Comments on: The point of paradox

By: Bill Carone

Bill Carone — Tue, 01 Jun 2004 16:19:54 +0000

Matt, “consider the k-switch problem; if I see k in the first envelope, do I want to switch envelopes? The answer to each k-switch problem is well-defined;” It isn’t as well-defined as I would like, as the limit diverges. “there’s only one limit to be taken, and when you take it you discover that it’s good to switch.” All I would conclude is that, as N increases, eventually it becomes good to switch. The limit doesn’t exist. So you can’t take the limit to get rid of N. If you can’t get rid of N, the dominance argument fails. If you could show that, as N increases, the number of switch values goes to zero, we would be on to something, I think. Then the dominance argument might work.

By: Matt Weiner

Matt Weiner — Sun, 30 May 2004 21:12:44 +0000

Bill—Your math looks sound to me, but it still seems to me to have paradoxical implications. The position you wind up with looks to me like it has a problem akin to w-inconsistency; as N goes to infinity there are more and more non-switch values; however, after you actually open the first envelope, there is no value that is in fact a non-switch value. That’s the issue I have; you know in advance that when you open the first envelope, it will have a finite number written on it. For any of these numbers, consider the k-switch problem; if I see k in the first envelope, do I want to switch envelopes? The answer to each k-switch problem is well-defined; there’s only one limit to be taken, and when you take it you discover that it’s good to switch. So it looks to me as though you can know in advance, for any of the numbers you see when you look in the envelope, that it will be rational to switch once you see it. There are an infinite number of k to be generalized over here, but I don’t see any more problem with that than with any other proof that applies to all finite numbers. (Have to finish in a hurry, soory.)

By: Matt Weiner

Matt Weiner — Sun, 30 May 2004 21:03:51 +0000

Jamie wrote in response to Bill: it seems to me that it is very difficult to see what the upper bound is supposed to be such that no matter what you were offered, you would not risk that upper bound for a 1/2 chance at the offer. I’m not sure that Bill or I needs to be able to answer this question, though. I certainly can’t think of anything that meets this criterion. But I think, to get the result that utilities are bounded, I only need the result that there is some thing such that, if I were offered it, there isn’t anything else that I would risk for a 1/2 chance at it. I can’t say what that is because I can’t survey all offers in advance—but since I can’t survey all offers in advance, it might be that I was in possession of this summum bonum without knowing it. That is, if I have X, and I wouldn’t in fact prefer X to a 50% chance of Y for any Y, I might not know this because I can’t run over all the Y in my head. It seems likely to me that there is some such X, but I couldn’t tell you what it was. Two or three provisos: Actually, what I think is something more like: utilities/day are bounded. For any X I might well prefer a 50% chance at the same thing, lasting more than twice as long. Also, I mentioned above that I don’t accept commensurability, and it seems to me that what happens when we double a certain number of times is that we invoke factors that I might not find commensurable. Would I prefer a 50% chance at spiritual enlightenment to guaranteed success in all my worldly endeavors? I’m not sure. (I think I got this from Putnam.) (And: Nothing is better than the summum bonum. A ham sandwich is better than nothing. QED.)

By: Bill Carone

Bill Carone — Fri, 28 May 2004 20:53:05 +0000

It seems as if some of my post has been garbled; some of the 2-to-the-N have been printed as 2N. Please take that into account, especially in the aside at the end. For example: “As an aside, the 2N has been bothering me; even according to Cantor, 2N is bigger than N when N is infinite, right?” should be “As an aside, the 2-to-the-N has been bothering me; even according to Cantor, 2-to-the-N is bigger than N when N is infinite, right?”

