I’m Harry Altman. I do strange sorts of math.
Posts I’d recommend:
A summary of Savage’s foundations for probability and utility—if the arguments used to ground probability and utility seem circular to you, here’s a non-circular way of doing it.
Sniffnoy
Karma: 4,152
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.)
Oh, I see. I misread your comment then. Yes, I am assuming one already has the ability to discern the structure of an argument and doesn’t need to hire someone else to do that for you...
What I said above. Sorry, to be clear here, by “argument structure” I don’t mean the structure of the individual arguments but rather the overall argument—what rebuts what.
(Edit: Looks like I misread the parent comment and this fails to respond to it; see below.)
This is a good point (the redemption movement comes to mind as an example), but I think the cases I’m thinking of and the cases you’re describing look quite different in other details. Like, the bored/annoyed expert tired of having to correct basic mistakes, vs. the salesman who wants to initiate you into a new, exciting secret. But yeah, this is only a quick-and-dirty heuristic, and even then only good for distinguishing snake oil; it might not be a good idea to put too much weight on it, and it definitely won’t help you in a real dispute (“Wait, both sides are annoyed that the other is getting basic points wrong!”). As Eliezer put it—you can’t learn physics by studying psychology!
Given a bunch of people who disagree, some of whom are actual experts and some of whom are selling snake oil, expertise yourself, there are some further quick-and-dirty heuristics you can use to tell which of the two groups is which. I think basically my suggestion can be best summarized at “look at argument structure”.
The real experts will likely spend a bunch of time correct popular misconceptions, which the fakers may subscribe to. By contrast, the fakers will generally not bother “correcting” the truth to their fakery, because why would they? They’re trying to sell to unreflective people who just believe the obvious-seeming thing; someone who actually bothered to read corrections to misconceptions at any point is likely too savvy to be their target audience.
Sometimes though you do get actual arguments. Fortunately, it’s easier to evaluate arguments than to determine truth oneself—of course, this is only any good if at least one of the parties is right! If everyone is wrong, heuristics like this will likely be no help. But in an experts-and-fakers situation, where one of the groups is right and the other pretty definitely wrong, you can often just use heuristics like “which side has arguments (that make some degree of sense) that the other side has no answer to (that makes any sense)?”. If we grant the assumption that one of the two sides is right, then it’s likely to be that one.
When you actually have a lot of back-and-forth arguing—as you might get in politics, or, as you might get in disputes between actual experts—the usefulness of this sort of thing can drop quickly, but if you’re just trying to sort out fakers from those with actual knowledge, I think it can work pretty well. (Although honestly, in a dispute between experts, I think the “left a key argument unanswered” is still a pretty big red flag.)
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]
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!
Here’s a quick issue I only just noticed but which fortunately is easily fixed:
Above I mentioned you probably want to restrict to a sigma-algebra of events and only allow measurable functions as actions. But, what does measurable mean here? Fortunately, the ordering on outcomes (even without utility) makes measurability meaningful. Except this puts a circularity in the setup, because the ordering on outcomes is induced from the ordering on actions.
Fortunately this is easily patched. You can start with the assumption of a total preorder on outcomes (considering the case of decisions without uncertainty), to make measurability meaningful and restrict actions to measurable functions (once we start considering decisions under uncertainty); then, for P3, instead of the current P3, you would strengthen the current P3 by saying that (on non-null sets) the induced ordering on outcomes actually matches the original ordering on outcomes. Then this should all be fine.
(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.
Apologies, but it sounds like you’ve gotten some things mixed up here? The issue is boundedness of utility functions, not whether they can take on infinity as a value. I don’t think anyone here is arguing that utility functions don’t need to be finite-valued. All the things you’re saying seem to be related to the latter question rather than the former, or you seem to be possibly conflating them?
In the second paragraph perhaps this is just an issue of language—when you say “infinitely high”, do you actually mean “aribtrarily high”? -- but in the first paragraph this does not seem to be the case.
I’m also not sure you understood the point of my question, so let me make it more explicit. Taking the idea of a utility function and modifying it as you describe is what I called “backwards reasoning” above—starting from the idea of a utility function, rather than starting from preferences. 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?
