A further short answer. In Savage’s formulation, from P1-P6 he derives Theorem 4 of section 2 of chapter 5 of his book, which is linear interpolation in any interval. Clearly, linear interpolation does not work on an interval such as [17,Inf], therefore there cannot be any infinitely valuable gambles. St. Petersburg-type gambles are therefore excluded from his formulation.
Savage does not actually prove bounded utility. Fishburn did this later, as Savage footnotes in the edition I’m looking at, so Fishburn must be tackled. Theorem 14.5 of Fishburn’s book derives bounded utility from Savage’s P1-P7. His proof seems to construct a St. Petersburg gamble from the supposition of unbounded utility, deriving a contradiction. I shall have to examine further how his construction works, to discern what in Savage’s axioms allows the construction, when P1-P6 have already excluded infinitely valuable gambles.
Savage does not actually prove bounded utility. Fishburn did this later, as Savage footnotes in the edition I’m looking at, so Fishburn must be tackled.
Yes, it was actually Fishburn that did that. Apologies if I carelessly implied it was Savage.
IIRC, Fishburn’s proof, formulated in Savage’s terms, is in Savage’s book, at least if you have the second edition. Which I think you must, because otherwise that footnote wouldn’t be there at all. But maybe I’m misremembering? I think it has to be though...
In Savage’s formulation, from P1-P6 he derives Theorem 4 of section 2 of chapter 5 of his book, which is linear interpolation in any interval.
I don’t have the book in front of me, but I don’t recall any discussion of anything that could be called linear interpolation, other than the conclusion that expected utility works for finite gambles. Could you explain what you mean? I also don’t see the relevance of intervals here? Having read (and written a summary of) that part of the book I simply don’t know what you’re talking about.
Clearly, linear interpolation does not work on an interval such as [17,Inf], therefore there cannot be any infinitely valuable gambles. St. Petersburg-type gambles are therefore excluded from his formulation.
I still don’t know what you’re talking about here, but I’m familiar enough with Savage’s formalism to say that you seem to have gotten quite lost somewhere, because this all sounds like nonsense.
From what you’re saying, the impression that I’m getting is that you’re treating Savage’s formalism like Fishburn’s, where there’s some a-prior set of actions under consideration, and so we need to know closure properties about that set. But, that’s not how Savage’s formalism works. Rather the way it works is that actions are just functions (possibly with a measurability condition—he doesn’t discuss this but you probably want it) from world-states to outcomes. If you can construct the action as a function, there’s no way to exclude it.
I shall have to examine further how his construction works, to discern what in Savage’s axioms allows the construction, when P1-P6 have already excluded infinitely valuable gambles.
Well, I’ve already described the construction above, but I’ll describe it again. Once again though, you’re simply wrong about that last part; that last statement is not only incorrect, but fundamentally incompatible with Savage’s whole approach.
Anyway. To restate the construction of how to make a St. Petersburg gamble. (This time with a little more detail.) An action is simply a function from world-states to outcomes.
By assumption, we have a sequence of outcomes a_i such that U(a_i) >= 2^i and such that U(a_i) is strictly increasing.
We can use P6 (which allows us to “flip coins”, so to speak) to construct events E_i (sets of world-states) with probability 1/2^i.
Then, the action G that takes on the value a_i on the set E_i is a St. Petersburg gamble.
For the particular construction, you take G as above, and also G’, which is the same except that G’ takes the value a_1 on E_0, instead of the value a_0.
Savage proves in the book (although I think the proof is due to Fishburn? I’m going by memory) that given two gambles, both of which are preferred to any essentially bounded gamble, the agent must be indifferent between them. (The proof uses P7, obviously—the same thing that proves that expected utility works for infinite gambles at all. I don’t recall the actual proof offhand and don’t feel like trying to reconstruct it right now, but anyway I think you have it in front of you from the sounds of it.) And we can show both these gambles are preferred to any essentially bounded gamble by comparing to truncated versions of themselves (using sure-thing principle) and using the fact that expected utility works for essentially bounded gambles. Thus the agent must be indifferent between G and G’. But also, by the sure-thing principle (P2 and P3), the agent must prefer G’ to G. That’s the contradiction.
Edit: Earlier version of this comment misstated how the proof goes
Oops, turns out I did misremember—Savage does not in fact put the proof in his book. You have to go to Fishburn’s book.
I’ve been reviewing all this recently and yeah—for anyone else who wants to get into this, I’d reccommend getting Fishburn’s book (“Utility Theory for Decision Making”) in addition to Savage’s “Foundations of Statistics”. Because in addition to the above, what I’d also forgotten is that Savage leaves out a bunch of the proofs. It’s really annoying. Thankfully in Fishburn’s treatment he went and actually elaborated all the proofs that Savage thought it OK to skip over...
