Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.
That appears to be an axiom that probabilities go to zero enough faster than utilities that total utility converges (in a setting in which the sure outcomes are a countable set). It lacks something in precision of formulation (e.g. what is being quantified over, and in what order?) but it is fairly clear what it is doing. There's nothing like it in VNM's book or the Wiki article, though. Where does it come from?
Yes, in the same way that VNM's axioms are just what is needed to get affine utilities, an axiom something like this will give you bounded utilities. Does the axiom have any intuitive appeal, separate from it providing that consequence? If not, the axiom does not provide a justification for bounded utilities, just an indirect way of getting them, and you might just as well add an axiom saying straight out that utilities are bounded.
None of which solves the problem that entirelyuseless cited. The above axiom forbids the Solomonoff prior (for which pi Mi grows with busy beaver fastness), but does not suggest any replacement universal prior.
No, the axiom doesn't put any constraints on the probability distribution. It merely constrains preferences, specifically it says that preferences for infinite lotteries should be the 'limits' of the preference for finite lotteries. One can think of it as a slightly stronger version of the following:
Axiom (5'): Let L = Sum(i=0...infinity, p_i M_i) with Sum(i=0...infinity, p_i)... (read more)