I think you’ve misunderstood a fair bit. I hope you don’t mind if I address this slightly out of order.
Or if infinite utilities are not immediately a problem, then by a more complicated argument, involving constructing multiple St. Petersburg-type combinations and demonstrating that the axioms imply that there both should and should not be a preference between them.
This is exactly what Fishburn does, as I mentioned above. (Well, OK, I didn’t attribute it to Fishburn, I kind of implicitly misattributed it to Savage, but it was actually Fishburn; I didn’t think that was worth going into.)
I haven’t studied the proof of boundedness in detail, but it seems to be that unbounded utilities allow St. Petersburg-type combinations of them with infinite utilities, but since each thing is supposed to have finite utility, that is a contradiction.
He does not give details, but the argument that I conjecture from his text is that if there are unbounded utilities then one can construct a convex combination of infinitely many of them that has infinite utility (and indeed one can), contradicting the proof from his axioms that the utility function is a total function to the real numbers.
What you describe in these two parts I’m quoting is, well, not how decision-theoretic utility functions work. A decision-theoretic utility function is a function on outcomes, not on gambles over outcomes. You take expected utility of a gamble; you don’t take utility of a gamble.
So, yes, if you have an unbounded decision-theoretic utility function, you can set up a St. Petersburg-style situation that will have infinite expected utility. But that is not by itself a problem! The gamble has infinite expected utility; no individual outcome has infinite utility. There’s no contradiction yet.
Of course, you then do get a contradiction when you attempt to compare two of these that have been appropriately set up, but...
But by a similar argument, one might establish that the real numbers must be bounded, when instead one actually concludes that not all series converge
What? I don’t know what one might plausibly assume that might imply the boundedness of the real numbers.
...oh, I think I see the analogy you’re going for here. But, it seems to rest on the misunderstanding of utility functions discussed above.
and that one cannot meaningfully compare the magnitudes of divergent infinite series.
Well, so, one must remember the goal here. So, let’s start with divergent series, per your analogy. (I’m assuming you’re discussing series of nonnegative numbers here, that diverge to infinity.)
So, well, there’s any number of ways we could compare divergent series. We could just say that they sum to infinity, and so are equal in magnitude. Or we could try to do a more detailed comparison of their growth rates. That might not always yield a well-defined result though. So yeah. There’s not any one universal way to compare magnitudes of divergent series, as you say; if someone asks, which of these two series is bigger, you might just have to say, that’s a meaningless question. All this is as you say.
But that’s not at all the situation we find ourselves in choosing between two gambles! If you reason backward, from the idea of utility functions, it might seem reasonable to say, oh, these two gambles are both divergent, so comparison is meaningless. But if you reason forward, from the idea of preferences… well, you have to pick one (or be indifferent). You can’t just leave it undefined. Or if you have some formalism where preferences can be undefined (in a way that is distinct from indifference), by all means explain it… (but what happens when you program these preferences into an FAI and it encounters this situation? It has to pick. Does it pick arbitrarily? How is that distinct from indifference?)
That we have preferences between gambles is the whole thing we’re starting from.
I note that in order to construct convex combinations of infinitely many states, Fishburn extends his axiom 0 to allow this. He does not label this extension separately as e.g. “Axiom 0*”. So if you were to ask which of his axioms to reject in order to retain unbounded utility, it could be none of those labelled as such, but the one that he does not name, at the end of the first paragraph on p.1055. Notice that the real numbers satisfy Axiom 0 but not Axiom 0*. It is that requirement that all infinite convex combinations exist that surfaces later as the boundedness of the range of the utility function.
Sorry, but looking through Fishburn’s paper I can’t see anything like this. The only place where any sort of infinite combination seems to be mentioned is section 9, which is not relevant. Axiom 0 means one thing throughout and allows only finite convex combinations. I simply don’t see where you’re getting this at all.
(Would you mind sticking to Savage’s formalism for simplicity? I can take the time to properly read Fishburn if for some reason you insist things have to be done this way, but otherwise for now I’m just going to put things in Savage’s terms.)