(Also, stating the obvious, but get the second edition of “Foundations of Statistics”, as it fixes some mistakes. You probably don’t want just Fishburn’s book, it’s fairly hard to read by itself.)
A further short answer. In Savage’s formulation, from P1-P6 he derives Theorem 4 of section 2 of chapter 5 of his book, which is linear interpolation in any interval. Clearly, linear interpolation does not work on an interval such as [17,Inf], therefore there cannot be any infinitely valuable gambles. St. Petersburg-type gambles are therefore excluded from his formulation.
Savage does not actually prove bounded utility. Fishburn did this later, as Savage footnotes in the edition I’m looking at, so Fishburn must be tackled. Theorem 14.5 of Fishburn’s book derives bounded utility from Savage’s P1-P7. His proof seems to construct a St. Petersburg gamble from the supposition of unbounded utility, deriving a contradiction. I shall have to examine further how his construction works, to discern what in Savage’s axioms allows the construction, when P1-P6 have already excluded infinitely valuable gambles.
Yes, it was actually Fishburn that did that. Apologies if I carelessly implied it was Savage.
IIRC, Fishburn’s proof, formulated in Savage’s terms, is in Savage’s book, at least if you have the second edition. Which I think you must, because otherwise that footnote wouldn’t be there at all. But maybe I’m misremembering? I think it has to be though...
I don’t have the book in front of me, but I don’t recall any discussion of anything that could be called linear interpolation, other than the conclusion that expected utility works for finite gambles. Could you explain what you mean? I also don’t see the relevance of intervals here? Having read (and written a summary of) that part of the book I simply don’t know what you’re talking about.
I still don’t know what you’re talking about here, but I’m familiar enough with Savage’s formalism to say that you seem to have gotten quite lost somewhere, because this all sounds like nonsense.
From what you’re saying, the impression that I’m getting is that you’re treating Savage’s formalism like Fishburn’s, where there’s some a-prior set of actions under consideration, and so we need to know closure properties about that set. But, that’s not how Savage’s formalism works. Rather the way it works is that actions are just functions (possibly with a measurability condition—he doesn’t discuss this but you probably want it) from world-states to outcomes. If you can construct the action as a function, there’s no way to exclude it.
Well, I’ve already described the construction above, but I’ll describe it again. Once again though, you’re simply wrong about that last part; that last statement is not only incorrect, but fundamentally incompatible with Savage’s whole approach.
Anyway. To restate the construction of how to make a St. Petersburg gamble. (This time with a little more detail.) An action is simply a function from world-states to outcomes.
By assumption, we have a sequence of outcomes a_i such that U(a_i) >= 2^i and such that U(a_i) is strictly increasing.
We can use P6 (which allows us to “flip coins”, so to speak) to construct events E_i (sets of world-states) with probability 1/2^i.
Then, the action G that takes on the value a_i on the set E_i is a St. Petersburg gamble.
For the particular construction, you take G as above, and also G’, which is the same except that G’ takes the value a_1 on E_0, instead of the value a_0.
Savage proves in the book (although I think the proof is due to Fishburn? I’m going by memory) that given two gambles, both of which are preferred to any essentially bounded gamble, the agent must be indifferent between them. (The proof uses P7, obviously—the same thing that proves that expected utility works for infinite gambles at all. I don’t recall the actual proof offhand and don’t feel like trying to reconstruct it right now, but anyway I think you have it in front of you from the sounds of it.) And we can show both these gambles are preferred to any essentially bounded gamble by comparing to truncated versions of themselves (using sure-thing principle) and using the fact that expected utility works for essentially bounded gambles. Thus the agent must be indifferent between G and G’. But also, by the sure-thing principle (P2 and P3), the agent must prefer G’ to G. That’s the contradiction.
Edit: Earlier version of this comment misstated how the proof goes
Oops, turns out I did misremember—Savage does not in fact put the proof in his book. You have to go to Fishburn’s book.
I’ve been reviewing all this recently and yeah—for anyone else who wants to get into this, I’d reccommend getting Fishburn’s book (“Utility Theory for Decision Making”) in addition to Savage’s “Foundations of Statistics”. Because in addition to the above, what I’d also forgotten is that Savage leaves out a bunch of the proofs. It’s really annoying. Thankfully in Fishburn’s treatment he went and actually elaborated all the proofs that Savage thought it OK to skip over...
(Also, stating the obvious, but get the second edition of “Foundations of Statistics”, as it fixes some mistakes. You probably don’t want just Fishburn’s book, it’s fairly hard to read by itself.)