I’ll certainly have more content that addresses these questions as the post develops. For now, I’ll simply respond to your misunderstanding about Arrow. The problem is not that there will always be an a posteori pivotal voter, but that (to satisfy the other criteria) there must be an a priori dictator. In other words, you would get the same election result by literally throwing away all ballots but one without ever looking at them. This is clearly not democracy.
This is kind of late and not exactly on point, but I wanted to add in two points about Arrow’s Theorem that I hope will help clarify things somewhat.
There is a generalization of Arrow’s Theorem to the case where the set of voters may be infinite. In this more general case, the conclusion of the theorem (where here I’m considering the premises to be IIA and unanimity) is not that there must be a dictator, but rather that the voting system is given by an ultrafilter on the set of voters. That is to say, there exists an ultrafilter U on the set of voters such that the candidate elected is precisely the candidate such that the set of people voting for that candidate is large with respect to U. The fact that if there are only finitely many voters there must be a dictator then follows as any ultrafilter on a finite set is principal.
In a nephew/niece comment, homunq mentions sortition (select a voter at random to be dictator) as an example. I don’t think this is correct? That is to say, Arrow’s Theorem only applies to deterministic voting systems, and “dictatorship” refers to having a pre-specified dictator. There may be a generalization to nondeterministic systems for all I know, but if so, I don’t think it’s normally considered part of the theorem per se, and I don’t think sortition is normally considered an example of dictatorship.
For now, I’ll simply respond to your misunderstanding about Arrow.
It’s not clear to me why you think that’s a misunderstanding; the statement of the theorem is not that the dictator is an a priori dictator, just that there never exists a situation where an individual can completely determine society’s preferences. The proof is a construction of a situation given the first two fairness axioms and at least three alternatives, where one voter will be a pivotal voter who can completely determine society’s preferences.
But if you don’t care about the third axiom, you don’t care about the proof. Okay, in a deeply divided but balanced situation, the one non-partisan can pick whether we go left, right, or to the middle; this isn’t a huge tragedy.
(The collapse of the scale of preferences is a huge tragedy.)
Again, you’re simply not understanding the theorem. If a system fails non-dictatorship, that really does mean that there is an a priori dictator. That could be that one vote is chosen by lot after the ballots are in, or it could be that everybody (or just some special group or person) knows beforehand that Mary’s vote will decide it. But it’s not that Mary just happens to turn out to be the pivotal voter between a sea of red on one side and blue on the other.
I realize that this is counterintuitive. Do you think I have to be clearer about it in the post?
I would say that Arrow simply excludes that possibility. As you said, he only considers systems that “consistently give the same winner...for the same voter preferences.” Nothing wrong with your analysis, but I think it’s an important disclaimer.
That could be that one vote is chosen by lot after the ballots are in
This is the case that doesn’t sound like an a-priori dictator to me, because you don’t know who the dictator will be, and thus can’t do anything to manipulate the outcome by dint of there being a dictator.
Under Arrow’s terms, this still counts as a dictator, as long as the other ballots have no effect. (Not “no net effect”, but no effect at all.)
In other words: if I voted for myself, and everyone else voted for Kanye, and my ballot happened to get chosen, then I would win, despite being 1 vote against 100 million.
It may not be the traditional definition of dictatorship, but it sure ain’t democracy.
There is obviously not an a priori dictationship for all voting environments under all aggregation rules that satisfy unanimity and IIA. For example, if 9 people prefer A>B>C, and 1 person prefers B>C>A, then society prefers A, regardless of how any specific individual changes their vote (so long as only one vote is changed).
(Note the counterfactual component of my statement- there needs to be an individual who can change the social preference function, not just identify the social preference function.)
But it’s not that Mary just happens to turn out to be the pivotal voter between a sea of red on one side and blue on the other.
Every proof of the theorem that I can see operates exactly this way; I’m still not seeing what specific step you think I misunderstand.
I’m sorry, you really are wrong here. You can’t make up just one scenario and its result and say that you have a voting rule; a rule must give results for all possible scenarios. And once you do, you’ll realize that the only ones which pass both unanimity and IIA are the ones with an a priori dictatorship. I’m not going to rewrite Arrow’s whole paper here but that’s really what he proved.
You can’t make up just one scenario and its result and say that you have a voting rule; a rule must give results for all possible scenarios.
