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 and you compare them by first comparing $u_{1}$ , and if and only if their $u_{1}$ s are the same compare $u_{2}$ , and so on, with probabilities multiplying through by each $u_{n}$ and payoffs being added element-wise. The VNM axioms exclude this by requiring continuity, with a payoff evaluated like this violating it because $(0, 0, \dots) ≺ (0, 1, \dots) ≺ (1, 0, \dots)$ but there is no probability $p$ for which a $p$ probability of a $(1, 0, \dots)$ payoff and a $1 - p$ probability of a $(0, 0, \dots)$ payoff is as exactly as good as a certainty of a $(0, 1, \dots)$ payoff.

Are there coherence theorems that exclude lexicographic preferences like this also?