I'm Harry Altman. I do strange sorts of math.

Highly upvoted posts:

- 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.
- Should correlation coefficients be expressed as angles?

A summary of Savage's foundations for probability and utility.

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.

Underappreciated points about utility functions (of both sorts)

(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.

Underappreciated points about utility functions (of both sorts)

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?

Underappreciated points about utility functions (of both sorts)

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.

Underappreciated points about utility functions (of both sorts)

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

Underappreciated points about utility functions (of both sorts)

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.)

Underappreciated points about utility functions (of both sorts)

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...

Underappreciated points about utility functions (of both sorts)

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.

Underappreciated points about utility functions (of both sorts)

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.

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

addaxioms, you'll only be able to concludemorethings, 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!