OK, time for the second half, where I get to the errors in the ones I initially skipped. And yes, I’m going to assert some philosophical positions which (for whatever reason) aren’t well-accepted on this site, but there’s still plenty of mathematical errors to go around even once you ignore any philosphical problems. And yeah, I’m still going to point out missing formalism, but I will try to focus on the more substantive errors, of which there are plenty.

So, let’s get those philosophical problems out of the way first, and quickly review utility functions and utilitarianism, because this applies to a bunch of what you discuss here. Like, this whole post takes a very naive view of the idea of “utility”, and this needs some breaking down. Apologies if you already know all of what I’m about to say, but I think given the context it bears repeating.

So: There are two different things meant by “utility function”. The first is decision-theoretic; an agent’s utility function is a function whose expected value it attempts to maximize. The second is the one used by utilitarianism, which involves (at present, poorly-defined) “E-utility” functions, which are *not* utility functions in the decision-theoretic sense, that are then somehow aggregated (maybe by addition? who knows?) into a decision-theoretic utility function. Yes, this terminology is terribly confusing. But these are two separate things and need to be kept separate.

Basically, any agent that satisfies appropriate rationality conditions has a utility function in the decision-theoretic sense (obviously such idealized agents don’t actually exist, but it’s still a useful abstraction). So you could say, roughly speaking, any rational consequentialist has a decision-theoretic utility function. Whereas E-utility is specifically a utilitarian notion, rather than a general consequentalist or purely descriptive notion like decision-theoretic utility (it’s also not at all clear how to define it).

Anyway, if you want surreal E-utility functions… well, I think that’s still probably pretty dumb for reasons I’ll get to, but since E-utility is so poorly defined that’s not *obviously* wrong. But let’s talk about decision-theoretic utility functions. These need to be real-valued for very good reasons.

Because, well, why use utility functions at all? What makes us think that a rational agent’s preferences can be described in terms of a utility function in the first place? Well, there’s an answer to that: Savage’s theorem. I’ve already described this above—it gives rationality conditions, phrased directly in terms of an agent’s preferences, that together suffice to guarantee that said preferences can be described by a utility function. And yes, it’s real-valued.

(And, OK, it’s real-valued because Savage includes an Archimedean assumption, but, well—do you think that’s a bad assumption? Let me repeat here a naive argument against infinite and infinitesimal utilities I’ve seen before on this site (I forget due to who; I think Eliezer maybe?). Suppose we go with a naive treatment of infinitesimal utilities, and A has infinitesimal utility compared to B. Then since any action you take at *all* has *some* positive (real, non-infinitesimal) probability of bringing about B, even sitting in your room waving your hand back and forth in the air, A simply has no effect on your decision making; all considerations of B, even stupid ones, completely wash it out. Which means that A’s infinitesimal utility does not, in fact, have any place in a decision-theoretic utility function. Do you really want to throw out that Archimedean assumption? Also if you do throw it out, I don’t think that actually gets you non-real-valued utilities, I think it just, y’know, doesn’t get you utilities. The agent’s preferences can’t necessarily be described with a utility function of any sort. Admittedly I could be wrong about that last part; I haven’t checked.)

In short, your philosophical mistake here is of a kind with your mathematical mistakes—in both cases, you’re starting from a system of numbers (surreals) and trying awkwardly to fit it to the problem, even when it blatantly does *not* fit, does not have the properties that are required; rather than seeing what requirements the problem actually *calls* for and finding something that meets those needs. As I’ve pointed out multiple times by now, you’re trying to make use of properties that the surreal numbers just don’t have. Work forward from the requirements, don’t try to force into them things that don’t meet them!

By the way, Savage’s theorem also shows that utility functions must be *bounded*. That utility functions must be bounded does not, for whatever reason, seem to be a well-accepted position on this site, but, well, it’s correct so I’m going to continue asserting it, including here. :P Now it’s true that the VNM theorem doesn’t prove this, but that’s due to a deficiency in the VNM theorem’s assumptions, and with that gap fixed it does. I don’t want to belabor this point here, so I’ll just refer you to this previous discussion.

(Also the VNM theorem is just a worse foundation generally because it assumes real-valued probabilities to begin with, but that’s a separate matter. Though maybe here it’s not—since you can’t claim to avoid the boundedness requirement by saying you’re justifying the use of utilities with VNM rather than Savage, since you seem to want to allow surreal-valued probabilities!)

Anyway, so, yes, utilities should be real-valued (and bounded) or else you have no good reason to use them—to use surreal-valued utilities is to start from the *assumption* that you should use utilities (a big assumption! why would one ever assume such a thing?) when it should be a conclusion (a conclusion of theorems that say it must be real-valued).