I think I see how the grandparent was confusing. I was assuming that the voting rule was something like plurality voting, with enough sophistication to make it a well-defined rule.
What I meant to do was define two dictatorship criteria which differ from Arrow’s, which apply to individuals under voting rules, rather than applying to rules. Plurality voting (with a bit more sophistication) is a voting rule. Bob choosing for everyone is a voting rule. But the rule where Bob chooses for everyone has an a priori dictator- Bob. (He’s also an a posteriori dictator, which isn’t surprising.)
Plurality voting as a voting rule does not empower an a priori dictator as I defined that in the grandparent. But it is possible to find a situation under plurality voting where an a posteriori dictator exists; that is, we cannot say that plurality voting is free from a posteriori dictators. That is what the nondictatorship criterion (which is applied to voting rules!) means- for a rule to satisfy nondictatorship, it must be impossible to construct a situation where that voting rule empowers an a posteriori dictator.
Because unanimity and IIA imply not nondictatorship, for any election which satisfies unanimity and IIA, you can carefully select a ballot and report just that ballot as the group preferences. But that’s because it’s impossible for the group to prefer A>B>C with no individual member preferring A>B>C, and so there is guaranteed to be an individual who mirrors the group, not an individual who determines the group. Since individuals determining group preferences is what is politically dangerous, I’m not worried about the ‘nondictatorship’ criterion, because I’m not worried about mirroring.
I’m not going to rewrite Arrow’s whole paper here but that’s really what he proved.
I’ve read it; I’ve read Yu’s proof; I’ve read Barbera’s proof, I’ve read Geanakoplos’s proof, I’ve read Hansen’s proof. (Hansen’s proof does follow a different strategy from the one I discussed, but I came across it after writing the grandparent.) I’m moderately confident I know what the theorem means. I’m almost certain that our disagreement stems from different uses of the phrase “a priori dictator,” and so hope that disagreement will disappear quickly.
Funnily enough, I asked Amartya Sen and Eric Maskin about this earlier today, in addition to a similarly extensive reading list. You definitely have your quantifiers backwards.
You claim that Arrow’s dictatorship criterion is “for all preferences, there exists some k such that the social preference is voter k’s preference”. In fact, the criterion is the stronger statement that “there exists some k such that for all preferences, the social preference is voter k’s preference”.
If you read the proof on Wikipedia carefully, you’ll notice that it proves that:
Pivotal Voter dictates society’s decision for B over C. That is, we show that no matter how the rest of society votes, if Pivotal Voter ranks B over C, then that is the societal outcome.
(Note the order of the quantifiers). They then show
I, too, hope that our disagreement will soon disappear. But as far as I can see, it’s clearly not a semantic disagreement; one of us is just wrong. I’d say it’s you.
So. Say there are 3 voters, and without loss of generality, voter 1 prefers A>B>C. Now, for every one of the 21 distinct combinations for the other two, you have to write down who wins, and I will find either an (a priori, determinative; not mirror) dictator or a non-IIA scenario.
ABC ABC: A
ABC ACB: A
ABC BAC: ?… you fill in these here
ABC BCA: ?
ABC CAB: .
ABC CBA: .
ACB ACB: .
ACB BAC:
ACB BCA:
ACB CAB:
ACB CBA:
BAC BAC:
BAC BCA:
BAC CAB:
BAC CBA:
BCA BCA:
BCA CAB: …. this one’s really the key, but please fill in the rest too.
BCA CBA:
CAB CAB:
CAB CBA:
CBA CBA:
Once you’ve copied these to your comment I will delete my copies.
I, too, hope that our disagreement will soon disappear. But as far as I can see, it’s clearly not a semantic disagreement; one of us is just wrong. I’d say it’s you.
I’m moderately confident I know what the theorem means. I’m almost certain that our disagreement stems from different uses of the phrase “a priori dictator,” and so hope that disagreement will disappear quickly.
I’m glad you’ve opened a discussion about this, BTW, even if it turns out you are wrong. I wondered myself what exactly the non-dictatorship criterion meant when I started reading the commentary on “Arrow’s Paradox” in Ken Binmore’s Playing Fair. After I read all of it I was fairly sure non-dictatorship referred to what you call an a priori dictator, but couldn’t be totally sure because I didn’t have the patience to sit down and puzzle through it systematically.
I’m glad you’ve opened a discussion about this, BTW, even if it turns out you are wrong.
