Cool. For me personally, I think that paying to avoid being given more options looks enough like being dominated that I’d want to keep the axiom of transitivity around, even if it’s not technically a money pump.
So in the case where we have transitivity but no completeness, it seems kind of like there might be a weaker coherence theorem, where the agent’s behaviour can be described by rolling a dice to pick a utility function before beginning a game, and then subsequently playing according to that utility function. Under this interpretation, if A > B then that means that A is preferred to B under all utility functions the agent could pick, while a preferential gap between A and B means that sometimes A will be ranked higher and sometimes B will be ranked higher, depending on which utility function the die roll happens to land on.
Does this match your intuition? Is there an obvious counterexample to this “coherence conjecture”?
This is cool. I don’t think violations of continuity are also in general exploitable, but I’d guess you should also be able to replace continuity with something weaker from Russell and Isaacs, 2020, just enough to rule out St. Petersburg-like lotteries, specifically any one of Countable Independence (which can also replace independence), the Extended Outcome Principle (which can also replace independence) or Limitedness, and then replace the real-valued utility functions with utility functions representable by “lexicographically ordered ordinal sequences of bounded real utilities”.
Cool. For me personally, I think that paying to avoid being given more options looks enough like being dominated that I’d want to keep the axiom of transitivity around, even if it’s not technically a money pump.
So in the case where we have transitivity but no completeness, it seems kind of like there might be a weaker coherence theorem, where the agent’s behaviour can be described by rolling a dice to pick a utility function before beginning a game, and then subsequently playing according to that utility function. Under this interpretation, if A > B then that means that A is preferred to B under all utility functions the agent could pick, while a preferential gap between A and B means that sometimes A will be ranked higher and sometimes B will be ranked higher, depending on which utility function the die roll happens to land on.
Does this match your intuition? Is there an obvious counterexample to this “coherence conjecture”?
Your coherence conjecture sounds good! It sounds like it roughly matches this theorem:
Screenshot is from this paper.
This is cool. I don’t think violations of continuity are also in general exploitable, but I’d guess you should also be able to replace continuity with something weaker from Russell and Isaacs, 2020, just enough to rule out St. Petersburg-like lotteries, specifically any one of Countable Independence (which can also replace independence), the Extended Outcome Principle (which can also replace independence) or Limitedness, and then replace the real-valued utility functions with utility functions representable by “lexicographically ordered ordinal sequences of bounded real utilities”.
This also looks like a generalization of stochastic dominance.
“paying to avoid being given more options looks enough like being dominated that I’d want to keep the axiom of transitivity around”
Maybe offtopic but paying to avoid being given more options is a common strategy in negotiation.