http://en.wikipedia.org/wiki/Voting_system has a nice comparison chart half way down.
Arrow's impossibility theorem
61–70 of 93 posts
Re: Arrow's impossibility theorem
#62It seems to me that concern over the dictatorship criterion is ill founded. Its failure merely implies that for any given set of alternatives there will always be one voter whose preferences, whatever they may be, determine the outcome. Now if everyone's preferences were known and constant then in theory you could find that person and they would be able to decide the outcome. In practice preferences don't stay constant and no-one knows them all, so as long as you have secret ballots the dictatorship criterion doesn't matter. So you can have the other two criteria which are far more important.
Re: Arrow's impossibility theorem
#63An 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…
"The most trivial example is that he simply changed his mind—but obviously this has nothing to do with IIA."
Delving further down this tangent... If that's what happened, he would be unlikely to have said "In that case, ...". That usually means "I am incorporating this new information, and it has changed my decision." More likely he would have said, "On second thought..." or "Actually..." or "Y'know..."
Not, of course, that this has much relevance outside of considering English pragmatics.
Re: Arrow's impossibility theorem
#64Earlier quoted context omitted.
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..
I don't see this. Could you elaborate?
Re: Arrow's impossibility theorem
#65A 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…
For #2, what if you relax the "no tactical voting" requirement to say that voters cannot predict how to vote tactically unless they have an impractically large quantity of information about other voters?
I think I'd be satisfied with that. Or ideally, "given attainable information, an honest vote is the most likely to make the output most agree with me."
Re: Arrow's impossibility theorem
#66A 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…
For #2, what if you relax the "no tactical voting" requirement to say that voters cannot predict how to vote tactically unless they have an impractically large quantity of information about other voters?
This is the insight. Proving "tactical voting is always theoretically possible" sounds bad, until you realize that there exist systems where tactical voting requires not only nearly perfect information about every other voters exact preferences ahead of time, but also solving cryptographically hard problems
Re: Arrow's impossibility theorem
#67An 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…
While amusing, the anecdote is off in a very important way: it is not a single person ordering dessert, but an entire party of people who represent different conflicting majorities. In such a real situation you can get whatever "absurd" result you want by just asking questions in the right order.
Ask your dinner party "Do we prefer Apple or Blueberry?", and a majority might reasonably answer Apple. Ask them "Blueberry or Cherry?", and a slightly different majority might reasonably answer Blueberry. Ask them "Cherry or Apple?" and a different majority would answer Cherry. You would get this situation, for instance, with the following voters: 2x A > B > C 2x B > C > A 1x C > A > B A beats B by 3:2, B beats C by 4:1, and C beats A by 3:2.
This is a rock paper scissors situation -- or a "condorcet cycle".
Now suppose instead of simple Sidney there were actually these 5 people ordering dessert. Is it really that unreasonable that the group as a whole might pick blueberry once cherry is on the table even though a majority prefers apple to blueberry?
Re: Arrow's impossibility theorem
#68Earlier quoted context omitted.
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…
No, you're combining a few different things that aren't related to IIA. For your first dining example, you've changed the relevant set of options to (A,A), (A,B), (A,C), (B,A), etc... where the first element is the diner's desert and the second is the partner's desert. IIA means that if (B,A) ≿ (A,B) in the original choice set, then it continues to hold when we add (Z,Z) as an option. For the second case, you've made…
> For the second case, you've made "good apple pie" and "bad apple pie" different elements of the choice set.
I don't think so. Arrow's theorem has a set of outcomes as a given. In this restaurant, one of the two, "good apple pie", "bad apple pie" is not actually an outcome, so it does not exist in the ranking.
Re: Arrow's impossibility theorem
#69Few theorems in economics, or political science in this case, are as disappointing as this one. Another one is Sonnenschein-Mantel-Debreu.
https://en.wikipedia.org/wiki/Sonnenschein%E2%80%93Mantel%E2...
SMD: rational individuals do not sum up to a rational aggregate.
Re: Arrow's impossibility theorem
#70An 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…
The problem is, this anecdote is completely misleading. IIA is about the group's preferences, not just your own.
Here's an easy example that shows the difference. The first is more like the anecdote, and evidently disproves it:
Given that there is no war to deal with, you prefer candidate A to B.
Given that there is a war to deal with, you prefer candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though ranked below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. You know this.
Therefore, your ranking of {A,B} is A > B, but your ranking of {A,B,C} is B > A > C. This "disproves" the pie anecdote. It does not, however, have anything to do with IIA.
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The second example is more like what IIA actually says:
Given that there is no war to deal with, everyone prefers candidate A to B.
Given that there is a war to deal with, everyone prefers candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though everyone ranks him below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. Everyone knows this.
Therefore, everyone's ranking of {A,B} is A > B, but everyone's ranking of {A,B,C} is B > A > C. Everyone votes accordingly.
IIA implies: the SOCIAL ranking of {A,B} should agree with the SOCIAL ranking of {A,B,C}, as long as the individual rankings for {A,B} all agree with the individual rankings of {A,B,C}.
Here, the conditional fails: the individual rankings are completely reversed. This example does not disprove IIA.