Comments on: Two More Envelopes

By: Bill Carone

Bill Carone — Sun, 23 May 2004 06:44:53 +0000

“I don’t think the problem is ill-posed as such.” All I mean by ill-posed is that – we can formulate different mathematical models that don’t conflict with the situation, and – those models give different answers. The idea is that anything left out of a problem must be irrelevant to the answer, or the problem is ill-posed. Similar to the treatment of the “random chord” problem in most probability texts. (However, that isn’t really ill-posed; see Jaynes’s “The Well-Posed Problem”). I don’t know what people do when ill-posed problems are tested by experiment. I don’t think Brian’s problem is ill-posed, as I think there are strong symmetry arguments to model the two envelopes as two finite deals with N possible outcomes each and then take N to infinity. So I think this is the one way to model the problem, and it gives the intuitive answer (that you should take the second “1 day smaller” deal). My position is that all the other ways suggested of looking at this problem use infinite sets directly without specifying limiting procedures, and I therefore reject them as models. Brian’s formulations have an infinite set of possible outcomes, and his calculations are based on that; specifically when he says that no matter what you see in the first deal, the other envelope still has an infinite expected value, so you should keep it. You can’t get the infinite expected value without calculating directly on the infinite set. And as you said before, you can use the same argument the other way, showing that no matter what the second one pays out, the first has an infinite expected value, so you should go with the second deal. This is often the case with infinite sets that aren’t limits; they can often prove A and not-A. This doesn’t happen with limits; as long as the limits are specified in advance, there will be no paradoxes. Of course, I might be wrong…

By: Bill Carone

Bill Carone — Sun, 23 May 2004 05:05:28 +0000

“In Brian’s problem, the two deals are exactly the same except one is one day less than the other. On the most obvious reading, that’s false;” You are clearly right; the deals have the same probabilities and payoffs, except that the payoffs of the second deal are always one less than the corresponding payoffs in the first. It isn’t that, e.g., if the first payout is 8, then the second is 7. This doesn’t change the argument that, if we are going to use limits, we should use the same N for both, right?

By: Luc

Luc — Sat, 22 May 2004 13:03:02 +0000

We still get the very odd result that if you open the envelope, you should prefer sticking with what you’ve got to taking the angel’s deal, no matter what you see. But it looks like you should be indifferent to, or perhaps even prefer, the deal before you look in the envelope. And that’s what’s at the core of the two-envelope paradox. I still don’t get it, i suppose. If you calculate the expectation, with or without the deal, it is infinite. The envelope is supposed not to contain an infinite number, so the content of an envelope is always less than the expectation (!). When comparing the expectation with and without the deal, there is no difference. Comparing an envelope with an expectation, selects the envelope. And only when comparing without using expectation, you can intuitively select the deal. The opening of the envelope doesn’t change anything, and doesn’t change the preference for the deal or not (?).

By: Matt Weiner

Matt Weiner — Fri, 21 May 2004 18:32:28 +0000

Bill, thanks for the response back. I don't think the problem is ill-posed as such. The mathematical calculations are ill-formulated, since they lead to paradoxical conclusions, but the situation Brian proposes is conceivable (if rather absurd)--God really could create such an atom, and I suppose He could put you in Purgatory for arbitrarily long stays if He wanted. (Note that in Brian's latest variant, there's never a question of an infinite stay, only of arbitrarily long stays.) In Brian’s problem, the two deals are exactly the same except one is one day less than the other. On the most obvious reading, that's false; deal 2 isn't calculated by subtracting 1 from deal 1. Rather, each of them is calculated by an independent process. If deal 2 were calculated by subtracting 1 from deal 1, it would be appropriate to use the same N for both limits. It would also be straightforwardly false that, no matter what you see in the first envelope, you would prefer it to the second envelope; if you saw 8 in the first envelope, you would know that the second envelope contains 7, and so on for every other number. That's more or less the case Brian discussed here; here's my variant of it. Sorry if this seems obvious--just wanted to make clear that the independence of the numbers in the envelopes is necessary for the appearance of paradox.