By: Bill Carone

Bill Carone — Fri, 28 May 2004 20:49:37 +0000

Matt, “But in the set-up I posed—which is not finite N, it’s meant to be the limit case—what k would that be?” Again, this is what I have a problem with. You are saying “First I’ll take the limit as N goes to infinity, then I’ll figure out what k is.” It’s very similar to “First I’ll put in and take out an infinite number of balls into the urn, then see if ball k is still in there.” I say, go all the way to your conclusion first. So, don’t switch if k exceeds N. As N goes to infinity, the number of “no-switch” possibilities increases (for example, when N=100, there are only 6 possibilities where you switch, and 94 when you don’t; the formula is something like N-log2N). So, you never can say that for all possible k you will switch, so the paradox disappears. You can’t argue that you should always switch, and therefore you are trading (St.Pete) for (St.Pete-1). I might also argue with the intuitiveness of not buying an option to switch; in most decisions, it would be valuable to have a “do-over.” For any N>3 or so it is worth it, so it should be the same in the limit. If I calculated it right, the actual value of the option diverges as N increases, so it is hard to conclude much about it; as the possible number of flips increases, the value of the option increases. One way to look at my position is to compare it to limit when N goes to zero. We know that, in the limit, f(N) doesn’t have to go to f(0); there might be a discontinuitiy there. One might argue the same thing about infinity, that the limit of f(N) as N goes to infinity doesn’t necessarily equal f(infinity). My position (and, I think, Gauss’s) is that that is an error; there is no such thing as f(infinity), it is defined as the limit of f(N). “There aren’t meant to be any hyperreals involved, so I require a lot of argument to be convinced that the answer can be a hyperreal.” My understanding is that in 1960 or so, someone rigorously established a set of hyperreal numbers; these could be used to model infinite and infinitesmal numbers in a rigorous manner. For example, there is no nonsense like N=N+1 when N is infinite; N+1 is greater than N, even though both are infinite. So, you have actually “introduced hyperreals” by insisting that we start with infinite N, then figure out k. Therefore the answer is simple; don’t switch when k exceeds N. Not possible because N is infinite? Nope; k can be as big as the infinite number 2^{N, which is greater than N.

As an aside, the 2}N has been bothering me; even according to Cantor, 2^N is bigger than N when N is infinite, right? So perhaps even Cantor would agree: in the limit there are an uncountable number of “no-switch” values and only a countable number of “switch” values. However, even if this is true it doesn’t matter; you can create, I think, a similar St. Pete’s paradox using a divergent harmonic series instead of the 1+1+1+1+… that wouldn’t suffer from this.

By: Bill Carone

Bill Carone — Fri, 28 May 2004 20:15:39 +0000

“it seems to me that it is very difficult to see what the upper bound is supposed to be such that no matter what you were offered, you would not risk that upper bound for a 1/2 chance at the offer.” What usually happens is that people grind to a halt, then come up with another idea and go off again. So they start with money, go on to length of life, then to quality of life, then to helping others. Even that grinds to a halt, as you start creating planets upon planets of happy people. The difficult points are different for different people. All I was saying was that if you want to see what people are talking about when they make claims about bounded utility, this exercise helps. I don’t think it proves that our utility is bounded, but it produces interesting internal reactions when we say to ourselves, “Gosh, I can’t think of anything.” for at least a moment or two.

By: Jamie

Jamie — Fri, 28 May 2004 19:36:15 +0000

Matt, “But anyway, it seems to me that if you’re an economist, you may only need to consider preferences that we could realistically encounter, and that those wouldn’t lead to unbounded utilities.” Ah, maybe so. Maybe even if you’re not an economist! Still, we want to be very careful here. If there are some goods that are really enormous compared to the ones we usually choose among, then the fact that they are also incredibly unlikely is not a reason to ignore them. ‘Unrealistic’ has to mean not just unlikely but ignorable. It’s not obvious that an upper bound on utility is established by restricting our attention to the realistic cases in the relevant sense.

By: Jamie

Jamie — Fri, 28 May 2004 19:28:44 +0000

Bill, “At some point it becomes very difficult to see what better prospect would be worth the risk, or to even think about the prospects rationally.” To the contrary, it seems to me that it is very difficult to see what the upper bound is supposed to be such that no matter what you were offered, you would not risk that upper bound for a 1/2 chance at the offer. Can you think of a plausible candidate?