In any case, in Savage’s formalism there’s no trouble in proving that the necessary actions exist—you don’t have to go taking convex combinations of anything, you simply directly construct the functions. You just need an appropriate partition of the set of world-states (provided by the Archimedean axiom he assumes, P6) and an appropriate set of outcomes (which comes from the assumption of unbounded utility). You don’t have to go constructing other things and then doing some fancy infinite convex combination of them.
If you don’t mind, I’d like to ask: could just tell me what in particular in Savage’s setup or axioms you find to be the probable weak point? If it’s P7 you object to, well, I already discussed that in the post; if you get rid of that, the utility function may be unbounded but it’s no longer guaranteed to give correct results when comparing infinite gambles.
While searching out the original sources, I found a paper indicating that at least in 1993, bounded utility theorems were seen as indicating a problem with Savage’s axioms: “Unbounded utility for Savage’s “Foundations of Statistics” and Other Models”, by Peter Wakker. There is another such paper from 2014. I haven’t read them, but they indicate that proofs of boundedness of utility are seen as problems for the axioms, not discoveries that utility must be bounded.
I realize a number of people see this as a problem. Evidently they have some intuition or argument that disagrees with the boundedness of utility. Whatever this intuition or argument is, I would be very surprised if it were as strong as the argument that utility must be bounded. There’s no question that assumptions can be bad. I just think the reasons to think these are bad that have been offered, are seriously flimsy compared to the reasons to think that they’re good. So I see this as basically a sort of refusal to take the math seriously. (Again: Which axiom should we throw out, or what part of the setup should we rework?)
Or if you have some formalism where preferences can be undefined (in a way that is distinct from indifference), by all means explain it… (but what happens when you program these preferences into an FAI and it encounters this situation? It has to pick. Does it pick arbitrarily? How is that distinct from indifference?)
A short answer to this (something longer later) is that an agent need not have preferences between things that it is impossible to encounter. The standard dissolution of the St. Petersberg paradox is that nobody can offer that gamble. Even though each possible outcome is finite, the offerer must be able to cover every possible outcome, requiring that they have infinite resources.
Since the gamble cannot be offered, no preferences between that gamble and any other need exist. If your axioms require both that preference must be total and that St. Petersburg gambles exist, I would say that that is a flaw in the axioms. Fishburn (op. cit., following Blackwell and Girschick, an inaccessible source) requires that the set of gambles be closed under infinitary convex combinations. I shall take a look at Savage’s axioms and see what in them is responsible for the same thing.
Looking at the argument from the other end, at what point in valuing numbers of intelligent lives does one approach an asymptote, bearing in mind the possibility of expansion to the accessible universe? What if we discover that the habitable universe is vastly larger than we currently believe? How would one discover the limits, if there are any, to one’s valuing?
Fishburn (op. cit., following Blackwell and Girschick, an inaccessible source) requires that the set of gambles be closed under infinitary convex combinations.
Again, I’m simply not seeing this in the paper you linked? As I said above, I simply do not see anything like that outside of section 9, which is irrelevant. Can you point to where you’re seeing this condition?
I shall take a look at Savage’s axioms and see what in them is responsible for the same thing.
In the case of Savage, it’s not any particular axiom, but rather the setup. An action is a function from world-states to outcomes. If you can construct the function, the action (gamble) exists. That’s all there is to it. And the relevant functions are easy enough to construct, as I described above; you use P6 (the Archimedean condition, which also allows flipping coins, basically) to construct the events, and we have the outcomes by assumption. You assign the one to the other and there you go.
(If you don’t want to go getting the book out, you may want to read the summary of Savage I wrote earlier!)
A short answer to this (something longer later) is that an agent need not have preferences between things that it is impossible to encounter. The standard dissolution of the St. Petersberg paradox is that nobody can offer that gamble. Even though each possible outcome is finite, the offerer must be able to cover every possible outcome, requiring that they have infinite resources. Since the gamble cannot be offered, no preferences between that gamble and any other need exist.
So, would it be fair to sum this up as “it is not necessary to have preferences between two gambles if one of them takes on unbounded utility values”? Interesting. That doesn’t strike me as wholly unworkable, but I’m skeptical. In particular:
Can we phrase this without reference to utility functions? It would say a lot more for the possibility if we can.