By: Tricky infinites

Tricky infinites — Fri, 21 May 2004 03:24:22 +0000

It is a problem with the infinites. The probability of an infinite is not zero but infinitely small. And when Brian encounters an infinite he should take the deal because then the odds for a smaller number are infinitely close to 1. For there to be a paradox there must be a chance of an infinite in the envelope, because if there wasn’t, there would be a maximum, which would imply that there is no paradox. But if there is a possibility for an infinite there is no paradox either, because in that case Brian would take the deal. So unless the paradox is hidden somewhere between finite and infinite there is none.

By: Bill Carone

Bill Carone — Fri, 21 May 2004 02:22:49 +0000

Matt, thanks for the response. “Also, on your account of what we should do, why should we stick with one value of N for both distributions?” I actually was going to harp on that originally: the problem is ill-posed if different limits give different answers. However, I decided that, to keep to the spirit of Brian’s problem, I should keep the same N for both. Here’s why. In Brian’s problem, the two deals are exactly the same except one is one day less than the other. The source of the paradox is preferring the former to the latter. If we start messing around with different Ns and Ms for each deal, that part of the problem is destroyed. So I decided, as a modeling decision, to use one N for both. (Aside: If we were using hyperreals, we would use the same infinite N for both as well, or the problem wouldn’t look paradoxical at all; the two deals would be completely different, not just off by a day). So I think Brian would agree that, if we are to use limits, we should use the same Ns for both. More later, about your favorite function. Don’t distress! :-)

By: Matt Weiner

Matt Weiner — Thu, 20 May 2004 21:14:33 +0000

Bill, do you think that it is impossible to define a mathematical function on an infinite set? One of my favorite functions is the one that, given x, returns: 2 if x = 1 4 if x = 2 6 if x = 3 etc. If I couldn’t calculate using that, I would be very much distressed. Also, on your account of what we should do, why should we stick with one value of N for both distributions? If we cap the original verdict at 1 flip, and then take the limit as the cap on the second verdict goes to infinity, then we get the result that you shouldn’t take the deal. If we cap the original verdict at 2 flips, and then take the limit as the cap on the second goes to infinity, we get the result that you shouldn’t take the deal. In fact, if we repeat this procedure as the cap on the original verdict goes to an infinite number of flips, we always get the result that you shouldn’t take the deal. Of course, we can create the opposite result by exchanging the order of the limits. So I don’t see how your approach guarantees the intuitive result, or any result whatsoever.

By: Bill Carone

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

Brian, We seem to be talking past each other; your new example has exactly the same problem as your others. Let me see if I can spell out the problem more clearly. “If y 0.5 or y = 0 then 2 days in purgatory If 0.25 y 0.5 then 4 days in purgatory If 0.125 y 0.25 then 8 days in purgatory If 0.0625 y 0.125 then 16 days in purgatory etc.” Note the “etc.” This is a typical signal of this type of problem (along with the dreaded “…”). You have created an infinite set in the middle of your calculations; the infinite set here is the set of all possible sentences. Your conclusion is based directly on this infinite set. Any paradox must be checked to see if it disappears when you model this correctly, as a limit. The correct way to handle this is to start with a finite set of N possible sentences (2 days, 4 days, 8 days, … 2-to-the-N days), calculate through all the way to your conclusion, then see what happens as N increases indefinitely (at the end of the problem, not the middle). If you had done so, you would see that your conclusion is false: “We still get the very odd result that if you open the envelope, you should prefer sticking with what you’ve got to taking the angel’s deal, no matter what you see.” Note that for any N, no matter how indefinitely large it gets, this isn’t true. So, there is no “odd result”; it is simply an artifact of your error of directly calculating on an infinite set.

By: dsquared

dsquared — Thu, 20 May 2004 07:09:19 +0000

I still think you’re pushing around a fundamental referential failure into ever cleverer hiding places. I’d now say that y doesn’t name a number, because as x tends to 1 second, the limit of y looks like it can be cut up to give the appropriate measure but can’t. I still think that this version of the paradox relies on cutting up time in the same way the Banach-Tarski paradox cuts up space; abusing a continuity assumption to make us think it has a straightforward physical interpretation.

