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?

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?

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.

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.)

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.

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.

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.)

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?

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?

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.

General comments: I discovered rangevoting.org years ago and have been a fan since. But it's not quite clear to me what the next action is, here. Yes, score voting is superior to plurality, but is it superior enough that people should join CRV? I use score voting to determine what restaurant to go to or what board game to play if there's any collective uncertainty, and I can see many applications for rationalist group houses or other organizations, but the statement "I would be happier if the US Presidential Election used score voting" misunderstands the nature of politics.

many less-wrong'ers

I would use "many LWers".

Arrows theorem shows that any voting system which can consistently give the same winner (or, in ties, winners) for the same voter preferences; which does not make one voter the effective dictator; which is sure to elect a candidate if all voters prefer them; and which will switch the results for two candidates if you switch their names on all the votes... must exhibit, in at least some situation, the pathology that befell the Rationalist Marching Band above, or in other words, must fail "independence of irrelevant alternatives".

"Arrow's". Later you use Arrows', which is also incorrect, and so I would do a search-replace on the whole thing.

[EDIT- I was wrong about Arrow's theorem. You can safely ignore the following thread, unless you had the same confusion I did, in which case I'll leave it as an example.]

Also, I never found this theorem that troublesome- the "dictator" axiom is necessary for the theorem but I find it untroubling. (The proof suggests that the "dictator" is not an individual selected before the voting process, but a 'pivotal voter'- the last person to move from coalition A to coalition B. But if you never allow a changed individual vote to change the group vote, then how can the group vote ever change?) It also doesn't apply to cardinal voting systems, just ordinal voting systems, so range voting is immune. (Which I see you bring up later.)

In one sense, score voting reducing to approval OK.

I think this sentence is missing an 'is'. (It's also much easier to proofread things in Google Docs, or similar software, than discussion posts.)

But in another sense, it's a problem. If one side of an issue is more inclined to be strategic than the other side, the more-strategic faction could win even if it's a minority. That clashes with many people's ideals of democracy; and worse, it encourages mind-killing political attitudes, where arguments are used as soldiers rather than as ways to seek the truth.

I think this is a feature, personally, since it allows you to encode how much you care about a particular issue. There are lots of ways to implement tradeoffs here, which can lead to overall more efficient decisions.