Via Juan at Philosophy617 (who doesn’t think much of the proffered solutions, and probably won’t like this one) I came back to this version of the two-envelope problem put forward by Brian, a bit before I joined CT.
In this case, once you observe that Brian’s angel is giving you faulty theology, it’s easy to show that you should reject his1 mathematics, and his offer. At the end of the problem, the angel says “It’s purgatory,” says the angel, “take all the time you want.” But the whole point of Purgatory is that it’s finite - you purge off your sins one at a time until they’re all paid off. Since we now have a finite problem, the solution is straightforward.
Recall that there are two envelopes, with numbers x and 2x representing remission of time in Purgatory, and that x is greater than 10. If your total time in Purgatory is M, we can assume that a just God is not going to give you more remission than that, so 2x is less than M, and x is less than M/2.
The trick in the problem is the apparent symmetry between the envelopes. If you pick one envelope, getting y, switching envelopes gives you y/2 or 2y with equal probability, which seems like a good bet. So it looks as though the angel can apply a Hell pump to you, with repeated offers to switch, paying a day in Hell each time.
The trick in the angel’s offer in the is that it’s not true, for any given y, that switching gives you even chances of y/2 or 2y. Suppose for example, you draw y greater than M/2. Then it’s certain that you’ve got the 2x envelope and that switching would be bad. Conversely, if you draw, say, 15 days, it’s obvious that you’ve got the x envelope and that switching would be good. Unfortunately, you can’t peek then decide whether to switch. If you could, the angel’s offer would probably be a good one. Since you can’t, and given any fixed distribution for x over the range [10,M/2], it’s easy to check that the expected gain from switching is zero.
It’s easy to extend the argument to allow for the case of a Bayesian soul, with a prior distribution that will be updated once the envelope is opened (of course, it’s too late to anything by then). You can also allow for some kinds of non-Bayesians but not too many, since the angel’s argument implicitly relies on the sure thing principle.
It’s also possible, in at least some cases, to refute the angel’s argument even when time in Purgatory may be infinite. All that’s really needed is a given probability distribution for remission time x with a finite mean.
1 I didn’t think angels were gendered, but the example uses male pronouns, and I’ll follow suit.
Yep if the theology requires that time in purgatory is finite (or at least has a finite expected mean) the argument doesn’t go through.
But why think that? Or, more the point, why let theological orthodoxy get in the way of a good puzzle?!
If you’re prepared to allow infinite expectations all sorts of puzzles arise, many of which are nicely catalogued here.
Envelopes shmenvelopes.
It’s a tenet of financial option pricing that stock prices are logarithmically distributed. Meaning, the price of a stock is as likely to double as it is to halve.
If there is equal likelihood of doubling and halving, and the current price is P, then the expected price in the future is the mean of 2P and P/2 … why, it’s 1.25*P!
Therefore all stocks are expected to rise in value. QED, the 90’s explained.
Thanks for the link, Brian. It was very interesting. The countable agglomerability principle referred to there is, I think, a special case of the sure thing principle I mentioned.
I should be more on top of countable/finite additivity issues than I am, but I think Peter Wakker has dealt with most of them, for example, in this paper
Wakker, Peter P. (1993), “Clarification of some Mathematical Misunderstandings about Savage’s Foundations of Statistics, 1954,” Mathematical Social Sciences 25, 199-202.
This note discusses some mathematical misunderstandings about Savage (1954). It is shown that in his model the probability measure cannot be countably additive, that the set of events must be a sigma-algebra and not just an algebra, that Savage did not characterize all atomless finitely additive probability measures, and that the state space in his model, while infinite, does not have to be uncountable.
and also in this self-described piece of persiflage
Wakker, Peter P. (1993), “Savage’s Axioms Usually Imply Violation of Strict Stochastic Dominance,” The Review of Economic Studies 60, 487-493.
Modern day math is taught using those elaborate stories. So I doubt many will fall for the sillyness of introducing y the wrong way.
Just write it out!
At start:
.5 chance of x
.5 chance of 2x
average 1.5 x
Change envelopes:
.5 chance of 2x
.5 chance of x
average is now 1.5 x
Now mix God, the angel, y and the rest of the story in and get confused again.
