The post Coherent decisions imply consistent utilities demonstrates some situations in which an agent that isn’t acting as if it is maximizing a real-valued utility function over lotteries is dominated by one that does, and promised that this applies in general.
However, one intuitively plausible way to make decisions that doesn’t involve a real-valued utility and that the arguments in the post don’t seem to rule out is to have lexicographic preferences; say, each lottery has payoffs represented as a sequence (u1,u2,u3,…) and you compare them by first comparing u1, and if and only if their u1s are the same compare u2, and so on, with probabilities multiplying through by each un and payoffs being added element-wise. The VNM axioms exclude this by requiring continuity, with a payoff evaluated like this violating it because (0,0,…)≺(0,1,…)≺(1,0,…) but there is no probability p for which a p probability of a (1,0,…) payoff and a 1−p probability of a (0,0,…) payoff is as exactly as good as a certainty of a (0,1,…) payoff.
Are there coherence theorems that exclude lexicographic preferences like this also?
[Question] Is there a “coherent decisions imply consistent utilities”-style argument for non-lexicographic preferences?
The post Coherent decisions imply consistent utilities demonstrates some situations in which an agent that isn’t acting as if it is maximizing a real-valued utility function over lotteries is dominated by one that does, and promised that this applies in general.
However, one intuitively plausible way to make decisions that doesn’t involve a real-valued utility and that the arguments in the post don’t seem to rule out is to have lexicographic preferences; say, each lottery has payoffs represented as a sequence (u1,u2,u3,…) and you compare them by first comparing u1, and if and only if their u1s are the same compare u2, and so on, with probabilities multiplying through by each un and payoffs being added element-wise. The VNM axioms exclude this by requiring continuity, with a payoff evaluated like this violating it because (0,0,…)≺(0,1,…)≺(1,0,…) but there is no probability p for which a p probability of a (1,0,…) payoff and a 1−p probability of a (0,0,…) payoff is as exactly as good as a certainty of a (0,1,…) payoff.
Are there coherence theorems that exclude lexicographic preferences like this also?