#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.
#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: