Arrow's theorem has to be one of the most overhyped and overplayed theorems in existence. Here, for instance, are three major things what it doesn't say:
1) That all voting systems are equivalent in any sort of meaningful sense
2) That the current voting system you are using is not awful
3) That a perfect class of voting system doesn't exist if you are willing to accept that rock-paper-scissors situations can happen among people's preferences.
I'm not being hyperbolic here. Number 1 and 2 are how people quickly use vague citations to Arrow's Theorem to shut down talk about voting reform even when the status quo consists of provably terrible systems like plurality voting.
Number 3 is the true result that if you relax the rather overly-strongly defined IIA criteria with a much more-reasonable criteria -- that the winner must remain among the top rock-paper-scissors loop of the voters -- then Arrow's theorem simply doesn't apply. This is well known: that "top loop" is the Smith Set and every Condorcet method of voting satisfies it.
There's also another interesting result: if voters have merely "single-peaked preferences", such as opinions about where to set a volume knob, then Arrow's theorem also doesn't apply since there will be no rock-paper-scissors set of equally fair options.
Number 3 is a valid criticism, but criticizing the theorem on the basis of people using it to support the status quo is like criticizing water because some people drown.
His theorem says nothing about the way things should be, but rather about how things are. I mentioned Poundstone's Gaming the Vote in another post, but that book in particular discusses the implications of the theorem and analyzes better alternatives to current systems.
The problem is that the flaw in thinking is everywhere. Even wikipedia unfairly lumps in Condorcet voting with its various "tiebreaker methods" - it's the tiebreaker methods that run afoul of Arrow's criteria, while the Condorcet method itself does not. And so the Condorcet method gets unfairly lumped together with voting techniques like plurality and IRV.
I disagree with a lot of that. Most people's criticism of a particular voting system takes the form of "your system sucks because it doesn't fulfill desirable criteria A." It's very appropriate to point out that no voting system can fulfill all desirable criteria. Obviously, it doesn't mean that certain voting systems aren't "better" than others, at least if we've agreed on what our goals are (and thus what constitutes "better").
For example, you confidently dismiss plurality voting, but you can't really do that unless you establish what the goals of your voting system are. These goals need to be clearly defined, you can't just say your goal is "making sure everyone has a voice" or "making sure the best candidate wins." That's really all Arrow's theorem says: there are several criteria that most of us agree are desirable, but you cannot have all of them.
Once you actually explain them to people, it's hard to argue that Arrow's theorem still lists desirable criteria. Some are surely reasonable sounding (eg non-dictatorship), but the "independence of irrelevant alternatives" is very deceptively-named: it requires that when voters preferences form a rock-paper-scissors relationship among a set of candidates, the system should never reflect that by having rock win once scissors enters the race.
Arrow calls scissors an "irrelevant alternative" to the race between paper and rock (since scissors doesn't win but makes paper lose). This is just unreasonable -- if there exists a rock-paper-scissors loop among the top preferences of voters, then the voting system should at least acknowledge that reality and make sure it picks someone within that loop (the particular winner is somewhat arbitrary).
And, indeed, if you instead use a more reasonable criteria that requires just that -- that the winner must come from the smith set -- then all of the "reasonable" criteria are satisfiable. Hence the overhyping of Arrow's theorem.
You are right, though, that basing analysis of voting systems on rigid criteria can lead you into trouble. Over-reliance on Arrow's theorem is just an example of that.
The entire point of Arrow's theorem, as I see it, is that criteria which apply to an individual's preferences do not apply to an aggregate of many individual's preferences. Since aggregating the preferences of many individuals is ostensibly the point of any election, it's extremely relevant to point out this rather unfortunate and counterintuitive fact.
IIA is an extremely reasonable quality of an individual's preferences. Although you could probably construct pathological cases where it might sort of make sense for an individual to swap his preference of X and Y when Z is introduced, I contend that it would be exceedingly rare. Showing that IIA is impossible to guarantee (at the same time as other criteria) in an aggregate of preferences is a pretty big deal.
Arrow's theorem does not say or logically imply that, if there is a cyclical Smith set, the voting system shouldn't acknowledge that and pick someone within that loop.