What if you’re playing against Nature? A gamble can be any action; and in a world of unbounded utility functions, why should one believe that any action must have some bound on how much utility it can get you? Sure, sure, second law of thermodynamics and all that, but that’s just a feature of the paticular universe we happen to live in, not something that reshapes your preferences. (And if we were taking account of that sort of thing, we’d probably just say, oh, utility is bounded after all, in a kind of stupid way.) Notionally, it could be discovered to be wrong! It won’t happen, but it’s not probability literally 0.
Or are you trying to cut out a more limited class of gambles as impossible? I’m not clear on this, although I’m not certain it affects the results.
Anyway, yeah, as I said, my main objection is that I see no reason to believe that, if you have an unbounded utility function, Nature cannot offer you a St. Petersburg game. Or I mean, to the extent I do see reasons to believe that, they’re facts about the particular universe we happen to live in, that notionally could be discovered to be wrong.
Looking at the argument from the other end, at what point in valuing numbers of intelligent lives does one approach an asymptote, bearing in mind the possibility of expansion to the accessible universe? What if we discover that the habitable universe is vastly larger than we currently believe? How would one discover the limits, if there are any, to one’s valuing?
This is exactly the sort of argument that I called “flimsy” above. My answer to these questions is that none of this is relevant.
Both of us are trying to extend our ideas about preferences from ordinary situations to extraordinary ones. (Like, I agree that some sort of total utilitarianism is a good heuristic for value under the conditions we’re familiar with.) This sort of extrapolation, to an unfamiliar realm, is always potentially dangerous. The question then becomes, what sort of tools can we expect to continue to work, without needing any sort of adjustment to the new conditions?
I do not expect speculation about the particular form preferences our would take under these unusual conditions to be trustworthy. Whereas basic coherence conditions had damn well better continue to hold, or else we’re barely even talking about sensible preferences anymore.
Or, to put it differently, my answer is, I don’t know, but the answer must satisfy basic coherence conditions. There’s simply no way that the idea that decision-theoretic utility has to increase linearly with number intelligent lives, is on anywhere near as solid ground as that! The mere fact that it’s stated in terms of a utility function in the first place, rather than in terms of something more basic, is something of a smell. Complicated statements we’re not even entirely sure how to formulate can easily break in a new context. Short simple statements that have to be true for reasons of simple coherence don’t break.
(Also, some of your questions don’t seem to actually appreciating what a bounded utility function would actually mean. It wouldn’t mean taking an unbounded utility function and then applying a cap to it. It would just mean something that naturally approaches 1 as things get better and 0 as things get worse. There is no point at which it approaches an asymptote; that’s not how asymptotes work. There is no limit to one’s valuing; presumably utility 1 does not actually occur. Or, at least, that’s how I infer it would have to work.)
Again, I’m simply not seeing this in the paper you linked? As I said above, I simply do not see anything like that outside of section 9, which is irrelevant. Can you point to where you’re seeing this condition?
In Fishburn’s “Bounded Expected Utility”, page 1055, end of first paragraph (as cited previously):
However, we shall for the present take ℘=℘d (for any σ -algebra that contains each x ) since this is the Blackwell-Girshick setting. Not only is ℘d an abstract convex set, but also if αi≥0 and Pi∈℘d for i=1,2,… and Σ∞i=1αi=1 , then Σ∞i=1αiPi∈℘d .
That depends on some earlier definitions, e.g. ℘d is a certain set of probability distributions (the “d” stands for “discrete”) defined with reference to some particular σ -algebra, but the important part is that last infinite sum: this is where all infinitary convex combinations are asserted to exist. Whether that is assigned to “background setup” or “axioms” does not matter. It has to be present, to allow the construction of St. Petersburg gambles.
(This is more properly a followup to my sibling comment, but posting it here so you’ll see it.)
I already said that I think that thinking in terms of infinitary convex combinations, as you’re doing, is the wrong way to go about it; but it took me a bit to put together why that’s definitely the wrong way.
Specifically, it assumes probability! Fishburn, in the paper you link, assumes probability, which is why he’s able to talk about why infinitary convex combinations are or are not allowed (I mean, that and the fact that he’s not necessarily arbitrary actions).
