[R] Two envelopes problem
Greg Snow
Greg.Snow at imail.org
Mon Aug 25 23:34:28 CEST 2008
I think it is like the girl problem, it is a matter of what you condition on (condition on at least 1 girl, or on a specific one being a girl).
In your simulations (at least from what I can see), you generate an x from a distribution, randomize the order of x and 2x, then look at the average of env1 and the average of env2, but you never condition on the information you get from env1, so you are answering a different problem.
Try this code:
tmpfun <- function() {
x <- sample( c(5,10), 1 )
x2 <- 2*x
sample( c(x,x2) )
}
out1 <- replicate( 100000, tmpfun() )
mean( out1[2, out1[1,] == 10] )
sum( out1[1,] == 10 )
Here we first generate the data such that one of the envelopes will have 10, then only look at the cases where env1 is 10, the average of env 2 in this case is about 12.5.
Now if we look at choosing env2 first:
mean( out1[1, out1[2,] == 10] )
We still have a mean of switching of about 12.5 (the paradox).
The first envelope gives us information which changes the problem. You are looking at the problem from the start, without the information, not conditioning on that information.
Imagine a situation where you have 4 cards, 2 black and 2 red. You shuffle the cards and randomly draw one out and place it face down in front of you (all backs are identical, probability of choosing each card is 1/4). Sitting at the table are 3 of your friends, you show friend #1 that one of the cards still in your hand (chosen randomly) is black, you show friend #2 that one of the cards in your hand (chosen randomly) is red, friend #3 sees none of your cards and they do not share any information about what they see. Now you ask them to each write down the probability that the face down card is black. Friend #1 will write down 1/3, #2 will write 2/3, and friend #3 will write 1/2. They are all looking at the same face down card, but each has different information to condition on (and if we simulate each friends case, we will find that their probabilities are correct for their knowledge).
The 2 envelope case is a subset of a large problem. Assume that there are n envelopes each with a different value in them (you know nothing about the distribution of the values). You can open an envelope and see what is inside, then either keep that amount (and the game ends), or chose another envelope, but once you choose to open a new envelope you can never go back to a previous one. The goal is to determine a strategy that will maximize your expected winings. The best strategy that I have heard (I saw the proof once, but don't remember the details) is to open about 1/3 of the envelopes (actually I think it was 1/e or maybe 1/pi, but 1/3 is a good approximation to both of those), then continue opening envelopes until the first one that is greater than the maximum of the 1st 1/3 and stop there (or stop at the last envelope if the maximum was in that first 1/3). As far as I know, nobody has come up with a better strategy yet (and some math people may have proven it is the best possible). Without any knowledge you don't know if the first envelope is high or low, but by opening the 1st 1/3 you get information that helps form the rest of the strategy. If n=2 (your 2 envelope case), then 1/3 of 2 rounds off to 1, select the first envelope, then go on to the next (which in this case also happens to be the last). Always switching follows this same strategy.
It is paradoxical, but in real life (and in your simulations) we generally have more information than what is in the puzzle.
--
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
(801) 408-8111
> -----Original Message-----
> From: Mario [mailto:mdosrei at nimr.mrc.ac.uk]
> Sent: Monday, August 25, 2008 2:51 PM
> To: Greg Snow; r-help at r-project.org
> Subject: Re: [R] Two envelopes problem
>
> No, no, no. I have solved the Monty Hall problem and the
> Girl's problem and this is quite different. Imagine this, I
> get the envelope and I open it and it has £A (A=10 or any
> other amount it doesn't matter), a third friend gets the
> other envelope, he opens it, it has £B, now £B could be
> either £2A or £A/2. He doesn't know what I have, he doesn't
> have any additional information. According to your logic, he
> should switch, as he has a 50% chance of having £2B and 50%
> chance of having £B/2. But the same logic applies to me. In
> conclusion, its advantageous for both of us to switch. But
> this is a paradox, if I'm expected to make a profit, then
> surely he's expected to make a loss! This is why this problem
> is so famous. If you look at the last lines of my simulation,
> I get, conditional on the first envelope having had £10, that
> the second envelope has £5 approximatedly 62.6% of the time
> and 37.4% for the second envelope. In fact, it doesn't matter
> what the original distribution of money in the envelopes is,
> conditional on the first having £10, you should exactly see
> 2/3 of the second envelopes having £5 and 1/3 having £20. But
> I'm getting a slight deviation from this ratio, which is
> consistent, and I don't know why.