Ah, but could infinities or infinitesimals appear in an E-utility function, that the utilitarians use? I’ve been ignoring those, after all. But, since they’re getting aggregated into a decision-theoretic utility function, which is real-valued (or maybe it’s not quite a decision-theoretic utility function, but it should still be real-valued by the naive argument above), unless this aggregation function can magnify an infinitesimal into a non-infinitesimal, the same problem will arise, the infinitesimals will still have no relevance, and thus should never have been included.

(Yeah, I suppose in what you write you consider “summing over an infinite number of people”. But: 1. such infinite sums with infinitesimals don’t actually work mathematically, for reasons I’ve already covered, and 2. you can’t actually have an infinite number of people, so it’s all moot anyway.)

Yikes, all that and I haven’t even gotten to examining in detail the particular mathematical problems in the remaining ones! You know what, I’ll end this here and split that comment out into a third post. Point is, now in these remaining ones, when I want to point out philosophical problems, I can just point back to this comment rather than repeating all this again.

OK, time to actually now get into what’s wrong with the ones I skipped initially. Already wrote the intro above so not repeating that. Time to just go.

Infinitarian paralysis: So, philosophical problems to start: As an actual decision theory problem this is all moot since you can’t actually have an infinite number of people. I.e. it’s not clear why this is a problem at all. Secondly, naive assumption of utilitarian aggregation as mentioned above, etc, not going over this again. Enough of this, let’s move on.So what are the mathematical problems here? Well, you haven’t said a lot here, but here’s what it’s look like to me. I think you’ve written one thing here that is essentially correct, which is that,

ifyou did have some system of surreal valued-utilities, it would indeed likely make the distinction you want.But, once again, that’s a big “if”, and not just for philosophical reasons but for the mathematical reasons I’ve already brought up so many times right now—you can’t do infinite sums in the surreals like you want, for reasons I’ve already covered. So there’s a reason I included the word “likely” above, because if you did find an appropriate way of doing such a sum, I can’t even necessarily guarantee that it would behave like you want (yes, finite sums should, but infinite sums require definition, and who knows if they’ll actually be compatible with finite sums like they should be?).

But the really jarring thing here, the thing that really exposes a serious error in your thought (well, OK, that does so to a greater extent), is not in your proposed solution—it’s in what you contrast it with.

Cardinalvalued-utilities? Nothing about that makes sense! That’s not a remotely well-defined alternative you can contrast with! And the thing that bugs me about this error is that it’s just sounforced—I mean, man, you could have said “extended reals” rather than cardinals, and made essentially the same point while making at least some sense! This is just demonstrating once again that not only do you not understand surreals, you do not understand cardinals or ordinals either.(Well, I suppose technically there’s the possibility that you do but expect your audience doesn’t and are talking down to them, but since you’re writing here on Less Wrong, I’m going to assume that’s not the case.)

Seriously, cardinals and utilities do not go together. I mean, cardinals and

real numbersdo not go together. Like surreals and utilities don’t go together either, but at least the surreals include the reals! At least you canattemptto treat it naively in special cases, as you’ve done in a number of these examples, even if the result probably isn’t meaningful! Cardinals you can’t even do that.And once again, there’s no reason anyone who understood cardinals would even

wantcardinal-valued utilities. That’s just not what cardinals are for! Cardinals are for counting how many there are of something. Utility calculations are not a “how many” problem.Sphere of suffering: Once again we have infinitely many people (so this whole problem is again a non-problem) and once again we have some sort of naive utility aggregation over those infinitely many people with all the mathematical problems that brings (only now it’s over time-slices as well?). Enough of this, moving on.Honestly I don’t have much new to say about the bad mathematics here, much of it is the same sort of mistakes as you made in the ones I covered in my initial comment. To cover those ones briefly:

Surreal numbers do not measure how far a grid extends (similar to examples I’ve already covered)

There’s not a

questionof how far the grid extends, allowing it to be a transfinite variable l is just changing the problem (similar to examples I’ve already covered)Surreal numbers also do not measure number of time steps, you want ordinals for that (similar to examples I’ve already covered)

Repeat #2 but for the time steps (similar to examples I’ve already covered)

But OK. The one new thing here, I guess, is that now you’re talking about a “majority” of the time slices? Yeah, that is once again not well-defined at all. Cardinality won’t help you here, obviously; are you putting a measure on this somehow? I think you’re going to have some problems there.

Trumped: Same problems I’ve discussed before. Surreal numbers do not count time steps, you’re changing the problem by introducing a variable, utility aggregation over an infinite set (this time of time-slices rather than people), you know the drill.But actually here you’re changing the problem in a different way, by supposing that Trump knows in advance the number of time steps? The original problem just had this as a repeated offer. Maybe that’s a philosophical rather than mathematical problem. Whatever. It’s changing the problem, is the point.

