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Sniffnoy comments on A summary of Savage's foundations for probability and utility. - Less Wrong
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Comments (88)
Numerical utility at all relies on this, so I'm not sure what you mean here.
Any particular finite gamble requires a finite number of parts. A St. Petersburg lottery requires an infinite number of parts.
Constructing a St. Petersburg lottery relies on this, but I don't see why that means "unbounded utility" depends on it; unbounded utility isn't even a consequence of these axioms, indeed the opposite is so.
In any case I don't even see how you state the notion of bounded (or unbounded) utility without numerical utility; we don't mean bounded in the sense of having internally a maximum and minimum, we mean corresponding to a bounded set of real numbers. No internal maximum or minimum is needed; how do you state that without setting up the correspondence? And to get real numbers you need an Archimedean assumption of some sort.
If there are only a finite number of options, utility can only be unbounded if at least one of the options has the possibility of utilities with arbitrarily large absolute value. It is hard to deal with an infinite number of options, but it might be possible depending on how that works with the other axioms, but this is irrelevant because P6 was not connected to the proof of bounded utility.
That is why I was initially concerned about P6.
I don't think real numbers are the best field to use for utility because of Pascal's mugging, some of the stuff described here, and this paper.
Okay, what field do you think works for utility that's better than real numbers?
The obvious candidates are surreal numbers or non-standard reals. Wikipedia says that the former doesn't have omega plus 1, where omega is the number of ordinary integers, but IIRC the latter does, so I'd try the latter first. I do not feel confident that it solves the problem, though.
The surreals do have ω+1 - see the "..And Beyond" section of the wiki page. If this is contradicted anywhere else on the page, tell me where and I'll correct it.
The surreals are probably the best to use for this, though they'll need to emerge naturally from some axioms, not just be proclaimed correct. From WP: "In a rigorous set theoretic sense, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals.", so even if the surreals are not necessary, they will probably be sufficient.
Conway used surreal numbers for go utilities. I discussed the virtues of surreal utilities here.
Those aren't really utilities because they aren't made for taking expectations, though any totally ordered set can be embedded in the surreals, so they are perfect for choosing from possibly-infinite sets of certain outcomes.
Checking with the definition of utility expectations do not seem critical.
Conway's move values may usefully be seen as utilities associated with possible moves.
Okay, that is not the kind of utility discussed in the post, but it is still a utility.
Are we agreed that bounded real-valued (or rational-valued) utility gets rid of Pascal's mugging?
Yes. Bounded utility solves tons of problems, it just doesn't, AFAICT, describe my preferences.
The bound would also have to be substantially less than 3^^^^3.
As you know, if there is a bound, without loss of generality we can say all utilities go from 0 to 1.
Repairing your claim to take that into account, if you're being mugged for $5, and the plausibility of the mugger's claim is 1/X where X is large, and the utility the mugger promises you is about 1, then you get mugged if your utility for $5 is less than 1/X, roughly. So I agree that there are utility functions that would result in the mugging, but they don't appear especially simple or especially consistent with observed human behavior, so the mugging doesn't seem likely.
Now, if the programming language used to compute the prior on the utility functions has a special instruction that loads 3^^^^3 into an accumulator with one byte, maybe the mugging will look likely. I don't see any way around that.