Comments
Arrow's theorem has to be one of the most overhyped and overplayed theorems in existence. Here, for instance, are three major things what it doesn't say: 1) That all voting systems are equivalent in any sort of meaningful sense 2) That the current voting system you are using is not awful 3) That a perfect class of voting system doesn't exist if you are willing to accept that rock-paper-scissors situations can happen among people's preferences.
I'm not being hyperbolic here. Number 1 and 2 are how people quickly use vague citations to Arrow's Theorem to shut down talk about voting reform even when the status quo consists of provably terrible systems like plurality voting.
Number 3 is the true result that if you relax the rather overly-strongly defined IIA criteria with a much more-reasonable criteria -- that the winner must remain among the top rock-paper-scissors loop of the voters -- then Arrow's theorem simply doesn't apply. This is well known: that "top loop" is the Smith Set and every Condorcet method of voting satisfies it.
There's also another interesting result: if voters have merely "single-peaked preferences", such as opinions about where to set a volume knob, then Arrow's theorem also doesn't apply since there will be no rock-paper-scissors set of equally fair options.
Number 3 is a valid criticism, but criticizing the theorem on the basis of people using it to support the status quo is like criticizing water because some people drown.
His theorem says nothing about the way things should be, but rather about how things are. I mentioned Poundstone's Gaming the Vote in another post, but that book in particular discusses the implications of the theorem and analyzes better alternatives to current systems.
The problem is people assume it means more than it does. It's like someone using Godel's incompleteness theorem to argue that math is pointless.
The problem is that the flaw in thinking is everywhere. Even wikipedia unfairly lumps in Condorcet voting with its various "tiebreaker methods" - it's the tiebreaker methods that run afoul of Arrow's criteria, while the Condorcet method itself does not. And so the Condorcet method gets unfairly lumped together with voting techniques like plurality and IRV.
Well he said it was overhyped, which isn't so much a criticism of the theorem, but a criticism of the hypers.
I disagree with a lot of that. Most people's criticism of a particular voting system takes the form of "your system sucks because it doesn't fulfill desirable criteria A." It's very appropriate to point out that no voting system can fulfill all desirable criteria. Obviously, it doesn't mean that certain voting systems aren't "better" than others, at least if we've agreed on what our goals are (and thus what constitutes "better").
For example, you confidently dismiss plurality voting, but you can't really do that unless you establish what the goals of your voting system are. These goals need to be clearly defined, you can't just say your goal is "making sure everyone has a voice" or "making sure the best candidate wins." That's really all Arrow's theorem says: there are several criteria that most of us agree are desirable, but you cannot have all of them.
Once you actually explain them to people, it's hard to argue that Arrow's theorem still lists desirable criteria. Some are surely reasonable sounding (eg non-dictatorship), but the "independence of irrelevant alternatives" is very deceptively-named: it requires that when voters preferences form a rock-paper-scissors relationship among a set of candidates, the system should never reflect that by having rock win once scissors enters the race.
Arrow calls scissors an "irrelevant alternative" to the race between paper and rock (since scissors doesn't win but makes paper lose). This is just unreasonable -- if there exists a rock-paper-scissors loop among the top preferences of voters, then the voting system should at least acknowledge that reality and make sure it picks someone within that loop (the particular winner is somewhat arbitrary).
And, indeed, if you instead use a more reasonable criteria that requires just that -- that the winner must come from the smith set -- then all of the "reasonable" criteria are satisfiable. Hence the overhyping of Arrow's theorem.
You are right, though, that basing analysis of voting systems on rigid criteria can lead you into trouble. Over-reliance on Arrow's theorem is just an example of that.
The entire point of Arrow's theorem, as I see it, is that criteria which apply to an individual's preferences do not apply to an aggregate of many individual's preferences. Since aggregating the preferences of many individuals is ostensibly the point of any election, it's extremely relevant to point out this rather unfortunate and counterintuitive fact.
IIA is an extremely reasonable quality of an individual's preferences. Although you could probably construct pathological cases where it might sort of make sense for an individual to swap his preference of X and Y when Z is introduced, I contend that it would be exceedingly rare. Showing that IIA is impossible to guarantee (at the same time as other criteria) in an aggregate of preferences is a pretty big deal.
Arrow's theorem does not say or logically imply that, if there is a cyclical Smith set, the voting system shouldn't acknowledge that and pick someone within that loop.