i haven't looked at your code and I'll try when I have time but, as
you stated, that's an EXTREMELY famous problem that has tried to be
posed
in a bayesian way and all sorts of other things have been done to try
solve it. Note that if you change the utility function so that its
log(X) rather than X
then it is seen that the expected values are the same and you don't get
the 1.5X versus X weirdness. When someone showed me that, I gave up and
just walked away from it by telling myself that it has something to do
with utility and percentage weirdness , sort of like when something in
the store is
marked down 50% and then up 50% but it doesn't get back to the original
price. ( that's not right but we spent like a month talking about the
problem and I got sick of it ).
Also, some have argued that the sample space is ill stated because once
you see X, that doesn't tell you about the chances of other X because of
the infinite sample space and that's not realistic. At the time, I
looked around a lot but couldn't find better answers than those. You're
brining back bad memories !!!!!
On Mon, Aug 25, 2008 at 3:40 PM, Mario wrote:
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@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.
______________________________________________
R-help@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.
