I’m having trouble with the solution to the dictatorship vs anarchy example. Is a 13 split actually maximal? Is “A dominates B” transitive?
Let’s say ϵ=0.1. Consider an unfair 50/50/0 split in which the first two potential dictators throw the third under the bus. As I understand, this dominates the equal split, since 2 out of the 3 voters prefer this new lottery.
Then say voter #3 counterproposes a 0/60/40 split. This dominates again. But then 50/0/50 dominates that… so domination is non-transitive. And I don’t see a maximal randomized outcome.
New confusion: in the dictator example, it seems like between option A, an even 13 split, and option B, a 50/50/0 split, neither dominates the other. I’m calculating P[v(A)>v(B)]=P[v(A)<v(B)]=29.
EDIT: Probably this is me discovering the difference between maximal lotteries and maximal lottery-lotteries...
I believe I’m starting to see my mistake; I was thinking there was an argmax step where there’s more like an expectation, due to sampling jointly, at the same time, across the lottery outcomes and voters.
I’m having trouble with the solution to the dictatorship vs anarchy example. Is a 13 split actually maximal? Is “A dominates B” transitive?
Let’s say ϵ=0.1. Consider an unfair 50/50/0 split in which the first two potential dictators throw the third under the bus. As I understand, this dominates the equal split, since 2 out of the 3 voters prefer this new lottery.
Then say voter #3 counterproposes a 0/60/40 split. This dominates again. But then 50/0/50 dominates that… so domination is non-transitive. And I don’t see a maximal randomized outcome.
Am I missing something here?
New confusion: in the dictator example, it seems like between option A, an even 13 split, and option B, a 50/50/0 split, neither dominates the other. I’m calculating P[v(A)>v(B)]=P[v(A)<v(B)]=29.
EDIT: Probably this is me discovering the difference between maximal lotteries and maximal lottery-lotteries...
Okay, this comment is now making more sense to me: https://www.lesswrong.com/posts/vwrNprXfEzeQ2cy3d/maximal-lottery-lotteries?commentId=Cw7TTF2oTXzvE5e6S
I believe I’m starting to see my mistake; I was thinking there was an argmax step where there’s more like an expectation, due to sampling jointly, at the same time, across the lottery outcomes and voters.