= 27d567bc2d48d9f40e9ccc20ea370b15)
jsteinhardt comments on Unconditionally Convergent Expected Utility - Less Wrong Discussion
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (32)
Do you also contest the Archimedean axiom for von Neumann's formulation of utility?
Yes. (Well, it's a bit more complicated than that; VNM utility theory doesn't extend to choices with an infinite number of possible outcomes, so I reject the whole system.) I discussed this in more detail in the comments in the linked article. In brief, there is a chance that my utility function is bounded, but I am definitely not willing to bet the universe on it.
VNM definitely does extend to the case of infinitely many outcomes. It requires a continuous utility function, and thus continuous preferences and a topology in outcome space. Why is this additional modeling assumption any more problematic than other VNM axioms?
In short, because utilities may not converge. The axioms do not assert themselves able to be applied an infinite number of times; if they did, they would run into all the usual problems with infinite series. There are modifications of the VNM theorem that extend infinitely, but they all either must only work for certain infinite sets or must require bounded utility.
This is exactly the stuff I was talking about. I mean, basic measure theory determines what functions you can even talk about. If you have a probability measure P, then utilities that are not in L^{1}_{P}(outcome domain) make no sense. You may need some more restrictions than that, but one can't talk about expected utility if the utility is not at least L1. You cannot define a function w.r.t. a probability measure than has a support set of infinite Lebesgue measure, is unbounded, and has a defined expectation (the L1 norm)... unless you know that the rate of growth of the unbounded utility function behaves in certain nice ways when compared to the decay of the probability measure. You might be already saying this, but this much simply can't be changed, no matter what you do. If your utility function is unbounded, then the probabilities for certain outcomes must decay faster than your utility grows. Since probabilities are given by nature and utilities (sort of) aren't, my guess would be that utilities have to decay quickly (or, conversely, probabilities have to decay super quickly).
Nature does not require that it is possible to make utility function converge at all. Also, nature neither requires that taking expectations be the only way of comparing choices, nor that utilities be real.
I totally agree and never meant to imply otherwise. But just as any consistent system of degrees of belief can be put into correspondence with the axioms of probability, so there are certain stipulations about what can reasonably called a utility function.
I would argue that if you meet a conscious agent and your model of their utility function says that it doesn't converge (in the appropriate L1 norm of the appropriate modeled probability space) then something's wrong with that model of utility function... not with the assumption that utility functions should converge. There are many subtleties, I'm sure, but non-integrable utility functions seem futile to me. If something can be well-modeled by a non-integrable utility function, then I'm fine updating my position, but in years of learning and teaching probability theory, I've never encountered anything that would convince me of that.
Doesn't this all assume that utility functions are real-valued?
No, all of the integrability theory (w.r.t. probability measures) extends straightforwardly to complex valued functions. See this and this.
Yes, good point. Is there any study of the most general objects to which integrability theory applies? Also, are you familiar with Martin Kruskal's work on generalizing calculus to the surreal numbers? I am having difficulty locating any of his papers.
Another point worth noting is that on a set D of finite measure (which any measurable subset of a probability space is), L^{N}(D) is contained in L^{N-1}(D), and so if the first moment fails to exist (non-integrable, no defined expectation) then all higher moments fail and computation of order statistics fails. Of course nature doesn't have to be modeled by statistics, but you'd be hard pressed to out-perform simple axiomatic formulations that just assume a topolgy, continuous preference functions, and get on with it and have access to higher order moments.
How do you construct utility without the VNM axioms? Are there less strong axioms for which a VNM-like result holds?
EDIT: Sorry if this is covered in the comments in the other article, I'm being a bit lazy here and not reading through all of your comments there in detail.
I don't yet. :) I have a few reason to think that it has a good chance of being possible, but it has not been done.
Okay. If you end up being successful, I would be quite interested to know about it. (A counterexample would also be interesting, actually probably more interesting since it is less expected.)