I don’t understand this part. How does probability mass constrain how “bad” the states can get? Could you rephrase this maybe?
The probability mass doesn’t constraint how “bad” the states can get; I was saying that the fact that there’s only 1 unit of probability mass means that the amount of probability mass on lower states is bounded (by 1).
Restricting the formalism to orderings means that there is no meaning to howbad a state is, only a meaning to whether it is better or worse than another state. (You can additionally decide on a measure of how bad, as long as it’s consistent with the ordering, but we don’t need that to analyze (this concept of) optimization.)
The probability mass doesn’t constraint how “bad” the states can get; I was saying that the fact that there’s only 1 unit of probability mass means that the amount of probability mass on lower states is bounded (by 1).
Restricting the formalism to orderings means that there is no meaning to how bad a state is, only a meaning to whether it is better or worse than another state. (You can additionally decide on a measure of how bad, as long as it’s consistent with the ordering, but we don’t need that to analyze (this concept of) optimization.)