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.
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
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.)
Huh. This would need some elaboration, but this is definitely the most plausible way around the problem I’ve seen.
Now (in Savage’s formalism) actions are just functions from world-states to outcomes (maybe with a measurability condition), so regardless of your prior it’s easy to construct the relevant St. Petersburg gambles if the utility function is unbounded. But seems like what you’re saying is, if we don’t allow arbitrary actions, then the prior could be such that, not only are none of the permitted actions St. Petersburg gambles, but also this remains the case even after future updates. Interesting! Yeah, that just might be workable...
OK, so going by that you’re suggesting, like, introducing varying caps and then taking limits as the cap goes to infinity? It’s an interesting idea, but I don’t see why one would expect it to have anything to do with preferences.
You should check out Abram’s post on complete class theorems. He specifically addresses some of the concerns you mentioned in the comments of Yudkowsky’s posts.
So, it looks to me like what Abrams is doing—once he gets past the original complete class theorem—is basically just inventing some new formalism along the lines of Savage. I think it is very misleading to refer to this as “the complete class theorem”—how on earth was I supposed to know that this was what was being referred to when “the complete class theorem” was mentioned, when it resembles the original theorem so little (and it’s the original theorem that was linked to)? -- and I don’t see why it was necessary to invent this anew, but sure, I can accept that it presumably works, even if the details aren’t spelled out.
But I must note that he starts out by saying that he’s only considering the case when there’s only a finite set of states of the world! I realize you weren’t making a point about bounded utility here; but from that point of view, it is quite significant...
Also, my inner model of Jaynes says that the right way to handle infinities is not to outlaw them, but to be explicit and consistent about what limits we’re taking.
I don’t really understand what that means in this context. It is already quite explicit what limits we’re taking: Given an action (a measurable function from states of the world to outcomes), take its expected utility, with regard to the [finitely-additive] probability on states of the world. (Which is implicitly a limit of sorts.)
I think this is another one of those comments that makes sense if you’re reasoning backward, starting from utility functions, but not if you’re reasoning forward, from preferences. If you look at things from a utility-functions-first point of view, then it looks like you’re outlawing infinities (well, unboundedness that leads to infinities). But from a preferences-first point of view, you’re not outlawing anything. You haven’t outlawed unbounded utility functions, rather they’ve just failed to satisfy fundamental assumptions about decision-making (remember, if you don’t have P7 your utility function is not guaranteed to return correct results about infinite gambles at all!) and so clearly do not reflect your idealized preferences. You didn’t get rid of the infinity, it was simply never there in the first place; the idea that it might have been turned out to be mistaken.
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?)
Is there a reason we can’t just solve this by proposing arbitrarily large bounds on utility instead of infinite bounds? For instance, if we posit that utility is bounded by some arbitrarily high value X, then the wager can only payout values X for probabilities below 1/X.
I’m not sure what you’re asking here. An individual decision-theoretic utility function can be bounded or it can be unbounded. Since decision-theoretic utility functions can be rescaled arbitrarily, naming a precise value for the bounds is meaningless; so like we could just assume the bounds are 0 below and 1 above.
So, I mean, yeah, you can make the problem go away by assuming bounded utility, but if you were trying to say something more than that, a bounded utility that is somehow “closer” to unbounded utility, then no such notion is meaningful.
Apologies if I’ve misunderstood what you’re trying to do.
Yes, thanks, I didn’t bother including it in the body of the post but that’s basically how it goes. Worth noting that this:
Both of these wagers have infinite expected utility, so we must be indifferent between them.
...is kind of shortcutting a bit (at least as Savage/Fishburn[0] does it; he proves indifference between things of infinite expected utility separately after proving that expected utility works when it’s finite), but that is the essence of it, yes.
(As for the actual argument… eh, I don’t have it in front of me and don’t feel like rederiving it...)
[0]I initially wrote Savage here, but I think this part is actually due to Fishburn. Don’t have the book in front of me right now though.
Sorry, but after reading this I’m not very clear on just what exactly the “Magic Formula” refers to. Could you state it explicitly?