Savage famously provide a representation theorem for subjective expected utility maximization, where both a utility over acts and a subjective probability measure are derived from preferences. The problem with his technique is that we are often interested in whether agents have probability distributions in mind – that is, whether they are “probabilistically sophisticated” – even if they are not EU maximizers. In this recent ReStud, Rostek provides a representation for quantile maximizers, such as those that maximize the median outcome, or the minimum outcome (“mixmax”), etc. It turns out such beliefs are probabilistically sophisticated when the quantile being maximized is not extreme – that is, when the agent is maximizing any percentile except 0 or 1.
Non-theorists have considered quantile maximization (particularly its treatment of downside risk), and this paper provides a useful behavioral characterization. I would only note that, in a normative sense, I still can’t think of any reason an individual would maximize quantiles. As the author notes, maximizing the median means ignoring all information (such as the spread) outside the median. For instance, a median maximizer (with u(x)=x) prefers the lottery (.49,$0,.51,$10000) to (1,$9999). As has been mentioned on this site before, counterexamples are not “refutations” of economic theories – we know theories in social science are always wrong. Nonetheless, it’s tough for me to see applications to applied work here. The examples given in the paper are interesting (for instance, a social planner should use the median rather than the mean when maximizing, in a utilitarian sense, public goods provision, since this helps overcome information problems), but don’t seem to necessitate a representation theorem for quantile maximization.
In any case, the proof technique is quite different from Savage (in particular, likelihood relations are not used), so the paper is worth a look for that alone.