

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 p…

Morgenbesser is always great, but a defense against criticisms of IIA is much simpler to mount in the context of voting: it simply amounts to the requirement that our voting procedure only takes into account information from the ballots.

Re: Arrow's impossibility theorem

#46

Post by grayclhn » Thu, Jun 04, 2015, 3:10 AM UTC

mcphage wrote: post #33

verteu wrote: post #24

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 p…

When you say it like that, IIA does sound pretty obvious. But if you change the terms a little bit, you can see why the IIA doesn't match up with how people actually vote: Say there's an election between a moderate democrat "blueberry pie" and a third party liberal "apple pie". As a liberal, Sidney would rather vote for the third party ("Sidney orders the apple pie"). However, if you introduce a republican candidate…

Arrow's impossibility theorem, and the IIA criterion, is about preferences not about uncertain actions. In particular, IIA doesn't mean that you'll vote differently than your preferences in some sort of election, it means that your preferences themselves don't change when you introduce other irrelevant options. In your example, it wouldn't be about how Sidney would vote in an election given the different menu of candidates, it's about who Sidney would prefer win the election.

(with the caveat that it has been a long time since I've thought about these results.)

Re: Arrow's impossibility theorem

#47

Post by logicchop » Thu, Jun 04, 2015, 3:25 AM UTC

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…

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 simple ranking (including rankings of indifference). This approach has always struck me as an unmotivated anti-empirical gimmick to get around this-or-that condition in the theorem(s).

Re: Arrow's impossibility theorem

#48

Post by logicchop » Thu, Jun 04, 2015, 3:27 AM UTC

mcphage wrote: post #33

verteu wrote: post #24

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 p…

When you say it like that, IIA does sound pretty obvious. But if you change the terms a little bit, you can see why the IIA doesn't match up with how people actually vote: Say there's an election between a moderate democrat "blueberry pie" and a third party liberal "apple pie". As a liberal, Sidney would rather vote for the third party ("Sidney orders the apple pie"). However, if you introduce a republican candidate…

In the context of voting, all IIA represents is the requirement that we only take into account the information on the ballots..

Re: Arrow's impossibility theorem

#49

Post by wtallis » Thu, Jun 04, 2015, 3:29 AM UTC

gjm11 wrote: post #38

SomeStupidPoint wrote: post #34

Earlier quoted context omitted.

I don't see why that anecdote makes it desirable -- can you explain?

The anecdote illustrates why it's desirable: Morgenbesser's change of mind doesn't make any sense, and the reason why it doesn't make any sense is that it violates IIA: whether they have cherry pie shouldn't make any difference to his preference between apple pie and blueberry pie. (Except that it might -- e.g., imagine that making cherry pie is incredibly difficult and most kitchens can't manage it, and that the spe…

> This is maybe just a tiny bit similar to, e.g., changing your vote from A to B when you learn that C is standing, because C is a terrible candidate but might win, and in scenarios where C is close to winning B is C's main rival.

That's just strategic voting, not an actual change in your preferences. How you go about voting strategically depends on the voting system in use. In your hypothetical scenario you make the unstated assumption that the voting system in use would not allow for you to express a preference for A over B without hurting B's ability to beat C. The inability of that voting system to fully capture your preferences is forcing you to vote misleadingly based on your knowledge of how others will probably vote.

Re: Arrow's impossibility theorem

#50

Post by logicchop » Thu, Jun 04, 2015, 3:34 AM UTC

c-slice wrote: post #2

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

Here's the version of the story with respect to voting: Democracy requires voting. Voting involves an objective measurement of group preference with respect to choices. Arrow's theorem and related theorems (see Gibbard-Satterthwaite theorem) show that there is no way to objectively measure group preferences.