I was wrong. Hansen’s 2002 proof was much, much easier for me to understand than the other ones, but I was reluctant to generalize it past 2 voters until I wrestled with homunq’s example. Even after seeing it in the 3 voter case, I had to resist an impulse that said “but surely there’s an n such that an n voter situation works!”
I’ll certainly have more content that addresses these questions as the post develops. For now, I’ll simply respond to your misunderstanding about Arrow. The problem is not that there will always be an a posteori pivotal voter, but that (to satisfy the other criteria) there must be an a priori dictator. In other words, you would get the same election result by literally throwing away all ballots but one without ever looking at them. This is clearly not democracy.
This is kind of late and not exactly on point, but I wanted to add in two points about Arrow’s Theorem that I hope will help clarify things somewhat.
There is a generalization of Arrow’s Theorem to the case where the set of voters may be infinite. In this more general case, the conclusion of the theorem (where here I’m considering the premises to be IIA and unanimity) is not that there must be a dictator, but rather that the voting system is given by an ultrafilter on the set of voters. That is to say, there exists an ultrafilter U on the set of voters such that the candidate elected is precisely the candidate such that the set of people voting for that candidate is large with respect to U. The fact that if there are only finitely many voters there must be a dictator then follows as any ultrafilter on a finite set is principal.
In a nephew/niece comment, homunq mentions sortition (select a voter at random to be dictator) as an example. I don’t think this is correct? That is to say, Arrow’s Theorem only applies to deterministic voting systems, and “dictatorship” refers to having a pre-specified dictator. There may be a generalization to nondeterministic systems for all I know, but if so, I don’t think it’s normally considered part of the theorem per se, and I don’t think sortition is normally considered an example of dictatorship.
It’s not clear to me why you think that’s a misunderstanding; the statement of the theorem is not that the dictator is an a priori dictator, just that there never exists a situation where an individual can completely determine society’s preferences. The proof is a construction of a situation given the first two fairness axioms and at least three alternatives, where one voter will be a pivotal voter who can completely determine society’s preferences.
But if you don’t care about the third axiom, you don’t care about the proof. Okay, in a deeply divided but balanced situation, the one non-partisan can pick whether we go left, right, or to the middle; this isn’t a huge tragedy.
(The collapse of the scale of preferences is a huge tragedy.)
Again, you’re simply not understanding the theorem. If a system fails non-dictatorship, that really does mean that there is an a priori dictator. That could be that one vote is chosen by lot after the ballots are in, or it could be that everybody (or just some special group or person) knows beforehand that Mary’s vote will decide it. But it’s not that Mary just happens to turn out to be the pivotal voter between a sea of red on one side and blue on the other.
I realize that this is counterintuitive. Do you think I have to be clearer about it in the post?
I would say that Arrow simply excludes that possibility. As you said, he only considers systems that “consistently give the same winner...for the same voter preferences.”
Nothing wrong with your analysis, but I think it’s an important disclaimer.
Yes, please.
This is the case that doesn’t sound like an a-priori dictator to me, because you don’t know who the dictator will be, and thus can’t do anything to manipulate the outcome by dint of there being a dictator.
Under Arrow’s terms, this still counts as a dictator, as long as the other ballots have no effect. (Not “no net effect”, but no effect at all.)
In other words: if I voted for myself, and everyone else voted for Kanye, and my ballot happened to get chosen, then I would win, despite being 1 vote against 100 million.
It may not be the traditional definition of dictatorship, but it sure ain’t democracy.
By an a priori dictatorship, I mean there is some individual 1 such that
By an a posteriori dictatorship, I mean there is some individual 1 such that
There is obviously not an a priori dictationship for all voting environments under all aggregation rules that satisfy unanimity and IIA. For example, if 9 people prefer A>B>C, and 1 person prefers B>C>A, then society prefers A, regardless of how any specific individual changes their vote (so long as only one vote is changed).
(Note the counterfactual component of my statement- there needs to be an individual who can change the social preference function, not just identify the social preference function.)
Every proof of the theorem that I can see operates exactly this way; I’m still not seeing what specific step you think I misunderstand.
I’m sorry, you really are wrong here. You can’t make up just one scenario and its result and say that you have a voting rule; a rule must give results for all possible scenarios. And once you do, you’ll realize that the only ones which pass both unanimity and IIA are the ones with an a priori dictatorship. I’m not going to rewrite Arrow’s whole paper here but that’s really what he proved.
