Yes, you can extend Arrow's theorem a bit. But again, it doesn't apply to people who can negotiate or compromise or who play repeatedly. And it also only applies to aggregating an ordering of preferences. It doesn't apply to eg filling up a parliament for proportional representation.
(Btw, the random dictatorship doesn't sound too bad. As a slightly modified form, I think it would be a good experiment to fill up parliament with a few hundred randomly selected people amongst all who are willing.)
> But again, it doesn't apply to people who can negotiate or compromise
You want hundreds of millions of people to negotiate and compromise with each other in a way that would eventually produce representatives that reflect the population's resulting preferences somehow? How would that work?
> or who play repeatedly.
I don't see why I should expect that to make the problem easier.
> I think it would be a good experiment to fill up parliament with a few hundred randomly selected people amongst all who are willing
That sounds like it could go incredibly wrong. Everyone who is willing will sell themselves out to the highest "bidder" (maybe bidding via money, maybe promises of future laws...), and the population unwilling or unable to become a member of parliament will have no say in the matter.
> You want hundreds of millions of people to negotiate and compromise with each other in a way that would eventually produce representatives that reflect the population's resulting preferences somehow? How would that work?
Tacit negotiations can work. And in practice, it's often your representatives that do the negotiations with other people's representatives.
>> or who play repeatedly.
> I don't see why I should expect that to make the problem easier.
Check out the repeated Prisoner's Dilemma for some inspiration for how repeated play can breed cooperation.
> That sounds like it could go incredibly wrong. Everyone who is willing will sell themselves out to the highest "bidder" (maybe bidding via money, maybe promises of future laws...), [...]
How is that different from people selling their vote today?
Just make sure that the legal system does not enforce these contracts, and you are good. (You can also make such contracts illegal completely, just like selling your vote today is illegal in many countries.)
> [...] and the population unwilling or unable to become a member of parliament will have no say in the matter.
You can cook up slightly more complicated versions: every voter nominates a (willing) candidate on their ballot. Nationwide, you collect 600 ballots and fill up parliament with the people named on them. Pick your favourite resolution method, in case the same person gets picked multiple times in your sample.
(Eg you could give that person more weight in parliament, or you could pick the voter's second choice, or you could pick the ballot of the guy who got picked twice to pick a replacement, etc.)
> Tacit negotiations can work. And in practice, it's often your representatives that do the negotiations with other people's representatives.
And now you're back to square one? How do you choose those representatives in a way that represents their constituents' views? That was literally the original problem.
> Check out the repeated Prisoner's Dilemma for some inspiration for how repeated play can breed cooperation.
Have you seen literature on this somewhere? On its face iterated prisoner's dilemma being more cooperative does not in any way suggest that iterates voting somehow admits an easier solution for finding collective preferences than non-repeated voting. The problems are drastically different so far as I can tell. If you've seen literature suggesting otherwise I would love a link or two.
> How is that different from people selling their vote today? Just make sure that the legal system does not enforce these contracts
You seem confused? The reason you can't sell your vote today isn't that it's illegal to sell your vote, but rather the fact that there's no way to prove how you voted, so you could just lie with no incriminating evidence.
Whereas it's pretty darn easy to see how the candidate who promised you tax breaks suddenly voted to raise them when he came into office.
> slightly more complicated version
I see nothing obvious suggesting that your (homemade?) scheme is better, so I'm gonna put the onus on you to explain it wouldn't suffer from similar problems...
> And now you're back to square one? How do you choose those representatives in a way that represents their constituents' views? That was literally the original problem.
No, why? Arrow's Theorem eg has nothing to say about proportional representation. And Arrow's Theorem only applies to aggregating orderings of a finite list of preferences. But the methods under investigation need to be 'generic', ie can't make use of any special properties of those preferences, either. (See eg https://people.mpi-sws.org/~dreyer/tor/papers/wadler.pdf for how being 'generic' limits what your methods can do.)
And to come back to iterated games: almost no matter how the representative was chosen in the first period, if she's standing for re-election, she has an incentive for keeping her represented happy.
Arrow's theorem just applies to a list of static choices; not to how the chosen might behave when trying to get re-elected.
> Have you seen literature on this somewhere?
I don't remember right now. But I think 'The Myth of the Rational Voter' might mention some research somewhere. (See https://en.wikipedia.org/wiki/The_Myth_of_the_Rational_Voter) That book mostly mentions this when it argues that the problem with democracy ain't that voters don't get their wishes, but the problem is that voters do get their wishes.
> Whereas it's pretty darn easy to see how the candidate who promised you tax breaks suddenly voted to raise them when he came into office.
Sure. But if millions of people are eligible to be drafted at random, you are going to have a hard time pre-emptively bribing them. That's equivalent to doing something nice for the entire country.
> I see nothing obvious suggesting that your (homemade?) scheme is better, so I'm gonna put the onus on you to explain it wouldn't suffer from similar problems...
Because eg people who don't want to stand for parliament still have a say? That was exactly one of the problem you brought up with naive sortition. Remember?
Well, if your theorem says A implies B; if A doesn't hold, your theorem doesn't apply, but B could still be true for other reasons. But you need a different argument or empirical data to convince people of B.
(Btw, the random dictatorship doesn't sound too bad. As a slightly modified form, I think it would be a good experiment to fill up parliament with a few hundred randomly selected people amongst all who are willing.)