CKDextHavn wrote:
Hey, don’t tell me. I got it!
I never said that. I just said that if you open the envelope and then switch in any case, no matter what you found (AuraSeer’s approach), you wouldn’t have had to peek in the first place because you didn’t let it influence your decision.
I think the two of us are pretty much in agreement. Just look back at the first postings from three month ago.
Yeah, sorry, Holg… time is at a premium, and I didn’t re-read the old posts. And sorry if I misread your recent comment.
Anyway, I think this one is finally put to bed?
I hope you don’t mind if I explain why I posted a few times on this thread (and still am!).
I agree that if there is extra information to be gained by opening the envelope (e.g. you’re on a quiz show called ‘The $64,000 question’ and the envelope contains $64,000!), then the probabilities change somewhat.
I did always specify that I was stating that there was no such background information to be gained.
I then had difficulty following December’s logic e.g.
‘3. If one assumes that the probability is always 50-50, then a mathematical contradiction occurs. The contradiction is precisely that you would both simultaneously be right. That is, you could prove that switching doesn’t help and also that it does help.’
‘“The conclusion that trading envelopes is always optimal is based on the assumption that there is no information obtained by observing the contents of the envelopes. From a Bayesian perspective, the key to a successful analysis is in recognizing the potential information to be gained from the observation.”’
I don’t follow either of the above - am I missing something?
CK, I’d love to put this one to bed, but I have my doubts. If only I were a moderator so I could abuse my power and close this thread!
glee, if you don’t know the probabilities of different envelope contents, that doesn’t mean they’re equal. It may seem to you that .5X and 2X are equally likely, but they don’t have to be. It’s then a problem of incomplete information, and stochastics can’t help you much.
Also, I have doubts whether it would even be possible to contruct a uniform distribution of probabilities over an unlimited (albeit discrete and enumerable) range of values. What would be the expected value of that?!
Dear Holg,
I think you’re being a bit hasty in saying you’d like to close the thread. After all, this is the ‘Straight Dope’ - founded to push back the barriers of ignorance (and a good thing too!). I’m unclear about something and I’d be grateful for help - why cut me off?
OK, first have I understood your last post? Are you saying you can’t realistically construct a truly random distribution using money values?
In any case, I’d still appreciate an explanation of two of December’s statements:
‘If one assumes that the probability is always 50-50, then a mathematical contradiction occurs. The contradiction is precisely that you would both simultaneously be right. That is, you could prove that switching doesn’t help and also that it does help.’
(How do you prove this contradiction?)
'“The conclusion that trading envelopes is always optimal is based on the assumption that there is no information obtained by observing the contents of the envelopes. From a Bayesian perspective, the key to a successful analysis is in recognizing the potential information to be gained from the observation.”
(I don’t get the first sentence - if you assume no information gained, why is switching optimal?)
Thanks for your forbearance - maybe we’ll meet sometime on a thread where I can help you!
Here’s an attempt at a formal description
(and solution) of this problem:
Assume X is uniformly distributed between
0 and M, where M is some deterministic
number.
Assume we have two envelopes: A and B
Assume a fair coin is tossed. If we get
heads, then X is put into envelope A and
2X is put into envelope B.
If we get tails, 2X is put into envelope
A and X is put into envelope B.
Assume we pick up envelope A, open it
and see amount Q in it.
Should we switch or not?
First, let’s assume that we know the
value of M, ie the maximum value for X
In this case, if Q > M, we know that Q=2X
and we should not switch.
On the other hand, if Q <= M, we know that
Q=X and Q=2X are equally likely and thus
must switch.
So, if we know the value of M, it doesn’t
matter how large M becomes:
opening the envelope does give us some
information about X, and we should act
accordingly.
Now, what happens if we do NOT know M ?
Then, of course, we do not know whether
Q > M or Q <= M, which means we cannot
make a formal decision whether to switch or
not.
So, if we don’t know the value of M,
opening the envelope does not give us any
useful information, and does
not enable us to make an informed decision
based on the amount found in envelope A.
Therefore, whether seeing the amount
in the envelope helps us decide to
switch or not, depends on how much
we know about X.
The above analysis (like most of the
discussions on this thread) assumed that
X was random. What would happen if we
assumed X to be deterministic but unknown?
I have heard that in Bayesian analysis,
there is a concept of an “improper prior”,
which basically enables Bayesian analysis
of problems with deterministic quantities.
So, if we take X as being deterministic,
and not as random, maybe an “improper prior”
analysis will tell us what the switching
strategy should be.
Based on the results from the analysis in
this message, I would venture to
guess that if X is deterministic but
unknown, opening the envelope will
not aid us in deciding whether to switch
or not. If anyone on this thread is familiar
with the concept of “improper priors”,
maybe they can hekp answer this.
