Most[1] problems with unbounded utility functions go away if you restrict yourself to
summable utility functions[2].
Summable utility functions can still be unbounded.
For example, if each planet in the universe gives you 1 utility, and P(universe has exactly n planets)=2−n for n≥1, then your utility function is unbounded but summable.
In such a universe it would be very unlikely for a casino to hand out a large number of planets.
Your proof relies on the assumption
assuming that the casino has unbounded utility to hand out.
A summable function is a measurable function for which the integral of its absolute value is finite (using the probability measure for the integral in this context).
This seems like it works but demands a very strange universal prior that penalizes big things and large numbers. I consider the original Pascal’s Mugging post to have settled the argument about this type of prior.
Most[1] problems with unbounded utility functions go away if you restrict yourself to summable utility functions[2]. Summable utility functions can still be unbounded.
For example, if each planet in the universe gives you 1 utility, and P(universe has exactly n planets)=2−n for n≥1, then your utility function is unbounded but summable. In such a universe it would be very unlikely for a casino to hand out a large number of planets.
Your proof relies on the assumption
and this assumption would be wrong in my example.
In fact, I do not know of an exception.
A summable function is a measurable function for which the integral of its absolute value is finite (using the probability measure for the integral in this context).
This seems like it works but demands a very strange universal prior that penalizes big things and large numbers. I consider the original Pascal’s Mugging post to have settled the argument about this type of prior.