Comment author: 30 October 2013 05:26:59PM -1 points [-]

Your probability theory here is flawed. The question is not about P(A&B), the probability that both are true, but about P(A|B), the probability that A is true given that B is true. If A is "has cancer" and B is "cancer test is positive", then we calculate P(A|B) as P(B|A)P(A)/P(B); that is, if there's a 1/1000 chance of cancer and and the test is right 99/100, then P(A|B) is .99.001/(.001.99+.999.01) which is about 1 in 10.

Comment author: 03 November 2013 11:08:30AM 0 points [-]

Can anyone explain why the parent was downvoted? I don't get it. I hope there's a better reason than the formatting fail.

Comment author: 31 October 2013 03:33:25PM 1 point [-]

Arrow's theorem considers your options holding others fixed - and does its analysis knowing them. But when you're actually filling out your ballot, you don't have access to that kind of information. So it doesn't prove that there aren't systems where the risk/reward from going strategic is poor under more realistic conditions.

Is there such a theorem?

Comment author: 31 October 2013 07:58:39PM *  -1 points [-]

This is a key question. The general answer is:

1. For realistic cases, there is no such theorem, and so the task of choosing a good system is a lot about choosing one which doesn't reward strategy in realistic cases.

2. Roughly speaking, my educated intuition is that strategic payoffs grow insofar as you know that the distinctions you care about are orthogonal to what the average/modal/median voter cares about. So insofar as you are average/modal/median, your strategic incentive should be low; which is a way of saying that a good voting system can have low strategy for most voters in most elections.

2a. It may be possible to make this intuition rigorous, and prove that no system can make strategy non-viable for the orthogonal-preferenced voter. However, that would involve a lot of statistics and random variables.... I guess that's what I'm learning in my PhD so eventually I may be up to taking on this proof.

1. The exception, the realistic case where there are a number of voters who have an interest that's orthogonal to the average voter, is a case called the chicken dilemma, which I'll talk about a lot more in section 6. Chicken strategy is by far the trickiest realistic strategy to design away.
Comment author: 31 October 2013 10:31:36AM 0 points [-]

This is a terrific post, worth chopping into several pieces and made a sequence of its own.
I just have one quibble: Arrovian instead of Arrowian?

Comment author: 31 October 2013 07:52:54PM 0 points [-]

Yup. That's what people say. I don't know what the general rule is, but it's definitely right for this case.

Comment author: 30 October 2013 10:05:13PM *  0 points [-]

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.

Comment author: 31 October 2013 02:23:08AM *  0 points [-]

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.

Comment author: 30 October 2013 08:02:15PM *  -1 points [-]

By an a priori dictatorship, I mean there is some individual 1 such that $F(R_1,R_2,\ldots,R_N)=R_1\ \forall\ (R_2,\ldots,R_N)\in L(A)^{N-1}$.

By an a posteriori dictatorship, I mean there is some individual 1 such that $\exists (R_2,\ldots,R_N)\in L(A)^N\ s.t.\ F(R_1,\ldots,R_N)=R_1\ \forall\ R_1$

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.

Comment author: 30 October 2013 08:25:58PM 1 point [-]

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.

Comment author: 30 October 2013 08:05:06PM -1 points [-]

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.

Comment author: 30 October 2013 08:22:15PM -1 points [-]

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.

Comment author: 30 October 2013 05:48:06PM *  0 points [-]

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.)

Comment author: 30 October 2013 06:29:57PM *  1 point [-]

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?

Comment author: 30 October 2013 06:11:00PM *  4 points [-]

I worry that this sort of analysis puts process ahead of results.

In large-scale decisionmaking, such as regional or national politics, most voters are confused and inattentive. I think this is inevitable and even proper. The world is too complicated for most people to have informed and thoughtful opinions on most topics. As a result, I don't particularly care if the process delivers results most voters want. Instead, I care if the process delivers decent results. And in particular, I want decent results for impatient voters and potentially-dishonest election apparatus. First-past-the-post has the important benefit that as a voter I have to indicate one preference, rather than an ordering. This requires strictly less input from me, and therefore probably less attention and thought, which is a Good Thing.

I would be interested to hear an argument for why all the voting theory stuff is useful in practice, given the constraints and goals of practical politics.

We have some examples of cities and countries that use systems other than first-past-the-post. Which of these actually are better governed as a result?

Comment author: 30 October 2013 06:25:13PM 1 point [-]

Wait until I get to explaining SODA; a voting system where you can vote for one and still get better results.

As for comparing different societies: there are of course societies with different electoral systems, and I think some systems do tend to lead to better governance than in the US/UK, but the evidence is weak and VERY confounded. It's certainly impossible to clearly demonstrate a causal effect; and would be, even assuming such an effect existed and were sizeable. I will talk about this more as I finish this post.

Comment author: 30 October 2013 04:46:28PM 1 point [-]

I would work on making the existing writing clearer before expanding further. There were a number of points where I did not follow you as a casual reader, like with the "circular money pump" and Sen, or things just appeared random, like going from has specific theoretical pitfall to BAD.

Second, if you are going to expand, you may want to address whether all this theory actually predicts much accurately. Beyond saying, yeah, there were these guys Duncan and Black who looked at specific cases--what were those cases?

Comment author: 30 October 2013 05:28:35PM 0 points [-]

Thanks, I'll work on that.

Comment author: 30 October 2013 04:26:58PM 0 points [-]

if other statements can increase credibility, they can also reduce it.

Sure, but it's utterly unsurprising that there exists a B such that P(A&B)<P(A). That there exists a B such that P(A&B) > P(A) is more surprising, which is why I'd asked for an example of what DanielLC had in mind by it.

Comment author: 30 October 2013 05:26:59PM -1 points [-]

Your probability theory here is flawed. The question is not about P(A&B), the probability that both are true, but about P(A|B), the probability that A is true given that B is true. If A is "has cancer" and B is "cancer test is positive", then we calculate P(A|B) as P(B|A)P(A)/P(B); that is, if there's a 1/1000 chance of cancer and and the test is right 99/100, then P(A|B) is .99.001/(.001.99+.999.01) which is about 1 in 10.

View more: Next