Ahh, thanks for clarifying. I think what happened was that your modus ponens was my modus tollens—so when I think about my preferences, I ask “what conditions do my preferences need to satisfy for me to avoid being exploited or undoing my own work?” whereas you ask something like “if my preferences need to correspond to a bounded utility function, what should they be?” [1]. As a result, I went on a tangent about infinity to begin exploring whether my modified notion of a utility function would break in ways that regular ones wouldn’t.
Why should one believe that modifying the idea of a utility function would result in something that is meaningful about preferences, without any sort of theorem to say that one’s preferences must be of this form?
I agree, one shouldn’t conclude anything without a theorem. Personally, I would approach the problem by looking at the infinite wager comparisons discussed earlier and trying to formalize them into additional rationality condition. We’d need
an axiom describing what it means for one infinite wager to be “strictly better” than another.
an axiom describing what kinds of infinite wagers it is rational to be indifferent towards
Then, I would try to find a decisioning-system that satisfies these new conditions as well as the VNM-rationality axioms (where VNM-rationality applies). If such a system exists, these axioms would probably bar it from being represented fully as a utility function. If it didn’t, that’d be interesting. In any case, whatever happens will tell us more about either the structure our preferences should follow or the structure that our rationality-axioms should follow (if we cannot find a system).
Of course, maybe my modification of the idea of a utility function turns out to show such a decisioning-system exists by construction. In this case, modifying the idea of a utility function would help tell me that my preferences should follow the structure of that modification as well.
Does that address the question?
[1] From your post:
We should say instead, preferences are not up for grabs—utility functions merely encode these, remember. But if we’re stating idealized preferences (including a moral theory), then these idealized preferences had better be consistent—and not literally just consistent, but obeying rationality axioms to avoid stupid stuff. Which, as already discussed above, means they’ll correspond to a bounded utility function.
Ahh, thanks for clarifying. I think what happened was that your modus ponens was my modus tollens—so when I think about my preferences, I ask “what conditions do my preferences need to satisfy for me to avoid being exploited or undoing my own work?” whereas you ask something like “if my preferences need to correspond to a bounded utility function, what should they be?” [1]
That doesn’t seem right. The whole point of what I’ve been saying is that we can write down some simple conditions that ought to be true in order to avoid being exploitable or otherwise incoherent, and then it follows as a conclusion that they have to correspond to a [bounded] utility function. I’m confused by your claim that you’re asking about conditions, when you haven’t been talking about conditions, but rather ways of modifying the idea of decision-theoretic utility.
Something seems to be backwards here.
I agree, one shouldn’t conclude anything without a theorem. Personally, I would approach the problem by looking at the infinite wager comparisons discussed earlier and trying to formalize them into additional rationality condition. We’d need
an axiom describing what it means for one infinite wager to be “strictly better” than another.
an axiom describing what kinds of infinite wagers it is rational to be indifferent towards
I’m confused here; it sounds like you’re just describing, in the VNM framework, the strong continuity requirement, or in Savage’s framework, P7? Of course Savage’s P7 doesn’t directly talk about these things, it just implies them as a consequence. I believe the VNM case is similar although I’m less familiar with that.
Then, I would try to find a decisioning-system that satisfies these new conditions as well as the VNM-rationality axioms (where VNM-rationality applies). If such a system exists, these axioms would probably bar it from being represented fully as a utility function.
That doesn’t make sense. If you add axioms, you’ll only be able to conclude more things, not fewer. Such a thing will necessarily be representable by a utility function (that is valid for finite gambles), since we have the VNM theorem; and then additional axioms will just add restrictions. Which is what P7 or strong continuity do!
Thanks for the reply. I re-read your post and your post on Savage’s proof and you’re right on all counts. For some reason, it didn’t actually click for me that P7 was introduced to address unbounded utility functions and boundedness was a consequence of taking the axioms to their logical conclusion.
Well, it’s worth noting that P7 is introduced to address gambles with infinitely many possible outcomes, regardless of whether those outcomes are bounded or not (which is the reason I argue above you can’t just get rid of it). But yeah. Glad that’s cleared up now! :)
Ahh, thanks for clarifying. I think what happened was that your modus ponens was my modus tollens—so when I think about my preferences, I ask “what conditions do my preferences need to satisfy for me to avoid being exploited or undoing my own work?” whereas you ask something like “if my preferences need to correspond to a bounded utility function, what should they be?” [1]. As a result, I went on a tangent about infinity to begin exploring whether my modified notion of a utility function would break in ways that regular ones wouldn’t.
I agree, one shouldn’t conclude anything without a theorem. Personally, I would approach the problem by looking at the infinite wager comparisons discussed earlier and trying to formalize them into additional rationality condition. We’d need
an axiom describing what it means for one infinite wager to be “strictly better” than another.
an axiom describing what kinds of infinite wagers it is rational to be indifferent towards
Then, I would try to find a decisioning-system that satisfies these new conditions as well as the VNM-rationality axioms (where VNM-rationality applies). If such a system exists, these axioms would probably bar it from being represented fully as a utility function. If it didn’t, that’d be interesting. In any case, whatever happens will tell us more about either the structure our preferences should follow or the structure that our rationality-axioms should follow (if we cannot find a system).
Of course, maybe my modification of the idea of a utility function turns out to show such a decisioning-system exists by construction. In this case, modifying the idea of a utility function would help tell me that my preferences should follow the structure of that modification as well.
Does that address the question?
[1] From your post:
That doesn’t seem right. The whole point of what I’ve been saying is that we can write down some simple conditions that ought to be true in order to avoid being exploitable or otherwise incoherent, and then it follows as a conclusion that they have to correspond to a [bounded] utility function. I’m confused by your claim that you’re asking about conditions, when you haven’t been talking about conditions, but rather ways of modifying the idea of decision-theoretic utility.
Something seems to be backwards here.
I’m confused here; it sounds like you’re just describing, in the VNM framework, the strong continuity requirement, or in Savage’s framework, P7? Of course Savage’s P7 doesn’t directly talk about these things, it just implies them as a consequence. I believe the VNM case is similar although I’m less familiar with that.
That doesn’t make sense. If you add axioms, you’ll only be able to conclude more things, not fewer. Such a thing will necessarily be representable by a utility function (that is valid for finite gambles), since we have the VNM theorem; and then additional axioms will just add restrictions. Which is what P7 or strong continuity do!
Thanks for the reply. I re-read your post and your post on Savage’s proof and you’re right on all counts. For some reason, it didn’t actually click for me that P7 was introduced to address unbounded utility functions and boundedness was a consequence of taking the axioms to their logical conclusion.
Well, it’s worth noting that P7 is introduced to address gambles with infinitely many possible outcomes, regardless of whether those outcomes are bounded or not (which is the reason I argue above you can’t just get rid of it). But yeah. Glad that’s cleared up now! :)