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 a scheme to discount the disutility of each person dying over threshold? Or just do the intuitive thing and ignore super low probability risks, no matter how much utility they promise. But otherwise keep the intuition that all lives are equally valuable and not discountable.
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.
Correct. VNM utility is not necessarily linear with respect to the intuitive strength of the preference. Your utility function is defined based on what bets you would accept, rather than being a way of telling you what bets you should accept.
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?
Nope; see above. You can define a notion of uti...
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.