It seems to me that the “continuity/Archimedean” property is the least intuitively necessary of the four axioms of the VNM utility theorem. One way of specifying preferences over lotteries that still obeys the other three axioms is assigning to each possible world two real numbers U1 and U2 instead of one, where U1 is a “top priority” and U2 is a “secondary priority”. If two lotteries have different ⟨U1⟩, the one with greater ⟨U1⟩ is ranked higher, and ⟨U2⟩ is used as a tie-breaker. One possible real-world example (with integer-valued U1 for deterministic outcomes) would be a parent whose top priority is minimizing the number of their children who die within the parent’s lifetime, with the rest of their utility function being secondary.
I’d be interested in whether there exist any preferences over lotteries quantifying our intuitive understanding of risk aversion while still obeying the other three axioms of the VNM theorem. I spent about an hour trying to construct an example without success, and suspect it might be impossible.
It seems to me that the “continuity/Archimedean” property is the least intuitively necessary of the four axioms of the VNM utility theorem. One way of specifying preferences over lotteries that still obeys the other three axioms is assigning to each possible world two real numbers U1 and U2 instead of one, where U1 is a “top priority” and U2 is a “secondary priority”. If two lotteries have different ⟨U1⟩, the one with greater ⟨U1⟩ is ranked higher, and ⟨U2⟩ is used as a tie-breaker. One possible real-world example (with integer-valued U1 for deterministic outcomes) would be a parent whose top priority is minimizing the number of their children who die within the parent’s lifetime, with the rest of their utility function being secondary.
I’d be interested in whether there exist any preferences over lotteries quantifying our intuitive understanding of risk aversion while still obeying the other three axioms of the VNM theorem. I spent about an hour trying to construct an example without success, and suspect it might be impossible.