I think Said Achmiz posted this somewhere, but there’s this article (pdf link)[1] about dropping completeness.
There they prove that with (VNM—completeness) you always[2] have a representation as a veto council of utility functions.
More interestingly: this sounds like the knightian uncertainty of infrabayesianism, where you want to have the worst case possibility under the uncertain set to be as best as possible. [Edit: important difference is that maximizing your worst case is different from having incomplete preferences. For example, if choice X gives Alice and Bob +1 and +3 while choice Y gives +3 +2 then maximin prefers the second even though Bob would veto that]. Unfortunately I don’t know a lick of infrabayes, so I don’t know if they also allow knightian uncertainty of the true utility function instead of just knightian uncertainty of the true distribution.
The nice thing with knightian uncertainty is that, maybe, you don’t have to worry about your subagents making deals with each other to complete the preferences, on account of there not actually being subagents?
in the finite case. in the infinite case, they prove that if the outcomes are a compact metric space then (VNM—completeness) <-> there’s a closed convex set of utility function for your veto council.
I think Said Achmiz posted this somewhere, but there’s this article (pdf link)[1] about dropping completeness.
There they prove that with (VNM—completeness) you always[2] have a representation as a veto council of utility functions.
More interestingly: this sounds like the knightian uncertainty of infrabayesianism, where you want to have the worst case possibility under the uncertain set to be as best as possible. [Edit: important difference is that maximizing your worst case is different from having incomplete preferences. For example, if choice X gives Alice and Bob +1 and +3 while choice Y gives +3 +2 then maximin prefers the second even though Bob would veto that]. Unfortunately I don’t know a lick of infrabayes, so I don’t know if they also allow knightian uncertainty of the true utility function instead of just knightian uncertainty of the true distribution.
The nice thing with knightian uncertainty is that, maybe, you don’t have to worry about your subagents making deals with each other to complete the preferences, on account of there not actually being subagents?
Expected utility theory without the completeness axiom
Juan Dubra, a Fabio Maccheroni, b and Efe A. Ok
https://doi.org/10.1016/S0022-0531(03)00166-2
in the finite case. in the infinite case, they prove that if the outcomes are a compact metric space then (VNM—completeness) <-> there’s a closed convex set of utility function for your veto council.