Savage doesn’t assume probability! So if you want to disallow certain actions… how do you specify them? Or if you want to talk about convex combinations of actions—not just infinitary ones, any ones—how do you even define these?
In Savage’s framework, you have to prove that if two actions can be described by the same probabilities and outcomes, then they’re equivalent. E.g., suppose action A results in outcome X with probability 1⁄2 and outcome Y with probability 1⁄2, and suppose action B meets that same description. Are A and B equivalent? Well, yes, but that requires proof, because maybe A and B take outcome X on different sets of probability 1⁄2. (OK, in the two-outcome case it doesn’t really require “proof”, rather it’s basically just his definition of probability; but the more general case requires proof.)
So, until you’ve established that theorem, that it’s meaningful to combine gambles like that, and that the particular events yielding the probabilities aren’t relevant, one can’t really meaningfully define convex combinations at all. This makes it pretty hard to incorporate them into the setup or axioms!
More generally this should apply not only to Savage’s particular formalism, but any formalism that attempts to ground probability as well as utility.
Anyway yeah. As I think I already said, I think we should think of this in terms not of, what combinations of actions yield permitted actions, but rather whether there should be forbidden actions at all. (Note btw in the usual VNM setup there aren’t any forbidden actions either! Although there infinite gambles are, while not forbidden, just kind of ignored.) But this is in particular why trying to put it it in terms of convex combinations as you’ve done doesn’t really work from a fundamentals point of view, where there is no probability yet, only preferences.
I already said that I think that thinking in terms of infinitary convex combinations, as you’re doing, is the wrong way to go about it; but it took me a bit to put together why that’s definitely the wrong way.
Specifically, it assumes probability! Fishburn, in the paper you link, assumes probability, which is why he’s able to talk about why infinitary convex combinations are or are not allowed (I mean, that and the fact that he’s not necessarily arbitrary actions).
Savage doesn’t assume probability!
Savage doesn’t assume probability or utility, but their construction is a mathematical consequence of the axioms. So although they come later in the exposition, they mathematically exist as soon as the axioms have been stated.
So if you want to disallow certain actions… how do you specify them?
I am still thinking about that, and may be some time.
As a general outline of the situation, you read P1-7 ⇒ bounded utility as modus ponens: you accept the axioms and therefore accept the conclusion. I read it as modus tollens: the conclusion seems wrong, so I believe there is a flaw in the axioms. In the same way, the axioms of Euclidean geometry seemed very plausible as a description of the physical space we find ourselves in, but conflicts emerged with phenomena of electromagnetism and gravity, and eventually they were superseded as descriptions of physical space by the geometry of differential manifolds.
It isn’t possible to answer the question “which of P1-7 would I reject?” What is needed to block the proof of bounded utility is a new set of axioms, which will no doubt imply large parts of P1-7, but might not imply the whole of any one of them. If and when such a set of axioms can be found, P1-7 can be re-examined in their light.
Oh, so that’s what you’re referring to. Well, if you look at the theorem statements, you’ll see that P=P_d is an axiom that is explicitly called out in the theorems where it’s assumed; it’s not implictly part of Axiom 0 like you asserted, nor is it more generally left implicit at all.
but the important part is that last infinite sum: this is where all infinitary convex combinations are asserted to exist. Whether that is assigned to “background setup” or “axioms” does not matter. It has to be present, to allow the construction of St. Petersburg gambles.
I really think that thinking in terms of infinitary convex combinations is the wrong way to go about this here. As I said above: You don’t get a St. Petersburg gamble by taking some fancy convex combination, you do it by just constructing the function. (Or, in Fishburn’s framework, you do it by just constructing the distribution; same effect.) I guess without P=P_d you do end up relying on closure properties in Fishburn’s framework, but Savage’s framework just doesn’t work that way at all; and Fishburn with P=P_d, well, that’s not a closure property. Rather what Savage’s setup, and P=P_d have in common, is that they’re, like, arbitrary-construction properties: If you can make a thing, you can compare it.
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.)
I think you’ve misunderstood a fair bit. I hope you don’t mind if I address this slightly out of order.