By: Jason

Jason — Thu, 20 May 2004 02:01:21 +0000

Forget my previous comment, I didn’t read the follow on post clearly enough.

By: Jason

Jason — Thu, 20 May 2004 01:55:38 +0000

Ummm, I am getting confused by all the comments, but I always thought this paradox was a Bayesian’s poke at Classical Statistics. It really begs the question – what do you mean by “random” in this context? You can’t have a uniform distribution over the reals (this is not just because it is “infinite” as so is the interval [0,1], but because its measure is infinite). With any actual distribution and a risk averse decision maker, you get that they shouldn’t switch. Your decay example is one such distribution, but it doesn’t matter which one you used (your one does have the additional nice property you described of never switching even after opening the envelope).

By: Nicholas Weininger

Nicholas Weininger — Thu, 20 May 2004 01:46:53 +0000

To continue matt’s theme of set-theoretic analysis: what happens if the angel is really an unreachable cardinal? “Well, I tried to call him up and tell him I’d decided to keep the envelope, but the secretary says he not only couldn’t be found, he couldn’t even be constructed—guess I’ll have to assume his existence axiomatically…”

By: Bill Carone

Bill Carone — Wed, 19 May 2004 22:48:06 +0000

I must point this out: to produce a decision theory paradox, you need direct computation on infinite sets, the Copenhagen interpretation of QM, angels, and God. I do like your style :-). However, since all of these things are rife with paradoxes of their own, it is hard to “blame” decision theory. Furthermore, all paradoxes disappear with two simple ideas: 1) infinite sets are limits, and 2) probability represents information (or even beliefs). “I’ll defer to (what seems to be) the vast majority of experts who say there is such a thing as quantum chance.” Perhaps some current sources? My understanding is that the Copenhagen interpretation has been on the way out for a while now, but I’m no expert (this is from my quantum physicist friend). In other words, you are assuming that the wave functions are probability functions, and I don’t think that is the majority view any more. To repeat and summarize: 1) You use infinite sets directly and get paradoxes. 2) I model the exact same situation using limits, and don’t get paradoxes. In fact, I get intuitively correct answers. 3)I conclude that the paradoxes are simply artifacts of your infinite sets. Remove them, and you remove the paradoxes.

By: Dave

Dave — Wed, 19 May 2004 21:34:13 +0000

And because HTML is allowed in comments, my expected utility value, , didn’t show up. But you know what I mean.

By: Dave

Dave — Wed, 19 May 2004 21:31:53 +0000

Let’s try to do a little mathematical analysis of the issue. First, the angel’s statement about the time required to do the flips is correct in the sense that arbitrarily many throws can be made in one second (Sum n from 1 to infinity of (1/2)^{n = 1). At least in colloquial speech, that’s what the angel is saying, so I have to problem with the statement.

Now it’s not so much an issue of expected time in purgatory, but rather the expected utility of that stay. We will say we have some function F(t) where F(t) is the utility of staying t days in purgatory before going to heaven. If F(t) = t, then = Sum n from 1 to infinity of -((1/2)*2)}n = -infinity. So clearly we would pick any result from the first envelope, because odds are the second envelope is very, very bad (note that the second envelope is never infinite, but can be arbitrarily large). Now, any function F that decreases linearly or superlinearly (in other words, each day in purgatory is at least as bad as the day before it) is going to make very bad. But if F(t) decreases at a less than linear rate, the expectation might be finite, and then we might pick the second envelope. For example, if F(t) = -Sqrt(t), then = Sum n from 1 to infinity of 2^(n/2) ~= -3.41. Note that in this case, if you got anything more than 16 days on the first try, you might consider switching. If F(t) = -ln(t), then is somewhere between -1.3 and -1.4. In this case, you probably want to switch if you got 4 or more days. In general, I don’t think I’d gamble on the second envelope. My F(t) probably decreases superlinearly (as I would guess does most people’s).