Arranging your probability estimates so that predictions of opposite utility cancel out is one way to satisfy the anti-mugging axiom. ... There's no rule that says that similar statements with positive and negative utilities have to have the same prior probabilities, unless you introduce it specifically for the purpose of anti-mugging defense.
I believe there is such a rule, which doesn't have to be introduced ad hoc, and which follows from the tenets of algorithmic information theory. Per the reasoning I gave in the linked post, an arbitrary complex conclusion you locate (like the one in Pascal's mugging) necessarily has a corresponding conclusion of equal complexity, but with the right predicate(s) inverted so that the inferred utility is reversed.
Because (by assumption) the conclusion is reached through arbitrary reasoning, disentangled from any real-world observation, you need no additional complexity for a hypothesis that critically inverts the first one. Since no other evidence supports either conclusion, their probability weights are determined by their complexity, and are thus equal.
That's why I don't think you need to introduce this reasoning as an additional axiom. However, as a separate matter (and whether or not you need it as an axiom), I thought this argument was refuted by the fact that the mugger, simply through assertion, introduces an arbitrarily small amount of evidence favoring one hypothesis over its inverse. If it refutes the defense I gave in the link, it should work against the anti-mugging axiom you're using as well.
For background, see here.
In a comment on the original Pascal's mugging post, Nick Tarleton writes:
Coming across this again recently, it occurred to me that there might be a way to generalize Vassar's suggestion in such a way as to deal with Tarleton's more abstract formulation of the problem. I'm curious about the extent to which folks have thought about this. (Looking further through the comments on the original post, I found essentially the same idea in a comment by g, but it wasn't discussed further.)
The idea is that the Kolmogorov complexity of "3^^^^3 units of disutility" should be much higher than the Kolmogorov complexity of the number 3^^^^3. That is, the utility function should grow only according to the complexity of the scenario being evaluated, and not (say) linearly in the number of people involved. Furthermore, the domain of the utility function should consist of low-level descriptions of the state of the world, which won't refer directly to words uttered by muggers, in such a way that a mere discussion of "3^^^^3 units of disutility" by a mugger will not typically be (anywhere near) enough evidence to promote an actual "3^^^^3-disutilon" hypothesis to attention.
This seems to imply that the intuition responsible for the problem is a kind of fake simplicity, ignoring the complexity of value (negative value in this case). A confusion of levels also appears implicated (talking about utility does not itself significantly affect utility; you don't suddenly make 3^^^^3-disutilon scenarios probable by talking about "3^^^^3 disutilons").
What do folks think of this? Any obvious problems?