>
> Cheers,
> Mario.
>
> Greg Snow wrote:
> > You are simulating the answer to a different question.
> >
> > Once you know that one envelope contains 10, then you know
> conditional on that information that either x=10 and the
> other envelope holds 20, or 2*x=10 and the other envelope
> holds 5. With no additional information and assuming random
> choice we can say that there is a 50% chance of each of
> those. A simple simulation (or the math) shows:
> >
> >
> >> tmp <- sample( c(5,20), 100000, replace=TRUE )
> >> mean(tmp)
> >>
> > [1] 12.5123
> >
> > Which is pretty close to the math answer of 12.5.
> >
> > If you have additional information (you believe it unlikely
> that there would be 20 in one of the envelopes, the envelope
> you opened has 15 in it and the other envelope can't have 7.5
> (because you know there are no coins and there is no such
> thing as a .5 bill in the local currency), etc.) then that
> will change the probabilities, but the puzzle says you have
> no additional information.
> >
> > Your friend is correct in that switching is the better strategy.
> >
> > Another similar puzzle that a lot of people get confused over is:
> >
> > "I have 2 children, one of them is a girl, what is the
> probability that the other is also a girl?"
> >
> > Or even the classic Monty Hall problem (which has many
> answers depending on the motivation of Monty).
> >
> > Hope this helps,
> >
> > (p.s., the above children puzzle is how I heard the puzzle,
> I actually have 4 children (but the 1st 2 are girls, so it
> was accurate for me for a while).
> >
> > --
> > Gregory (Greg) L. Snow Ph.D.
> > Statistical Data Center
> > Intermountain Healthcare
> > greg.snow at imail.org
> > (801) 408-8111
> >
> >
> >
> >
> >> -----Original Message-----
> >> From: r-help-bounces at r-project.org
> >> [mailto:r-help-bounces at r-project.org] On Behalf Of Mario
> >> Sent: Monday, August 25, 2008 1:41 PM
> >> To: r-help at r-project.org
> >> Subject: [R] Two envelopes problem
> >>
> >> A friend of mine came to me with the two envelopes
> problem, I hadn't
> >> heard of this problem before and it goes like this:
> >> someone puts an amount `x' in an envelope and an amount `2x'
> >> in another. You choose one envelope randomly, you open it,
> and there
> >> are inside, say £10. Now, should you keep the £10 or swap
> envelopes
> >> and keep whatever is inside the other envelope? I told my
> friend that
> >> swapping is irrelevant since your expected earnings are
> 1.5x whether
> >> you swap or not. He said that you should swap, since if
> you have £10
> >> in your hands, then there's a 50% chance of the other
> envelope having
> >> £20 and 5% chance of it having £5, so your expected earnings are
> >> £12.5 which is more than £10 justifying the swap. I told my friend
> >> that he was talking non-sense. I then proceeded to write a
> simple R
> >> script (below) to simulate random money in the envelopes and it
> >> convinced me that the expected earnings are simply
> >> 1.5 * E(x) where E(x) is the expected value of x, a random
> variable
> >> whose distribution can be set arbitrarily. I later found out that
> >> this is quite an old and well understood problem, so I got
> back to my
> >> friend to explain to him why he was wrong, and then he
> insisted that
> >> in the definition of the problem he specifically said that you
> >> happened to have £10 and no other values, so is still
> better to swap.
> >> I thought that it would be simply to prove in my
> simulation that from
> >> those instances in which £10 happened to be the value seen in the
> >> first envelope, then the expected value in the second
> envelope would
> >> still be £10. I run the simulation and surprisingly, I'm getting a
> >> very slight edge when I swap, contrary to my intuition. I think
> >> something in my code might be wrong. I have attached it below for
> >> whoever wants to play with it. I'd be grateful for any feedback.
