In the same way that we work with delta functions as though they were ordinary functions, we can work with lexicographic preferences as though they were ordinary utility-function preferences.
There's a pattern which comes up in math sometimes where some rules hold for all the functions in some set, and the rules also apply "in the limit" to things "on the boundary" of the set, even if the "boundary" is kinda weird - e.g. a boundary at infinity. Delta functions are a prototypical example: we usually define them as a limit of ordinary functions, and then work with them as though they were ordinary functions. (Though this is not necessarily the best way to think of delta functions - the operator formalism is cleaner is some ways.)
Lexicographic preferences fit this pattern: they are a limiting case of ordinary utility functions. Specifically, given the lexicographic utility sequence and some real number a, we can construct a single utility function . In the limit as , this converges to the same preferences as the lexicographic utility. So, just like we can work with delta functions like ordinary functions and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere), we can work with lexicographic preferences like ordinary utilities and then take the limit later if we have to (or, more often, just leave the limit implicit everywhere).
To answer your exact question: lexicographic utilities are consistent with the coherence theorems, as long as we drop smoothness/boundedness assumptions. They shouldn't be excluded by coherence considerations.
If one uses surreals for chances then one can provide the required chance to make continuity happen.
There might be multiple ways to map a lexiographic weight to 1 + w + w^2 + w^3... but I would expect that simiarly that real functions can be scaled for no effect using different transfinites would just be a matter of consistency. Ie whether you map (1,2,0) to 1+2w or 1+ 2w^2 is a choice that can be made freely if it is just sticked to.
Then you can have a 1/w=e where 0<e<1 which can function as the p to make the lottery mix exactly ambipreferable.
Consider the following game: At any time , you may say "stop!", in which case you'll get the lottery that resolves to an outcome you value at with probability , and to an outcome you value at with probability . If you don't say "stop!" in that time period, we set .
Let's say at every instant in you can decide to either say "stop!" or to wait a little longer. (A dubious assumption, as it lets you make infinitely many decisions in finite time.) Then you'll naturally wait until and get a payoff of . It would have been better for you to say "stop!" at , in which case you'd get .
You can similarly argue that it's irrational for your utility to be discontinuous in the amount of wine in your glass: Otherwise you'll let the waiter fill up your glass and then be disappointed the instant it's full.
I imagine you would be able to solve this by replacing real-valued utilities with utilities in a numbering system that contains infinitesimals. However, it seems to me that it would not matter much in practice, since if you had even a tiny chance of affecting the leading term, then that chance would outweigh all of the other terms in the calculation, and so in practice only the leading term would matter for the decisions.
Or to give an argument from reflection, there's a cost to making decisions, so an agent with a lexicographic ordering would probably want to self-modify into an agent that only considers the leading term in the preference ordering, since then it doesn't have to spend resources on the other terms.
The "leading term will dominate" effect is broken if there are infinidesimal chances around.
It might be sensible for an agent for some purposes to assume away some risks ie treat them as 0 chance. However it might come about that in some circumstances those risks can't be assumed away. So a transformation in the other direction of turning a dirty hack into an actual engine that can accurately work on edgecases might be warranted.
Not sure if relevant, but we might want to view lexicographic preferences as a formalization of satisficers with different weights combining into one preference structure.
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 , and if and only if their s are the same compare , and so on, with probabilities multiplying through by each and payoffs being added element-wise. The VNM axioms exclude this by requiring continuity, with a payoff evaluated like this violating it because but there is no probability for which a probability of a payoff and a probability of a payoff is as exactly as good as a certainty of a payoff.
Are there coherence theorems that exclude lexicographic preferences like this also?