The surreal version of the VNM representation theorem in “Surreal Decisions” (https://arxiv.org/abs/2111.00862) seems to still have a surreal version of the Archimedean axiom.
Re the parent example, utility function (or its evaluations) changing in an expectable way seems problematic to rational optimizing. If you know you prefer A to B, and know that you will prefer B to A in future even given only current context (so no “waiter must run back and forth”), then you don’t reflectively endorse either decision.
So would it be accurate to say that a preference over lotteries (where each lottery involves only real-valued probabilities) satisfies the axioms of the VNM theorem (except for the Archimedean property) if and only if that preference is equivalent to maximizing the expectation value of a surreal-valued utility function?
Re the parent example, I agree that changing in an expectable way is problematic to rational optimizing, but I think “what kind of agent am I happy about being?” is a distinct question from “what kinds of agents exist among minds in the world?”.
That’s right! However it is not really a problem unless we can obtain surreal probabilities from the real world; and if all our priors and evidence are just real numbers, updates won’t lead us into the surreal area. (And it seems non-real-valued probabilities don’t help us in infinite domains, as I’ve written in https://www.lesswrong.com/posts/sZneDLRBaDndHJxa7/open-thread-fall-2024?commentId=LcDJFixRCChZimc7t.)
Re the parent example, utility function (or its evaluations) changing in an expectable way seems problematic to rational optimizing. If you know you prefer A to B, and know that you will prefer B to A in future even given only current context (so no “waiter must run back and forth”), then you don’t reflectively endorse either decision.
So would it be accurate to say that a preference over lotteries (where each lottery involves only real-valued probabilities) satisfies the axioms of the VNM theorem (except for the Archimedean property) if and only if that preference is equivalent to maximizing the expectation value of a surreal-valued utility function?
Re the parent example, I agree that changing in an expectable way is problematic to rational optimizing, but I think “what kind of agent am I happy about being?” is a distinct question from “what kinds of agents exist among minds in the world?”.