Thus, for example, intransitivity requires giving up on an especially plausible Stochastic Dominance principle, namely: if, for every outcome o and probability of that outcome p in Lottery A, Lottery B gives a better outcome with at least p probability, then Lottery B is better (this is very similar to “If Lottery B is better than Lottery A no matter what happens, choose Lottery B” – except it doesn’t care about what outcomes get paired with heads, and which with tails).
This principle is phrased incorrectly. Taken literally, it would imply that the mixed outcome “utility 0 with probability 0.5, utility 1 with probability 0.5” is dominated by “utility 2 with probability 0.5, utility −100 with probability 0.5″. What you probably want to do is to add the condition that the function f mapping each outcome o to a better outcome f(o) is injective (or equivalently, bijective). But in that case, it is impossible for f(o) to occur with probability strictly greater than P(o), since
P(o)≤P(f(o))=1−∑o′≠oP(f(o′))≤1−∑o′≠oP(o′)=P(o).
This principle is phrased incorrectly. Taken literally, it would imply that the mixed outcome “utility 0 with probability 0.5, utility 1 with probability 0.5” is dominated by “utility 2 with probability 0.5, utility −100 with probability 0.5″. What you probably want to do is to add the condition that the function f mapping each outcome o to a better outcome f(o) is injective (or equivalently, bijective). But in that case, it is impossible for f(o) to occur with probability strictly greater than P(o), since P(o)≤P(f(o))=1−∑o′≠oP(f(o′))≤1−∑o′≠oP(o′)=P(o).
Oops! You’re right, this isn’t the right formulation of the relevant principle. Will edit to reflect.