[R] Two envelopes problem

[Mario]

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