Do you want to select a distribution that contains the best average future? Do you want to select from a distribution that contains the best median future?
That question doesn't quite make sense as stated. Utilities are ways of modeling our preferences, not features of the universe for us to have preferences about. So it makes sense to ask what ways of choosing between probability distributions over utilities are compatible with what we actually want, but not which way is what we actually want.
Well that's just the thing. With median utility, you don't actually need to put a number on it at all. You just need a preference ordering of outcomes.
I am actually somewhat confused on how to assign utility to outcomes with expected utility. Just because you think an outcome is a thousand times more desirable, doesn't necessarily mean you would accept a 1,000:1 bet for it. Or does it? I do not know.
Like, 1,000 people dying seems like it should objectively be 1,000 worse than 1 person dying. Does that mean I should pay the mugger? Or should I come up with ...
In a previous post, I looked at some of the properties of using the median rather than the mean.
Inspired by Househalter's comment, it seems we might be able to take a compromise between median and mean. It seems to me that simply taking the mean of the lower quartile, median, and upper quartile would also have the nice features I described, and would likely be closer to the mean.
Furthermore, there's no reason to stop there. We can take the mean of the n-1 n-quantiles.
Two questions:
Note the unlike the median approach, for large enough n, this maximiser will pay Pascal's mugger.