# Liron comments on VNM expected utility theory: uses, abuses, and interpretation - Less Wrong

20 17 April 2010 08:23PM

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Comment author: 18 April 2010 01:38:13AM 3 points [-]

I don't see that you've offered a better alternative to VNM utility.

Your examples assert that VNM does a bad job of capturing subtleties in the structure of human preference. But they can all be fixed by using the appropriate outcome-space as your utility function's domain.

For example, if I prefer to decide which child gets a car with a fair coin flip, then I can represent my utility function like this:

``````U(son gets car after coin flip) = 10.1 U(daughter gets car after coin flip) = 10.1 U(son gets car after my arbitrary decision) = 10 U(daughter gets car after my arbitrary decision) = 10
``````

To me this seems like an elegant enough way to capture my value of fairness. And this kind of VNM formulation is so precise that I think it is useful for defining exactly how I value fairness.

I think the alternate utility theories introduce complexity without improving decision theory.

Comment author: 18 April 2010 02:01:03AM *  2 points [-]

Unfortunately, what you describe deviates from the VNM model, which does not allow utilities that depend on the source of "objective randomness" in the lotteries; in this case, the coin. If we allow this dependence, it leads to difficulty in defining what probability is, or at least characterizing it behaviorally, and in proving the VNM theorem itself. This is all explained Bolker (1967) - A simultaneous axiomatization of utility and probability, which is probably my favorite paper on utility theory.

The key to building an axiomatic theory where your approach makes sense, due to Bolker, is to axiomatize expectation itself. I originally planned to post on Bolker's theory, but unfortunately, the axioms themselves are a bit abstract, so I decided I should at least start with VNM utility. But Bolker's axioms allow one to work in the exactly the way you describe, which is why I like it so much.

You didn't mention what I think is most important, though, and why I wrote this post: what do you think of the issues regarding the Archimedean axiom?

Comment author: 18 April 2010 05:49:12AM 4 points [-]

When it comes to biting VNM axioms, the Archimedian/continuity axiom is a non-bullet. Why do you have to insist that 1/3^^^3 human lives is an insultingly high price for green socks? I think it's a bargain. Like I said, this infinitesimal business adds complexity with no decision theory payoff.

I don't see why the son-daughter-car formulation I described isn't good enough as is. The domain of my utility function is the set of spans of future time, and the function works by analyzing the causal relationship between my brain and the person who gets the car.

And I don't see why "objective randomness" or anything else needs to come into the picture. I already have a structure that captures what I intuitively mean by "preference" for this example, and an algorithm that makes good decisions accordingly.

Comment author: 18 April 2010 11:56:00AM *  2 points [-]

Why do you have to insist that 1/3^^^3 human lives is an insultingly high price for green socks?

At no point have I insisted that, or anything analogous. But that's not the question at hand. I haven't found any of my values to be non-Archimedean. The normative question is whether they should be outlawed for rational agents. From the post:

I'd say you're allowed to choose the green socks over the red socks every time, while exhibiting no willingness to sacrifice Big context expected utility for it, and still be considered "rational".

What's your take? Do you think it should be normatively illegal to have non-Archimedian values (like say, a pair of Hausner utilities) and be considered rational? Please share you thoughts on this thread if you're interested.

I don't see why the son-daughter-car formulation I described isn't good enough as is

As far as I'm concerned, it is good enough. So then the challenge is to provide reasonable assumptions which

1. allow the kind of analysis you describe, and

2. still manage to imply mathematical theorems comparable in strength to those of VNM.

VNM utility theory isn't just an intuitive model. It's special: it's actually a consistent mathematical theory, with theorems. And the task of weakening assumptions while maintaining strong theorems is formidable. Luckily, Bolker has done this, by axiomatizing expectation itself, so when we want rigorous theorems to fall back on while reasoning as you describe, they're to be found.

And I don't see why "objective randomness" or anything else needs to come into the picture

AFAIC, it doesn't. But whether we like it or not, the VNM theory needs the source of randomness to not be a source of utility in order for the proof of the VNM utility theorem to work. I find this unsatisfactory, as you seem to. Luckily, Bolker's theory doesn't require this, which is great. Instead, Bolker pays a different price:

• his utility is no longer unique up to two constants, as it is VNM.
• probabilities themselves become behaviorally ambiguous.

This doesn't bother me much, nor you probably; I consider it a price easily worth paying to pass from VNM to Bolker.

I admit this comment is not as in-depth an explanation of some concepts as I'd like; if I find it ties together with enough interesting topics, and I think it will, then I'll write a top level post better explaining this stuff.

Comment author: 18 April 2010 01:38:50AM 1 point [-]

But I did upvote this post, because I think it's a good discussion to have.