we might be able to take a compromise between median and mean.
I'll repeat my comment to Househalter's post: in taking your compromise, what are you optimizing for? On the basis of which criteria will you distinguish a "good" compromise from a "bad" compromise?
It's somewhat subjective. There is no law of the universe that says one approach is right and the other is long (so long as you avoid inconsistencies, which the policy selection idea does.) There is no objective way to compress an entire distribution down to a single value.
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? Or do you want something in between? What do we, humans, actually want?
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.