So now we know what Purgatory is like. If Heaven’s Angels are so bad at presenting puzzles, then it really sounds like Purgatory! I suppose we can excuse them since Angels would have been theology majors, not mathematics majors. I wonder if the Devil’s helpers are any smarter?
A deeper reading is that smart people deserve to spend less time in Hell than stupid people. This is the underlying reward system that we have in our rule-based society anyway, not racialism, not genderism, but “smart-ism” or “talent-ism”. A philosophy supported by smart talented people who run things, and that stupid people have fewer opportunities and “rights”.
“If you’re prepared to allow infinite expectations all sorts of puzzles arise, many of which are nicely catalogued here.”
I want to challenge you on this.
If you treat infinities correctly, as limits of finite sets, all the puzzles disappear and you get intuitively correct results.
The St. Petersburg in the paper Brian references, for example, is simply a mathematical error; you can’t start by summing an “infinite number” of rows, then switch to summing columns instead. It is like saying that
(23 - 23) + (23 - 23) + … = 0
23 + (-23 + 23) + (-23 + 23) + … = 23
Therefore 0 = 23.
It is exactly the same incorrect reasoning that is used in the paper. If you use limits, these problems don’t occur, and you get the correct, intuitive answer.
The way to solve problems with infinite sets is to solve the entire problem, all the way to the conclusion, using a finite set, then take the limit and see if it is well-behaved.
This is the method Gauss used: “I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without retriction.”
“Or, more the point, why let theological orthodoxy get in the way of a good puzzle?!”
Using only finite sets and well-behaved limits of finite sets is a “mathematical orthodoxy”; it gets in the way of a good puzzle because the puzzles are illusory.
PS Even if you treat infinities using hyperreals, the paradoxes disappear (Leibniz’s method).
For example, take the standard numbered ball and urn example:
1. Put balls labelled 1 through 10 in, take the ball labelled 1 out.
2. Put balls labelled 11 through 20 in, take the ball labelled 2 out.
…
Some will say that, in the limit, there are zero balls in the urn, since each ball is taken out. This is incorrect.
Clearly, if you take the limit, you see that it diverges, as the number of balls increases without limit.
If you use hyperreals, you get the same (correct) answer; if you do this N times, where N is an infinite integer, you still have 9N balls left in the urn. All the finite numbered balls are out, but 9N infinite numbered balls are in. Again, no paradox.
Trickster,
“Just write it out!”
I think your formulation proves too much.
Say you are playing the standard two-envelope game with me, but you know beforehand that I only have $100 to play with.
If you open up one envelope and see $75, you won’t switch, right?
Your formulation seems to apply to my version as well as the original, so there must be something wrong with it. Do you disagree?
I’m with Bill on this. The sensible notion of a limit is one that maintains properties that are true as you approach the limit. For example if f(x)>=g(x) for all x near some limit, you want limf>=limg.
Paradoxes about infinity, beginning with Zeno, are about taking limits the wrong way.
Bill,
Of course I disagree!
From the original problem:
Unfortunately, you can’t peek then decide whether to switch.
In your example, by looking in the envelope and knowing there is only $100 in total, the odds are no longer fixed at 1/2. But the construction remains evident for all x.
The problem in the story is in the wrongfull introduction of y. (And of course in portraying an Angel as a bait and switch con artist)
Note also that in the “three boxes and one prize” problem it is also the added info of knowing the content of one (empty) box that makes you switch.
I’m old enough to think that a short visit to hell would be a good time to catch up with some old friends.
I basically agree with Trickster. A full explanation of the paradox (and why we fall for it) is on my blog.
Do let me know if you see any flaws in my argument there!
This angel that is doing the “con”: how do we know he was not a fallen Angel sent from Hell (Lucifer?), as he is unreliable? It could be a trick to get you to go to Hell.
It reminds me of the old problem of the guilty prisoner told by a judge:
(1) We will execute you in the next 7 days and
(2) you won’t know the night before that you will be executed the next day.
The answer is of course that the judge is possibly a sadist and cannot be not relied upon, but I leave others to work out why we know this!
I would come at this one from measure theory and point out that the move from “the envelope is either y or 2y” to “the envelope is equally likely to be y or 2y” is only valid if y was a random draw from a well-behaved Borel set, which a fixed distribution over [10,M/2] is but the set [10,) isn’t. Same point as Bill and John really.
