BTW, this is off-topic, but since you link to that Arbital page, and I don’t know where else to comment on that—the theorem you’re looking for, that grounds both utility and probability simultaneously, in a non-circular fashion, and without any assumption baked in that R is the correct system of numbers to use[0], is not the complete class theorem. It is Savage’s theorem.
[0]Yes, Savage’s theorem includes an Archimedean assumption, which you could argue is the same thing as baking in R; but I’d say it’s not, because this Archimedean assumption is a direct statement about the agent’s preferences, whereas it’s not immediately clear what picking R as your number system means about the agent’s preferences (and I suspect that most people have used R more on the basis of convenience/familiarity rather than because they recognized the necessity of an Archimedean condition).
Savage’s Theorem isn’t going to convince anyone who doesn’t start out believing that preference ought to be a total preorder. Coherence theorems are talking to anyone who starts out believing that they’d rather have more apples.
If one is talking about one’s preferences over number of apples, then the statement that it is a total preorder, is a weaker statement than the statement that more is better. (Also, you know, real number assumptions all over the place.) If one is talking about preferences not just over number of apples but in general, then even so it seems to me that the complete class theorem seems to be making some very strong assumptions, much stronger than the assumption of a total preorder! (Again, look at all those real number assumptions.)
Moreover it’s not even clear to me that the complete class theorem does what you claim it does, like, at all. Like it starts out assuming the notion of probability. How can it ground probability when it starts out assuming it? And perhaps I’m misunderstanding, but are the “risk functions” it discusses not in utility? It sure looks like expected values of them are being taken with the intent that smaller is better (this seems to be implicit in the definition of r(θ), that r(θ) is measured by expected value when T isn’t a pure strategy). Is that mistaken?
(Possible source of error here: I can’t seem to find a statement of the complete class theorem that fits neatly into Savage/VNM/Cox/etc-style formalism and I’m having some trouble translating it to such, so I may be misunderstanding. The most sense I’m making of it at the moment is that it’s something like your examples for why probabilities must sum to one—i.e., it’s saying, if you already believe in utility, and something almost like probability, it must actually be probability. Is that accurate, or am I off?)
(Edit: Also if you’re taking issue with the preorder assumption, does this mean that you no longer consider VNM to be a good grounding of the notion of utility for those who already accept the idea of probability?)
BTW, this is off-topic, but since you link to that Arbital page, and I don’t know where else to comment on that—the theorem you’re looking for, that grounds both utility and probability simultaneously, in a non-circular fashion, and without any assumption baked in that R is the correct system of numbers to use[0], is not the complete class theorem. It is Savage’s theorem.
[0]Yes, Savage’s theorem includes an Archimedean assumption, which you could argue is the same thing as baking in R; but I’d say it’s not, because this Archimedean assumption is a direct statement about the agent’s preferences, whereas it’s not immediately clear what picking R as your number system means about the agent’s preferences (and I suspect that most people have used R more on the basis of convenience/familiarity rather than because they recognized the necessity of an Archimedean condition).
Savage’s Theorem isn’t going to convince anyone who doesn’t start out believing that preference ought to be a total preorder. Coherence theorems are talking to anyone who starts out believing that they’d rather have more apples.
I can’t make sense of this comment.
If one is talking about one’s preferences over number of apples, then the statement that it is a total preorder, is a weaker statement than the statement that more is better. (Also, you know, real number assumptions all over the place.) If one is talking about preferences not just over number of apples but in general, then even so it seems to me that the complete class theorem seems to be making some very strong assumptions, much stronger than the assumption of a total preorder! (Again, look at all those real number assumptions.)
Moreover it’s not even clear to me that the complete class theorem does what you claim it does, like, at all. Like it starts out assuming the notion of probability. How can it ground probability when it starts out assuming it? And perhaps I’m misunderstanding, but are the “risk functions” it discusses not in utility? It sure looks like expected values of them are being taken with the intent that smaller is better (this seems to be implicit in the definition of r(θ), that r(θ) is measured by expected value when T isn’t a pure strategy). Is that mistaken?
(Possible source of error here: I can’t seem to find a statement of the complete class theorem that fits neatly into Savage/VNM/Cox/etc-style formalism and I’m having some trouble translating it to such, so I may be misunderstanding. The most sense I’m making of it at the moment is that it’s something like your examples for why probabilities must sum to one—i.e., it’s saying, if you already believe in utility, and something almost like probability, it must actually be probability. Is that accurate, or am I off?)
(Edit: Also if you’re taking issue with the preorder assumption, does this mean that you no longer consider VNM to be a good grounding of the notion of utility for those who already accept the idea of probability?)