I think I see how the grandparent was confusing. I was assuming that the voting rule was something like plurality voting, with enough sophistication to make it a well-defined rule.
What I meant to do was define two dictatorship criteria which differ from Arrow’s, which apply to individuals under voting rules, rather than applying to rules. Plurality voting (with a bit more sophistication) is a voting rule. Bob choosing for everyone is a voting rule. But the rule where Bob chooses for everyone has an a priori dictator- Bob. (He’s also an a posteriori dictator, which isn’t surprising.)
Plurality voting as a voting rule does not empower an a priori dictator as I defined that in the grandparent. But it is possible to find a situation under plurality voting where an a posteriori dictator exists; that is, we cannot say that plurality voting is free from a posteriori dictators. That is what the nondictatorship criterion (which is applied to voting rules!) means- for a rule to satisfy nondictatorship, it must be impossible to construct a situation where that voting rule empowers an a posteriori dictator.
Because unanimity and IIA imply not nondictatorship, for any election which satisfies unanimity and IIA, you can carefully select a ballot and report just that ballot as the group preferences. But that’s because it’s impossible for the group to prefer A>B>C with no individual member preferring A>B>C, and so there is guaranteed to be an individual who mirrors the group, not an individual who determines the group. Since individuals determining group preferences is what is politically dangerous, I’m not worried about the ‘nondictatorship’ criterion, because I’m not worried about mirroring.
I’ve read it; I’ve read Yu’s proof; I’ve read Barbera’s proof, I’ve read Geanakoplos’s proof, I’ve read Hansen’s proof. (Hansen’s proof does follow a different strategy from the one I discussed, but I came across it after writing the grandparent.) I’m moderately confident I know what the theorem means. I’m almost certain that our disagreement stems from different uses of the phrase “a priori dictator,” and so hope that disagreement will disappear quickly.
Funnily enough, I asked Amartya Sen and Eric Maskin about this earlier today, in addition to a similarly extensive reading list. You definitely have your quantifiers backwards.
You claim that Arrow’s dictatorship criterion is “for all preferences, there exists some k such that the social preference is voter k’s preference”. In fact, the criterion is the stronger statement that “there exists some k such that for all preferences, the social preference is voter k’s preference”.
If you read the proof on Wikipedia carefully, you’ll notice that it proves that:
(Note the order of the quantifiers). They then show
I’m hella envious.
And also convinced. If Sen says Arrow meant non-dictatorship in the strong sense, I’m pretty much willing to take Sen’s word for it.
I, too, hope that our disagreement will soon disappear. But as far as I can see, it’s clearly not a semantic disagreement; one of us is just wrong. I’d say it’s you.
So. Say there are 3 voters, and without loss of generality, voter 1 prefers A>B>C. Now, for every one of the 21 distinct combinations for the other two, you have to write down who wins, and I will find either an (a priori, determinative; not mirror) dictator or a non-IIA scenario.
ABC ABC: A
ABC ACB: A
ABC BAC: ?… you fill in these here
ABC BCA: ?
ABC CAB: .
ABC CBA: .
ACB ACB: .
ACB BAC:
ACB BCA:
ACB CAB:
ACB CBA:
BAC BAC:
BAC BCA:
BAC CAB:
BAC CBA:
BCA BCA:
BCA CAB: …. this one’s really the key, but please fill in the rest too.
BCA CBA:
CAB CAB:
CAB CBA:
CBA CBA:
Once you’ve copied these to your comment I will delete my copies.
Thanks; I see it now. Editing my earlier posts.
I’m glad you’ve opened a discussion about this, BTW, even if it turns out you are wrong. I wondered myself what exactly the non-dictatorship criterion meant when I started reading the commentary on “Arrow’s Paradox” in Ken Binmore’s Playing Fair. After I read all of it I was fairly sure non-dictatorship referred to what you call an a priori dictator, but couldn’t be totally sure because I didn’t have the patience to sit down and puzzle through it systematically.
I was wrong. Hansen’s 2002 proof was much, much easier for me to understand than the other ones, but I was reluctant to generalize it past 2 voters until I wrestled with homunq’s example. Even after seeing it in the 3 voter case, I had to resist an impulse that said “but surely there’s an n such that an n voter situation works!”