You can cut up an orange with a finite number of cuts and reassemble it into a sphere the size of Jupiter if you’re prepared to assume that you can define arbitrary sets on a continuous field but you can’t.
You can cut up an orange with a finite number of cuts and reassemble it into a sphere the size of Jupiter if you’re prepared to assume that you can define arbitrary sets on a continuous field but you can’t.
Oh, yes I can. But it doesn’t work with an orange because I can’t cut it fine enough with the knife that it is that I have. Now, Zorn’s lemon, on the other hand…
(runs away and hides)
The best way to visualize it intuitively is to consider an alternate hypothetical where first the Angel gives you an envelope with “X” written on it, and then gives you the opportunity to choose between two SEPARATE envelopes, on of which is guaranteed to be 2X, and the other of which is guaranteed to be X/2.
In this case, it is mathematically worthwhile to make the switch.
The faulty math in the actual problem is in trying to convince you that you are really in this 3 envelope world, when you are really in the 2 envelope world.
“You can cut up an orange with a finite number of cuts and reassemble it into a sphere the size of Jupiter if you’re prepared to assume that you can define arbitrary sets on a continuous field”
Notice that you can’t do the above if you treat your continuous field as a limit of a finite set.
Measure theory is a great tool for dealing with complicated probability calculations, but it suffers from the same nonsense that happens when you start with infinite sets without specifying the limiting procedure used to create them.
“Zorn’s lemon”, jesus. I begin to see how it is that people get hunted down and killed for message board posts :-)
I think that at its deepest level, though, this paradox has a lot in common with the Banach-Tarski one. You’re being encouraged to cut up the real number line in a way which gives you a measure that looks like it ought to work as a probability measure but doesn’t, in the same way in which B-T invites you to divide up a continous 3-space with a measure that looks like it ought to have a volume but doesn’t.
This is a sucker fallacy, but it’s a math geek’s fallacy. You’d never catch the check-out girl at your local grocery store in this one.
Consider the money in the envelope version of the problem, since it eliminates infinities: Either you have x in your hand or you have 2x in your hand. Average return from not switching: 1.5x. If you switch envelopes, you have either x or 2x. Average return from switching: 1.5x. I imagine some stereotyped Ms Trailer Park would come to that conclusion quickly enough: “Now, lemme get this straight. If I open this here envelope, either I get $100 or I get $200. If I open that there envelope, I get either $100 or $200. An’ you wanna know if I wanna change envelopes? You tryin’ t’test me for ESP er sumpthin’?”
““Now, lemme get this straight. If I open this here envelope, either I get $100 or I get $200. If I open that there envelope, I get either $100 or $200.”
That isn’t what we know in the problem, though. If we did, then you would be right.
What we know is that one envelope has twice the amount as the other, and we’ve opened one and seen, say $100. This isn’t the same situation as you describe.
Both the answers “Always switch” and “It doesn’t matter” are wrong. It is more complicated than that.
Define x as the “small” amount in the envelope (so the two envelopes have x and 2x in them). You don’t know what x is, so you assign a probability distribution to describe your information about x.
For example, my distribution for x would be different if I were playing with Bill Gates than if I were playing with my professor.
You now take one envelope and look inside. Define y as the amount you see.
Now, either y=x or y=2x. The probability isn’t necessarily 50%; you need to calculate it using your initial distribution for x.
For example, if you know that I have decided to limit my losses to $100, your probability for x will be zero for any x>$50. If you observe y=$75, then you won’t switch, since seeing the $75 has told you that you have the higher envelope for sure (since I wouldn’t risk $150, the other must have $37.50).
After you see y, you can use standard probability calculations to find the probability that the other envelope has y/2 (call it p1). You should switch only when p1 is less than (2/3).
In practice, here is how it works: if I open the envelope and see $100, I think “Before I saw this, what were my probabilities for x=$50 and for x=$100? If the former was less than twice the latter, I should switch.”
So, depending on your initial distribution for x, you might want to switch or you might not, depending on what you see in your envelope.
Some will say that, in the limit, there are zero balls in the urn, since each ball is taken out. This is incorrect.
I was with you on everything else but you have to be more careful here: you’re using an English formulation which implicitly implies commutativity of cardinality and limits. Or, to look at it another way, you’re talking about a limit without specifiying what your limiting process really is.