> >>
> >> # Envelopes simulation:
> >> #
> >> # There are two envelopes, one has certain amount of money
> `x', and
> >> the other an # amount `r*x', where `r' is a positive constant
> >> (usaully r=2 or r=0.5).
> >> You are
> >> # allowed to choose one of the envelopes and open it.
> After you know
> >> the amount # of money inside the envelope you are given
> two options:
> >> keep the money from # the current envelope or switch envelopes and
> >> keep the money from the second # envelope. What's the best
> strategy?
> >> To switch or not to switch?
> >> #
> >> # Naive explanation: imagine r=2, then you should switch
> since there
> >> is a 50% # chance for the other envelope having 2x and 50% of it
> >> having x/2, then your # expected earnings are E = 0.5*2x +
> 0.5x/2 =
> >> 1.25x, since 1.25x > x you # should switch! But, is this
> explanation
> >> right?
> >> #
> >> # August 2008, Mario dos Reis
> >>
> >> # Function to generate the envelopes and their money # r:
> >> constant, so that x is the amount of money in one envelop
> and r*x is
> >> the
> >> # amount of money in the second envelope
> >> # rdist: a random distribution for the amount x # n: number of
> >> envelope pairs to generate # ...: additional parameters for the
> >> random distribution # The function returns a 2xn matrix containing
> >> the (randomized) pairs # of envelopes generateenv <- function (r,
> >> rdist, n, ...) {
> >> env <- matrix(0, ncol=2, nrow=n)
> >> env[,1] <- rdist(n, ...) # first envelope has `x'
> >> env[,2] <- r*env[,1] # second envelope has `r*x'
> >>
> >> # randomize de envelopes, so we don't know which one from
> >> # the pair has `x' or `r*x'
> >> i <- as.logical(rbinom(n, 1, 0.5))
> >> renv <- env
> >> renv[i,1] <- env[i,2]
> >> renv[i,2] <- env[i,1]
> >>
> >> return(renv) # return the randomized envelopes }
> >>
> >> # example, `x' follows an exponential distribution with
> E(x) = 10 #
> >> we do one million simulations n=1e6) env <- generateenv(r=2, rexp,
> >> n=1e6, rate=1/10)
> >> mean(env[,1]) # you keep the randomly assigned first envelope
> >> mean(env[,2]) # you always switch and keep the second
> >>
> >> # example, `x' follows a gamma distributin, r=0.5 env <-
> >> generateenv(r=.5, rgamma, n=1e6, shape=1, rate=1/20)
> >> mean(env[,1]) # you keep the randomly assigned first envelope
> >> mean(env[,2]) # you always switch and keep the second
> >>
> >> # example, a positive 'normal' distribution # First write your won
> >> function:
> >> rposnorm <- function (n, ...)
> >> {
> >> return(abs(rnorm(n, ...)))
> >> }
> >> env <- generateenv(r=2, rposnorm, n=1e6, mean=20, sd=10)
> >> mean(env[,1]) # you keep the randomly assigned first envelope
> >> mean(env[,2]) # you always switch and keep the second
> >>
> >> # example, exponential approximated as an integer rintexp <-
> >> function(n, ...) return (ceiling(rexp(n, ...))) # we use
> ceiling as
> >> we don't want zeroes env <- generateenv(r=2, rintexp, n=1e6,
> >> rate=1/10)
> >> mean(env[,1]) # you keep the randomly assigned first envelope
> >> mean(env[,2]) # you always switch and keep the second i10 <-
> >> which(env[,1]==10)
> >> mean(env[i10,1]) # Exactly 10
> >> mean(env[i10,2]) # ~ 10.58 - 10.69 after several trials
> >>
> >> ______________________________________________
> >> R-help at r-project.org mailing list
> >> https://stat.ethz.ch/mailman/listinfo/r-help
> >> PLEASE do read the posting guide
> >> http://www.R-project.org/posting-guide.html
> >> and provide commented, minimal, self-contained, reproducible code.
> >>
> >>
> >
> >
>
>
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