Sendos
glee, I was kidding about closing the thread! I don’t want to cut you off. But I have to say I can hardly think of any more ways of explaining what I mean.
I think it might be impossible to contruct a random distribution that is (a) uniform and (b) of unlimited range. I can easily think of a uniform one for a limited range ($1 - $n) or of a non-uniform one for an unlimited range (e.g. decreasing exponentially). These would all imply that there is some expected value that a contestant could aim for if he knew or could guess it.
To paraphrase: The conclusion that .5X and 2X are equally likely for the other envelope if you picked the one containing X could only be valid (in my opinion, anyway) with a uniform, unlimited distribution. If that doesn’t exist, the paradox is resolved.
I should make clear that I’m not sure about the non-existence of that distribution. This may just be a lack of imagination on my part. And, of course, this has nothing to do with whether we’re talking of money or anything else.
The contradiction is obvious: One way of calculating shows that you can expect a gain from switching, the other shows you can’t. I think you’re overinterpreting this one.
I’m not quite sure what december means here. My guess is the following: The conclusion that switching is always optimal means that your decision does not really depend on what you found in the envelope, i.e. opening the envelope did not give you any information to base that decision on. I’ve always stated that this can’t be true (and I think december means the same): Given that the distribution of possible values is not uniform and unlimited (see above), seeing the amount in the envelope will always give you a clue as to whether you should switch or not. What may be true, however, is that you (as a contestant) have no idea of the distribution. In that case, you just can’t make an informed decision and switching or not doesn’t matter.
No sweat, that’s what we’re here for. As I said, I was kidding earlier. But I am running out of paraphrases…
Holg,
I’ve got it!
(Thanks)
Quote:
> I think it might be impossible to contruct
> a random distribution that is (a) uniform
> and (b) of unlimited range.
This is precisely what the term “improper
prior” describes. This improper distribution
has been used by statisticians for a while
now. Unfortunately, that’s about all I know
about it.
Maybe it has some usefulness for the envelope
problem.
sendos
Sendos, your conclusion doesn’t follow from your analysis. You say: << In this case, if Q > M, we know that Q=2X and we should not switch. >>
Agreed. Similarly, if there is a minimum value m (for instance, $1), we would have information on the other side: if Q < 2m, then we know that we have opened the low value Q=X and we should switch.
But you go on: << On the other hand, if Q <= M, we know that Q=X and Q=2X are equally likely and thus must switch. >>
That’s an incorrect conclusion. We have opened the envelope, found Q. We do NOT know whether Q = X or Q = 2X; we know they are equally likely outcomes. But why does that mean we must switch? If Q = 2X, then switching lowers our value; if Q = X, then switching raises our value. The expected value is 1.5X (but we don’t know what X is and so we don’t know the value for 1.5X; it could be more than Q or could be less than Q.)
So I don’t see where your conclusion follows from your analysis.
sendos wrote:
It is? Dang, I hadn’t made that connection. I must admit I’m not familiar with the term.
So, if statisticians use it, what does that mean? Is it a useful tool? Or do they use it wrongly? In any case, I assume it’s something you have to be very careful with. (Don’t walk around pointing it at people 
)
I think I have the best solution…
1 envelope (1x) is better than no envelopes (0x). So be happy with whatever you get - assume that the amount in the envelope you took is higher. By switching, you have as much chance to lose as to gain; since you have already gotten money for nothing, why risk it?
“Minds are like parachutes; they work best when open.”
-Lord Thomas Dewar
Holg – You’re right.
There is no probability distribution on a countable infinite set of elements with equal probability on each elements. (Proof: Note that the total probability must be 1. No matter how small the probability on each element, when you sum a sufficient of them, the total probability would exceed 1. If the probability of each element were zero, then the total would be zero.)
Therefore you cannot assume equal probability for all pairs of form X, 2X. As you point out, that disposes of the contradiction.
If you can’t assume that all amounts were equally likely, then what SHOULD you assume?
Your assumption ought to be based on the amount of money you saw as well as any other information you have. In particular, you would know how you had been informed about the money in the other envelope.
There does not seem to be a single clear answer, although some assumptions seem more reasonable than others…
December,
What is a countable infinite set?
Glee –
A countable infinite set is a set like" the positive integers, {1, 2, 3, 4,…}.
An example of an uncountable infinite set is the set of real numbers.
I can’t believe it!! The contradiction… solved? I mean - what do I do now?! No endless discussion anymore, nothing to spend my time at work! I’ll be so… lonely…
I’ll miss you all! (Sob!)