And then on top of that your solution doesn’t even make any sense. Let’s suppose you meant an ordinal number of days rather than a surreal number of days, since that is what you’d actually use in this context. OK. Suppose for example then that the number of days is ω (which is, after all, the original problem before you changed it). So your solution says that Trump should accept the deal so long as the day number is less than the surreal number ω/3. Except, oops! Every ordinal less than ω is also less than ω/3. Trump always accepts the deal, we’re back at the original problem.

I.e., even granting that you can somehow make all the formalism work, this is still just wrong.

St. Petersburg paradox: OK, so, there’s a lot wrong here. Let me get the philosophical problem out of the way first—the real solution to the St. Petersburg paradox is that you must look not at expected money, but at expected utility, and utility functions must be bounded, so this problem can’t arise. But let’s get to the math, because, like I said, there’s a lot wrong here.Let’s get the easy-to-describe problems out of the way first: You are once again using surreals where you should be using ordinals; you are once again assuming some sort of theory of infinite sums of surreals; getting infinitely many heads has zero probability, not infinitesimal (probabilities are real-valued, you could try to introduce a theory of surreal probabilities but that will have problems already discussed), what happens in that case is irrelevant; you are once again changing the problem by allowing things to go on beyond ω steps; and, minor point, but where on earth did the function n |-> n comes from? Don’t you mean n |-> 2^n?

OK, that’s largely stuff I’ve said before. But the thing that puzzled me the most in your claimed solution is the first sentence:

What? I mean, yeah, sure, the surreals have multiple infinities while, say, the extended nonnegative reals have only one, no question there. But that sentence still makes no sense! It, like, seems to reveal a fundamental misunderstanding so great I’m having trouble comprehending it. But I will give it my best shot.So the thing is, that—ignoring the issue of unbounded utility and what’s the correct decision—the original setup has no ambiguities. You can’t choose to make it different by changing what system of numbers you

describeit with. Now, I don’t know if you’re making the mistake I think you’re making, because who knows what mistake you might be making, but it looks to me that you are confusing numbers that are part of the actual problem specification, with auxiliary numbers just used todescribethe problem.Like, what’s actually going on here is that there is a set of coin flips, right? The elements of that set will be indexed by the natural numbers, and will form a (possibly improper, though with probability 0) initial segment of it—those numbers are part of the actual problem specification. The idea though that there might be infinitely many coin flips… that’s just a description. When I say “With probability 0, the set of flips will be infinite”, that’s just another way of saying, “With probability 0, the set of flips will be

N.” It doesn’t make sense to ask “Ah, but what system of numbers are you using to measure its infinitude?” It doesn’t matter! The set I’m describing isN! (And in any case I just said it was an infiniteset, although I suppose you could say I was implicitly using cardinals.)This is, I suppose, an idea that’s shown up over and over in your claimed solutions, but since I skipped over this particular one before, I guess I never got it so explicitly before. Again, I’m having to guess what you think, but it looks to me like you think that the

numbersare what’s primary, rather than the actualobjectsthe problems are about, and so you can just change the numbers system and get a different version of the same problem. I mean, OK, often the numbersareprimary and you can do that! But sometimes they’re just descriptive.Oy. I have no idea whether I’ve correctly described what your misunderstanding, but whatever it is, it’s pretty big. Let’s just move on.

Trouble in St. Petersburg: Can I first just complain that your numbers don’t seem to match up with your text? 13 is not 9*2+3. I’m just going to assume you meant 21 rather than 13, because none of the other interpretations I can come up with make sense.Also this problem once again relies on unbounded utilities, but I don’t need to go on about that. (Although if you were to somehow reformulate it without those—though that doesn’t seem possible in this coin-flip formulation—then the problem would be basically similar to Satan’s Apple. I have my own thoughts on that problem, but, well, I’m not going to go into it here because that’s not the point.)

Anyway, let’s get to the surreal abuse! Well, OK, again I don’t have much new to say here, it’s the same sort of surreal abuse as you’ve made before. Namely: Using surreals where they don’t make sense (time steps should be counted by ordinals); changing the problem by introducing a transfinite variable; thinking that all ordinals are successor ordinals (sorry, but with n=ω, i.e. the original problem, there’s still no last step).

Ultimately you don’t offer any solution? Whatever. The errors above still stand.

The headache: More naive aggregation and thinking you can do infinite sums and etc. Or at least so I’m gathering from your claimed solution. Anyway that’s boring.The surreal abuse here though is also boring, same types as we’ve seen before—using surreals where they make no sense but where ordinals would; ignoring the existence of limit ordinals; and of course the aforementioned infinite sums and such.

OK. That’s all of them. I’m stopping there. I think the first comment was really enough to demonstrate my point, but now I can honestly claim to have addressed every one of your examples. Time to go sleep now.