This is exactly what Fishburn does, as I mentioned above. (Well, OK, I didn’t attribute it to Fishburn, I kind of implicitly misattributed it to Savage, but it was actually Fishburn; I didn’t think that was worth going into.)
What you describe in these two parts I’m quoting is, well, not how decision-theoretic utility functions work. A decision-theoretic utility function is a function on outcomes, not on gambles over outcomes. You take expected utility of a gamble; you don’t take utility of a gamble.
So, yes, if you have an unbounded decision-theoretic utility function, you can set up a St. Petersburg-style situation that will have infinite expected utility. But that is not by itself a problem! The gamble has infinite expected utility; no individual outcome has infinite utility. There’s no contradiction yet.
Of course, you then do get a contradiction when you attempt to compare two of these that have been appropriately set up, but...
What? I don’t know what one might plausibly assume that might imply the boundedness of the real numbers.
...oh, I think I see the analogy you’re going for here. But, it seems to rest on the misunderstanding of utility functions discussed above.
Well, so, one must remember the goal here. So, let’s start with divergent series, per your analogy. (I’m assuming you’re discussing series of nonnegative numbers here, that diverge to infinity.)
So, well, there’s any number of ways we could compare divergent series. We could just say that they sum to infinity, and so are equal in magnitude. Or we could try to do a more detailed comparison of their growth rates. That might not always yield a well-defined result though. So yeah. There’s not any one universal way to compare magnitudes of divergent series, as you say; if someone asks, which of these two series is bigger, you might just have to say, that’s a meaningless question. All this is as you say.
But that’s not at all the situation we find ourselves in choosing between two gambles! If you reason backward, from the idea of utility functions, it might seem reasonable to say, oh, these two gambles are both divergent, so comparison is meaningless. But if you reason forward, from the idea of preferences… well, you have to pick one (or be indifferent). You can’t just leave it undefined. Or if you have some formalism where preferences can be undefined (in a way that is distinct from indifference), by all means explain it… (but what happens when you program these preferences into an FAI and it encounters this situation? It has to pick. Does it pick arbitrarily? How is that distinct from indifference?)
That we have preferences between gambles is the whole thing we’re starting from.
Sorry, but looking through Fishburn’s paper I can’t see anything like this. The only place where any sort of infinite combination seems to be mentioned is section 9, which is not relevant. Axiom 0 means one thing throughout and allows only finite convex combinations. I simply don’t see where you’re getting this at all.
(Would you mind sticking to Savage’s formalism for simplicity? I can take the time to properly read Fishburn if for some reason you insist things have to be done this way, but otherwise for now I’m just going to put things in Savage’s terms.)
In any case, in Savage’s formalism there’s no trouble in proving that the necessary actions exist—you don’t have to go taking convex combinations of anything, you simply directly construct the functions. You just need an appropriate partition of the set of world-states (provided by the Archimedean axiom he assumes, P6) and an appropriate set of outcomes (which comes from the assumption of unbounded utility). You don’t have to go constructing other things and then doing some fancy infinite convex combination of them.
If you don’t mind, I’d like to ask: could just tell me what in particular in Savage’s setup or axioms you find to be the probable weak point? If it’s P7 you object to, well, I already discussed that in the post; if you get rid of that, the utility function may be unbounded but it’s no longer guaranteed to give correct results when comparing infinite gambles.
I realize a number of people see this as a problem. Evidently they have some intuition or argument that disagrees with the boundedness of utility. Whatever this intuition or argument is, I would be very surprised if it were as strong as the argument that utility must be bounded. There’s no question that assumptions can be bad. I just think the reasons to think these are bad that have been offered, are seriously flimsy compared to the reasons to think that they’re good. So I see this as basically a sort of refusal to take the math seriously. (Again: Which axiom should we throw out, or what part of the setup should we rework?)
A short answer to this (something longer later) is that an agent need not have preferences between things that it is impossible to encounter. The standard dissolution of the St. Petersberg paradox is that nobody can offer that gamble. Even though each possible outcome is finite, the offerer must be able to cover every possible outcome, requiring that they have infinite resources.
