Utility need not be bounded
A few months ago I disagreed with Sniffnoy about whether the theorem regarding Savage’s axioms for probability and utility, that utility must be bounded, is a good reason for believing that utility must be bounded. Sniffnoy said yes, because it follows from the axioms, I said no, therefore there is a flaw in the axioms. (The fact that the theorem does follow from the axioms is not an issue.) I concluded that conversation by saying I’d have to think about it.
This I have done. I have followed Jaynes’ dictum that when infinities lead to problems, one must examine the limiting process by which those infinities were arrived at, which almost invariably dissolves the problem. The flaw in Savage’s system is easy to find, easy to describe, and easy to rectify. I have devised a new set of axioms such that:
Probability and utility are constructed from the preference relation by the same method as Savage.
Every model of Savage’s axioms is a model of these axioms and constructs the same probability measure and utility function.
The new axioms also have models with acts and outcomes of unbounded utility.
Acts of infinite utility (such as the St. Petersburg game) are admitted as second-class citizens, in much the same way that measurable functions with infinite integral are in measure theory.
More pathological infinite games (such as St. Petersburg with every other payout in the series reversed in sign) are excluded from the start, but without having to exclude them by any criterion involving utility. (Utility is constructed from the axioms, so cannot be mentioned within them.) Like measurable functions that have no integral, well, that’s just what they are. There’s no point in demanding that they all should.
This removes all force from the argument that because Savage’s axioms imply bounded utility, utility must be bounded. (There are other axiom systems that have that consequence, but I believe that my construction would apply equally to them all.) If one prefers Savage’s axioms because they have that consequence, one must have some other reason for believing that utility must be bounded, or the argument is circular.
There are a few details of proofs still to be filled in, but I don’t think there will be any problems there. Any expert on measure theory could probably dispose of them with a theorem off the shelf. Because of this I don’t want to stick it on arXiv yet, but I would welcome interested readers. Anyone interested, ask me for a copy and give me a way of sending you a PDF.
Despite the title of this post, the only argument for bounded utility I am addressing here is the argument that it follows from various axiom systems. For other, more informal reasons people have for believing in bounded utility, Eliezer (an unbounded fun theorist) has had plenty to say about that in the past, so I’ll just refer people to the Fun Theory Sequence. Because they are informal, you can chew over them forever, which I find an un-fun activity.