For definiteness’ sake let A_i be the set of balls in the urn at the i’th stage, let |X| be the cardinality function, and let A be the “limit” of the A_i, i.e. A = { x | there is an N such that x \in A_i for all i > N }, that being the most natural notion of a “limit” of a countable sequence of sets.
There are then two ways of parsing the statement “in the limit there are zero balls in the urn”:
1) lim_{i \to \infty} |A_i| = 0
2) | lim_{i \to \infty} A_i | = 0
The first translation is incorrect by exactly the argument you gave (either by the standard or non-standard approach). The problem is that the second translation is not incorrect, because lim_{i \to \infty} A_i = A = 0, and I’d argue that your English formulation of the problem could be appropriately translated in either way.
[In fact, my original post was going to be a screed to the effect that #2 was the only way to translate your English statement until I, uh, woke up.]
Usual disclaimers apply, of course, the most important of which is YMMV.
You’re being encouraged to cut up the real number line in a way which gives you a measure that looks like it ought to work as a probability measure but doesn’t, in the same way in which B-T invites you to divide up a continous 3-space with a measure that looks like it ought to have a volume but doesn’t.
IIRC, the whole point of B-T is that it relies on cutting up R^3 into something that isn’t a measure. [My knowledge of group actions and orbifolds is somewhat, well, nonexistent, however, so take that with a grain of salt.] The two paradoxes are thus, IMO, unrelated: the two-envelope problem shows the perils of treating infinite-valued [i.e. extended] measures as if they were finite-valued, while the B-T paradox shows the wackiness that can ensue from the existence of non-measurable sets — or more specifically, from the application of choice to the continuum.
Put another way, if one works in the Solovay model of ZF + DC + “Every set of reals is nice” (IIRC, you get that every set of reals is measurable, has the perfect set property and is Baire) then the B-T paradox disappears, but the two-envelope problems retains its full force in any model of ZF, let alone ZFC.
Anarch, thanks for the response.
“There are then two ways of parsing the statement “in the limit there are zero balls in the urn”:”
I agree; the limit of the sum vs. the sum of the limits.
My understanding of Gauss’s dictum is always to take limits as late as possible. So I instinctively chose to take the limit of the sum.
Is there a reason why I wouldn’t want to follow this advice?
No problem, Bill.
The short answer is that you’d not want to follow Gauss’ dictum when the behavior of the
“infinitary object” isn’t well-approximated by the behavior of the “finitary objects”. This comes up all the time in logic, a lot of times in topology, infrequently in algebra and almost never (that I know of) in analysis — the latter because infinitary gadgets in analysis are almost always defined as their finitary approximations.
[Extensive examples available upon request, but I should warn you that the post I wrote to that effect is, uh, long.]
The other way to answer your question, btw, is “whichever way the problem asks you to do it.” If it’s talking about something inherently infinite (see my response to you in the next thread) then you have to deal with the full object, using finitary approximations only when explicitly licensed by theorems; if it’s talking about the finitary approximations then you have to use those, regardless of the behavior of the infinitary object.
All that should be festooned with IMO and YMMV, as appropriate.
“[Extensive examples available upon request, but I should warn you that the post I wrote to that effect is, uh, long.]”
I certainly would like to hear them, if you have the time and inclination.
“The other way to answer your question, btw, is “whichever way the problem asks you to do it.””
I certainly agree with that.
However, I am finding more and more that, when I take limits as late as possible, paradoxes tend to disappear. Is this something that others have found? Or am I just lucky in my choice of problems?
Anarch,
“If it’s talking about something inherently infinite (see my response to you in the next thread)”
I may be several kinds of idiot, but I can’t seem to find the response referred to here. Could you repost, please?
My current position would be that there is no such thing as something that is inherently infinite except when it can be defined as a limit of a sequence of finite things. I don’t think “infinity” should be allowed in mathematics, except when used as a figure of speech that really means a limit. Gauss agrees, Cantor doesn’t (I think).
For example, I find the infinite set of natural numbers quite useful, but I think of it as a limit of a finite set {1,..N} as N goes to infinity.
I think of real numbers as very useful, but I think of them as limits of rational numbers with finite decimal expansions.
Both of these things are “inherently infinite”, no?
So, it would be great if you could provide examples of infinite objects that cannot be represented as such limits, or that we would grossly misunderstand if we did look at them merely as limits.
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