As a final comment, there is almost a motte-bailey thing happening where Rationalists will say that, obviously, the VNM axioms describe the optimal framework in which to make decisions, and then proceed to never ever actually use the VNM axioms to make decisions.
This is a misunderstanding. The vNM axioms constrain the shape of an agent’s preferences, they say nothing about how to make decisions, and I don’t think many people ever claimed this—maybe nobody ever claimed this[1]? The vNM axioms specify that your utilities should be linear in probability your utility indifference currves should form parallel hyperplanes in all dimensions of the probability simplex on the available options. That’s it. Preferences conforming to the vNM axioms may be necessary for making “good” (i.e. unexploitable) decisions, but not sufficient.
A similar misunderstanding would be to say “Rationalists will say that, obviously, the Peano axioms describe the optimal framework in which to perform arithmetic, and then proceed to never ever actually invoke any of the Peano axioms when doing their taxes.”
I do agree that there was an implicit promise of “this piece of math will be applicable to your life”, which was fulfilled less for the vNM axioms than for Bayes theorem.
The vNM axioms specify that your utilities should be linear in probability. That’s it.
I don’t think this is right. You are perhaps thinking of the continuity axiom here? But the completeness axiom is not about this (indeed, one cannot even construct a unique utility function to represent incomplete preferences, so there is nothing which may be linear or non-linear in probability).
The vNM axioms constrain the shape of an agent’s preferences, they say nothing about how to make decisions
Suppose your decision in a particular situation comes down to choosing between some number of lotteries (with specific estimated probabilities over their outcomes) and there’s no complexity/nuance/tricks on top of that. In that case, vNM says that you should choose the one with the highest expected utility as this is the one you prefer the most.
At least assuming that choice is the right operationalization of preferences but if it isn’t, then the Dutch book / money-pump arguments don’t follow.
ETA: I guess I could just say:
What are your preferences if not your idealized evaluations of decision-worthiness of options (modulo “being a corrupted piece of software running on corrupted hardware”)?
This is a misunderstanding. The vNM axioms constrain the shape of an agent’s preferences, they say nothing about how to make decisions, and I don’t think many people ever claimed this—maybe nobody ever claimed this[1]? The vNM axioms specify that
your utilities should be linear in probabilityyour utility indifference currves should form parallel hyperplanes in all dimensions of the probability simplex on the available options. That’s it. Preferences conforming to the vNM axioms may be necessary for making “good” (i.e. unexploitable) decisions, but not sufficient.A similar misunderstanding would be to say “Rationalists will say that, obviously, the Peano axioms describe the optimal framework in which to perform arithmetic, and then proceed to never ever actually invoke any of the Peano axioms when doing their taxes.”
I do agree that there was an implicit promise of “this piece of math will be applicable to your life”, which was fulfilled less for the vNM axioms than for Bayes theorem.
I welcome examples where people claimed this.
I don’t think this is right. You are perhaps thinking of the continuity axiom here? But the completeness axiom is not about this (indeed, one cannot even construct a unique utility function to represent incomplete preferences, so there is nothing which may be linear or non-linear in probability).
Oops, you’re of course right. I’ll change my comment.
Suppose your decision in a particular situation comes down to choosing between some number of lotteries (with specific estimated probabilities over their outcomes) and there’s no complexity/nuance/tricks on top of that. In that case, vNM says that you should choose the one with the highest expected utility as this is the one you prefer the most.
At least assuming that choice is the right operationalization of preferences but if it isn’t, then the Dutch book / money-pump arguments don’t follow.
ETA: I guess I could just say: