I am no longer confident in the arguments presented in two parts of this paper:
1) Section 7.1, when I argue that it is impossible to translate all of the paradox regions outside the unit simplex. As I try to visualize it, I am sure that the statement is correct, but I don't know that the method of argument is correct.
2) Section 8, where I argue that Type 1b stages must be point systems with quotas. I argue that if we think of it as a Type 1 stage augmented with Type 2 inequalities, then the Type 1 stage must always return a winner to avoid FBC violations, in accordance with the results of section 7.1. (Note that my concerns in section 8 are independent of the validity of the theorem in section 7.1. Even if the theorem is correct, I might not be applying it correctly.) Suppose that the Type 1 stage sometimes returns no winner, but only in cases where the Type 2 inequalities don't return a winner either? Then it's the non-satisfaction of the Type 2 inequalities that moves the method to the next stage, not the non-satisfaction of the Type 1 inequalities. So I have to figure out whether this is possible. I can prove that it's impossible in a restricted case, but I don't know about the general case.