Since the gamble cannot be offered, no preferences between that gamble and any other need exist. If your axioms require both that preference must be total and that St. Petersburg gambles exist, I would say that that is a flaw in the axioms. Fishburn (op. cit., following Blackwell and Girschick, an inaccessible source) requires that the set of gambles be closed under infinitary convex combinations. I shall take a look at Savage’s axioms and see what in them is responsible for the same thing.
Looking at the argument from the other end, at what point in valuing numbers of intelligent lives does one approach an asymptote, bearing in mind the possibility of expansion to the accessible universe? What if we discover that the habitable universe is vastly larger than we currently believe? How would one discover the limits, if there are any, to one’s valuing?
Again, I’m simply not seeing this in the paper you linked? As I said above, I simply do not see anything like that outside of section 9, which is irrelevant. Can you point to where you’re seeing this condition?
In the case of Savage, it’s not any particular axiom, but rather the setup. An action is a function from world-states to outcomes. If you can construct the function, the action (gamble) exists. That’s all there is to it. And the relevant functions are easy enough to construct, as I described above; you use P6 (the Archimedean condition, which also allows flipping coins, basically) to construct the events, and we have the outcomes by assumption. You assign the one to the other and there you go.
(If you don’t want to go getting the book out, you may want to read the summary of Savage I wrote earlier!)
So, would it be fair to sum this up as “it is not necessary to have preferences between two gambles if one of them takes on unbounded utility values”? Interesting. That doesn’t strike me as wholly unworkable, but I’m skeptical. In particular:
Can we phrase this without reference to utility functions? It would say a lot more for the possibility if we can.
What if you’re playing against Nature? A gamble can be any action; and in a world of unbounded utility functions, why should one believe that any action must have some bound on how much utility it can get you? Sure, sure, second law of thermodynamics and all that, but that’s just a feature of the paticular universe we happen to live in, not something that reshapes your preferences. (And if we were taking account of that sort of thing, we’d probably just say, oh, utility is bounded after all, in a kind of stupid way.) Notionally, it could be discovered to be wrong! It won’t happen, but it’s not probability literally 0.
Or are you trying to cut out a more limited class of gambles as impossible? I’m not clear on this, although I’m not certain it affects the results.
Anyway, yeah, as I said, my main objection is that I see no reason to believe that, if you have an unbounded utility function, Nature cannot offer you a St. Petersburg game. Or I mean, to the extent I do see reasons to believe that, they’re facts about the particular universe we happen to live in, that notionally could be discovered to be wrong.
This is exactly the sort of argument that I called “flimsy” above. My answer to these questions is that none of this is relevant.
Both of us are trying to extend our ideas about preferences from ordinary situations to extraordinary ones. (Like, I agree that some sort of total utilitarianism is a good heuristic for value under the conditions we’re familiar with.) This sort of extrapolation, to an unfamiliar realm, is always potentially dangerous. The question then becomes, what sort of tools can we expect to continue to work, without needing any sort of adjustment to the new conditions?
I do not expect speculation about the particular form preferences our would take under these unusual conditions to be trustworthy. Whereas basic coherence conditions had damn well better continue to hold, or else we’re barely even talking about sensible preferences anymore.
Or, to put it differently, my answer is, I don’t know, but the answer must satisfy basic coherence conditions. There’s simply no way that the idea that decision-theoretic utility has to increase linearly with number intelligent lives, is on anywhere near as solid ground as that! The mere fact that it’s stated in terms of a utility function in the first place, rather than in terms of something more basic, is something of a smell. Complicated statements we’re not even entirely sure how to formulate can easily break in a new context. Short simple statements that have to be true for reasons of simple coherence don’t break.
(Also, some of your questions don’t seem to actually appreciating what a bounded utility function would actually mean. It wouldn’t mean taking an unbounded utility function and then applying a cap to it. It would just mean something that naturally approaches 1 as things get better and 0 as things get worse. There is no point at which it approaches an asymptote; that’s not how asymptotes work. There is no limit to one’s valuing; presumably utility 1 does not actually occur. Or, at least, that’s how I infer it would have to work.)