By: Matt Weiner

Matt Weiner — Fri, 28 May 2004 18:46:01 +0000

Jamie—No summum bonum in mind; actually I reject several of the Savage axioms (commensurability and the sure-thing principle, for two), which means I don’t think utility functions are a good way to discuss rationality at all, so I’m perhaps not the best person to discuss this. If we’re talking about conceivable prospects, I suppose I prefer N+1 days in heaven to N days in heaven for any N; but I prefer being in heaven forever to anything else, which is a summum bonum; but the way to treat that is probably by assigning it infinite utility (sorry, Bill), so that won’t work much. Next, I prove that whatever the summum bonum is, a ham sandwich is better.—But anyway, it seems to me that if you’re an economist, you may only need to consider preferences that we could realistically encounter, and that those wouldn’t lead to unbounded utilities. I can’t quite spell it out, but maybe my living for 150 years and being happy all the time and world peace being attained and a few other things would serve as a cap with respect to the choices anyone would reasonably take. Bill—ISTM that the result of the “no sure thing” argument is equally paradoxical to anything we might come up with. The argument is supposed to yield that it’s not true that for all k you see, you’ll want to switch. But in the set-up I posed—which is not finite N, it’s meant to be the limit case—what k would that be? There aren’t meant to be any hyperreals involved, so I require a lot of argument to be convinced that the answer can be a hyperreal.

By: Bill Carone

Bill Carone — Fri, 28 May 2004 15:51:06 +0000

Matt, “One thing is that I’m skeptical that there is any way of specifying the problem-solving procedures in advance that won’t exclude some intuitively obvious answers (or include some paradoxes)” My position (probably wrong, but FWIW): – Paradoxes won’t happen as long as you treat infinities as limits and carefully specify how the limits are to be taken. – Intuitive answers, when such exist, seem to correspond to the idea of taking any limit as late as possible. In other words, do the entire calculation, all the way to the end, in terms of, say, N, then let N go to infinity. – When limits can be taken many different ways (even when we take them as late as possible) and different limits give different answers, the problem is ill-posed, and we should refuse to give an answer in those cases. I would probably agree that, when infinities are involved, there is no method that guarantees an answer that is non-paradoxical and intuitive.

By: Bill Carone

Bill Carone — Fri, 28 May 2004 15:21:26 +0000

Jamie, “Just speaking personally, I can’t think of what prospect that would be. For me, I mean.” Say I’ll give you $2 or a 50-50 shot at X or nothing. What X makes you indifferent? Repeat (I’ll give you X or a 50-50 chance at Y or nothing, what is Y?). Most people find that, at some point, they can’t think of an amount of money that is worth the risk (or they realize that there just isn’t that much material wealth in the world). When that happens, start putting in other prospects instead of money (longer life, better quality of life, helping others, curing diseases, etc.) This is a quick way to see where your own personal “boundedness” starts to show. At some point it becomes very difficult to see what better prospect would be worth the risk, or to even think about the prospects rationally. This isn’t a proof, but an experiment I find interesting.

By: Jamie

Jamie — Fri, 28 May 2004 13:58:18 +0000

It’s not clear to me that human beings do have bounded utility functions. Is there an argument that we do? “I don’t think there’s anything contingent about the fact that we are finite beings with a finite capacity to consume (earthly) goods and bads. So, for the purposes of decision theory, I don’t think there’s anything problematic about rejecting unbounded payments.” But utility is not a matter of one’s capacity to consume. It is a matter of one’s preferences. I very often have preferences over things I do not, will not, cannot consume. If your utility is bounded, that doesn’t necessarily mean that you have a most preferred prospect, but it means you can find a prospect such that there is no prospect you prefer to it by much (say, by more than you prefer one extra drop of espresso in this morning’s latte). Just speaking personally, I can’t think of what prospect that would be. For me, I mean. Does Matt or John Q. have a ‘summum bonum’ in mind?

