Continuity and independence.
Continuity: Consider the scenario where each of the [LMN] bets refer to one (guaranteed) outcome, which we’ll also call L, M and N for simplicity.
Let U(L) = 0, U(M) = 1, U(N) = 10**100
For a simple EU maximizer, you can then satisfy continuity by picking p=(1-1/10**100). A PESTI agent, OTOH, may just discard a (1-p) of 1/10**100, which leaves no other options to satisfy it.
The 10**100 value is chosen without loss of generality. For PESTI agents that still track probabilities of this magnitude, increase it until they don’t.
Independence: Set p to a number small enough that it’s Small Enough To Ignore. At that point, the terms for getting L and M by that probability become zero, and you get equality between both sides.
That still sounds wrong. You appear to be deciding on what to precompute for purely by probability, without considering that some possible futures will give you the chance to shift more utility around.
If I don’t know anything about Newcomb’s problem and estimate a 10% chance of Omega showing up and posing it to me tomorrow, I’ll definitely spend more than 10% of my planning time for tomorrow reading up on and thinking about it. Why? Because I’ll be able to make far more money in that possible future than the others, which means that the expected utility differentials are larger, and so it makes sense to spend more resources on preparing for it.
The I-am-undetectably-insane case is the opposite of this, a scenario that it’s pretty much impossible to usefully prepare for.
And a PM scenario is (at least for an expected-utility maximizer) a more extreme variant of my first scenario—low probabilities of ridiculously large outcomes, that are because of that still worth thinking about.