In Fishburn’s “Bounded Expected Utility”, page 1055, end of first paragraph (as cited previously):
That depends on some earlier definitions, e.g. ℘d is a certain set of probability distributions (the “d” stands for “discrete”) defined with reference to some particular σ -algebra, but the important part is that last infinite sum: this is where all infinitary convex combinations are asserted to exist. Whether that is assigned to “background setup” or “axioms” does not matter. It has to be present, to allow the construction of St. Petersburg gambles.
Will address the rest of your comments later.
(This is more properly a followup to my sibling comment, but posting it here so you’ll see it.)
I already said that I think that thinking in terms of infinitary convex combinations, as you’re doing, is the wrong way to go about it; but it took me a bit to put together why that’s definitely the wrong way.
Specifically, it assumes probability! Fishburn, in the paper you link, assumes probability, which is why he’s able to talk about why infinitary convex combinations are or are not allowed (I mean, that and the fact that he’s not necessarily arbitrary actions).
Savage doesn’t assume probability! So if you want to disallow certain actions… how do you specify them? Or if you want to talk about convex combinations of actions—not just infinitary ones, any ones—how do you even define these?
In Savage’s framework, you have to prove that if two actions can be described by the same probabilities and outcomes, then they’re equivalent. E.g., suppose action A results in outcome X with probability 1⁄2 and outcome Y with probability 1⁄2, and suppose action B meets that same description. Are A and B equivalent? Well, yes, but that requires proof, because maybe A and B take outcome X on different sets of probability 1⁄2. (OK, in the two-outcome case it doesn’t really require “proof”, rather it’s basically just his definition of probability; but the more general case requires proof.)
So, until you’ve established that theorem, that it’s meaningful to combine gambles like that, and that the particular events yielding the probabilities aren’t relevant, one can’t really meaningfully define convex combinations at all. This makes it pretty hard to incorporate them into the setup or axioms!
More generally this should apply not only to Savage’s particular formalism, but any formalism that attempts to ground probability as well as utility.
Anyway yeah. As I think I already said, I think we should think of this in terms not of, what combinations of actions yield permitted actions, but rather whether there should be forbidden actions at all. (Note btw in the usual VNM setup there aren’t any forbidden actions either! Although there infinite gambles are, while not forbidden, just kind of ignored.) But this is in particular why trying to put it it in terms of convex combinations as you’ve done doesn’t really work from a fundamentals point of view, where there is no probability yet, only preferences.
Savage doesn’t assume probability or utility, but their construction is a mathematical consequence of the axioms. So although they come later in the exposition, they mathematically exist as soon as the axioms have been stated.
I am still thinking about that, and may be some time.
As a general outline of the situation, you read P1-7 ⇒ bounded utility as modus ponens: you accept the axioms and therefore accept the conclusion. I read it as modus tollens: the conclusion seems wrong, so I believe there is a flaw in the axioms. In the same way, the axioms of Euclidean geometry seemed very plausible as a description of the physical space we find ourselves in, but conflicts emerged with phenomena of electromagnetism and gravity, and eventually they were superseded as descriptions of physical space by the geometry of differential manifolds.
It isn’t possible to answer the question “which of P1-7 would I reject?” What is needed to block the proof of bounded utility is a new set of axioms, which will no doubt imply large parts of P1-7, but might not imply the whole of any one of them. If and when such a set of axioms can be found, P1-7 can be re-examined in their light.
Oh, so that’s what you’re referring to. Well, if you look at the theorem statements, you’ll see that P=P_d is an axiom that is explicitly called out in the theorems where it’s assumed; it’s not implictly part of Axiom 0 like you asserted, nor is it more generally left implicit at all.
I really think that thinking in terms of infinitary convex combinations is the wrong way to go about this here. As I said above: You don’t get a St. Petersburg gamble by taking some fancy convex combination, you do it by just constructing the function. (Or, in Fishburn’s framework, you do it by just constructing the distribution; same effect.) I guess without P=P_d you do end up relying on closure properties in Fishburn’s framework, but Savage’s framework just doesn’t work that way at all; and Fishburn with P=P_d, well, that’s not a closure property. Rather what Savage’s setup, and P=P_d have in common, is that they’re, like, arbitrary-construction properties: If you can make a thing, you can compare it.
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.)