By: Bill Carone

Bill Carone — Fri, 28 May 2004 06:05:42 +0000

Matt, “I think if we model this with both coin flips capped at N, for all finite N we’ll find that it’s advantageous to retain the option to switch. There will be values in the first envelope for which you don’t want to switch” True, and in fact the number of such “non-switch” values increases as N increases (although the probability of any of these occurring decreases). However, “(As N, the number of flips, goes to infinity there will be an N past which it’s advantageous to switch.)” In other words, any particular value k will, as N increases, be one of those that make you want to switch (a “switch” value). Uh oh, we’ve been here before; this is similar to the infinite ball and urn example. There, the number of balls in the urn increases with N, but any particular ball eventually gets taken out as N increases. Here, the number of “non-switch” values increases with N, but each particular value eventually becomes a “switch” value as N gets high enough. You see now what I will say; take the limit at the end; there is no “sure-thing” argument that lets you say “you know you’ll switch no matter what you see in the second envelope,” So I don’t think it is established that no matter what k is, you’ll want to switch. In fact, for any N no matter how large, there are many possible values of k that will be “non-switch” values. This is similar as Brian’s Paradox 2 (and Bill’s Non-paradox 2) above. So the above dominance argument that you use doesn’t work, and therefore it isn’t intuitively obvious that you shouldn’t take the option.

By: Matt Weiner

Matt Weiner — Thu, 27 May 2004 22:37:13 +0000

Second, all I mean by ill-posed is that we can create more than one mathematical model consistent with the problem statement and different models give different answers. [etc.] I don’t have any real objection to this statement; the question is going to be whether your technique exhausts the ways we can model the problem. (Keep in mind that you and Brian agree at least in part about what goes wrong with the original St. Petersburg two-envelope problem; it involves rearranging the terms of a divergent sequence.) One thing is that I’m skeptical that there is any way of specifying the problem-solving procedures in advance that won’t exclude some intuitively obvious answers (or include some paradoxes). In particular, I’d like some more information about when we take the same N for two different limits—here there are obvious symmetry considerations, but can those considerations be specified more generally? I also have a new problem: God puts a number in each envelope (without subtracting 1 from the second), representing days in heaven (so you want to maximize your number). He then gives you one envelope, and the following offer: Before opening the envelope, you can make the decision in advance to keep what you have. He’ll then destroy the second envelope. Or, you can retain the option to buy the second envelope for 1 day. So after you open the envelope, you’ll have the option to get the what’s in the second envelope minus 1 extra day in heaven instead of what’s in the first envelope. Now, I think it’s already established that for any k in the first envelope, if you’ve retained the option to switch you should. (As N, the number of flips, goes to infinity there will be an N past which it’s advantageous to switch.) I also think it’s intuitively obvious that, if given this option, you should decide in advance to keep the number in the first envelope. If you know you’ll switch no matter what you see in the second envelope, then keeping the option is giving yourself a St. Petersburg ~~1 instead of a St. Petersburg. But~~-I think if we model this with both coin flips capped at N, for all finite N we’ll find that it’s advantageous to retain the option to switch. There will be values in the first envelope for which you don’t want to switch, and it’ll be straightforwardly advantageous to retain the option to switch. Opinions?

By: armando

armando — Thu, 27 May 2004 16:47:11 +0000

You are adressing a different mathematical problem which you feel is more in the spirit of the question raised. Mathematically, this is a non-issue, except in a modelling sense. Your gripe doesn’t seem to be about infinities, per se, but the appropriate use of infinities in certain problems. Bill’s paradoxes lend some weight to this view, but otherwise I don’t really see the problem. (For instance, I think that paradox 2 relies on an intuitive use of “expected” that isn’t supported by the maths.)