Ok, here are some questions to help me understand/poke holes in this proof. (Don’t think too hard on these questions. If the answers are not obvious to you, then I am asking the wrong questions.
Does the argument (or a simple refactorization of the argument you also believe) decompose through “If B strictly dominates A∞, then there is a B′ that also strictly dominates A∞ such that the probability of any voter being indifferent between something sampled from B′ and something sampled from A∞ is 0 (or negligable).”
If Yes to 1, do you believe the above lemma is also true for an arbitrary A?
If Yes to 1, do you believe the above lemma is true if we replace “strictly dominates” with “f(B,A)>x” for some fixed x>0.
If No to 1, is there some minor modification that will give me a similar looking lemma the argument does decompose through?
#1: It doesn’t. The previous version implied that there was a B′ for which the probability of ties was arbitrarily low, but the new version can have lots of voters who are indifferent. If B puts its mass in the interior of a face F, then we redistribute probability mass within the interior of F, but some voters assign the same utility to everything in F.
#4: The current lemma is:
If B strictly dominates A, then there is a face F of the simplex and a B’ which is continuous over F such that B’ strictly dominates A.
Ok, here are some questions to help me understand/poke holes in this proof. (Don’t think too hard on these questions. If the answers are not obvious to you, then I am asking the wrong questions.
Does the argument (or a simple refactorization of the argument you also believe) decompose through “If B strictly dominates A∞, then there is a B′ that also strictly dominates A∞ such that the probability of any voter being indifferent between something sampled from B′ and something sampled from A∞ is 0 (or negligable).”
If Yes to 1, do you believe the above lemma is also true for an arbitrary A?
If Yes to 1, do you believe the above lemma is true if we replace “strictly dominates” with “f(B,A)>x” for some fixed x>0.
If No to 1, is there some minor modification that will give me a similar looking lemma the argument does decompose through?
#1: It doesn’t. The previous version implied that there was a B′ for which the probability of ties was arbitrarily low, but the new version can have lots of voters who are indifferent. If B puts its mass in the interior of a face F, then we redistribute probability mass within the interior of F, but some voters assign the same utility to everything in F.
#4: The current lemma is: