# On the Von Neumann Morgenstern Utility Theorem, Part 1

A question I occasionally get asked is “What do you have against the Von Neumann Morgenstern Utility Theorem, David?”. Or the more specific variant “Which of its premises do you disagree with?”.

Well, it would be faster for me to list what aspects of the theorem I do agree with.

Which is, err.

Um.

Well I haven’t come up with a convincing objection to the independence hypothesis. That’s not to say that I necessarily think it’s right, only that I haven’t got a strong opinion on it either way.

On top of its explicit premises, I also reject the implicit ones that this is a sensible way to reason about peoples’ behaviours, and the conclusion would say anything useful even if I were to grant all of the other premises.

So, yeah, I have a lot of objections. Far too many to write only one article about it. So instead I’m going to break this up and write it as and when I feel like it. Advance warning: When I write series of articles, they tend not to be finished. I will try to make each of these self contained so you don’t need to hold your breath waiting for the next one.

I will start with the most obvious place to begin this series: Somewhere in the middle picked almost at random.

#### Preferences over lotteries are discontinuous

Amongst the “reasonable hypotheses” of the VNM theorem is a continuity assumption. It can be formulated as this:

Suppose we have lotteries A, B, C with A < B < C. There is some probability p such that the lottery $$L_p = pA + (1-p)C$$ is equivalently preferable to B. There are other, weaker, formulations of this but that doesn't matter very much: If continuity is not a hypothesis of your variant of the VNM theorem, it's still a conclusion of it, so its truth value is relevant.

I think continuity is simply false. Consider the following example: C is “I have a sandwich”, B is “I have a kitten to play with” and A is “The world is destroyed”.

We have $$L_0 = A < B < C = L_p$$, certainly. We also have that for $$p < p'$$, $$L_p > L_{p’}$$ (I will always prefer to increase the probability of my getting a sandwich versus destroying the world).

There is no probability with which I will accept a sandwich as a fair exchange for some probability of destroying the world. No matter how tasty the sandwich is, or how cute the kitten I’d get in the non world destroying chance.

But I certainly prefer kittens to sandwiches (not to eat of course. To cuddle). If the probability of destroying the world is functionally indistinct from 0, I will pick $$L_p$$ – give me a kitten with probability 1. If it is functionally distinct from 0, I will always pick $$B$$ – the possibility of the kitten isn’t worth it, give me the sandwich instead.

You might argue that there is some true tiny probability with which I would consider them equal, but I’m skeptical. Further, I think that even if there were such a probability in theory (which I am not convinced of), it is so small as to not be plausible to estimate or work with.

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## 5 thoughts on “On the Von Neumann Morgenstern Utility Theorem, Part 1”

1. Paul Crowley

This doesn’t really make a big difference – if you take away the Archimedian assumption, you end up with something very much like VNM rationality, but instead of utility being a real number it’s something more like a map from ordinals to real numbers.

1. david Post author

That’s a fairly huge difference given that basically every single model based on top of utility theory lives and breathes real valued utility functions, and that one of my major objections to most of this reasoning strategy is that utility functions are taken to be real valued.

1. Paul Crowley

(FTR I think you’re mistaken to reject the Archimedian assumption, but that’s a longer discussion.)

I don’t think it actually makes a big difference to how you might use VNM in practice to make decisions. Most decisions will be made at the level of the first ordinal; that looks exactly like VNM. Then you might have decisions where you’re perfectly indifferent at that ordinal, but the next ordinal makes your mind up for you. All the rest of the framework, of multiplying by probabilities and summing to get the expected utility, stays exactly the same.

2. david Post author

I think in practice the sort of decisions this leads to are quite different than those lead to from real valued numbers, but I need to think through it a bit further before I say that with any degree of confidence.