Many people have heard about the
Monty Hall problem. A similar (but less well known and more mathematically interesting) problem is the
two envelopes problem, which Wikipedia describes as follows:
“You
are given two indistinguishable envelopes, each containing money, one
contains twice as much as the other. You may pick one envelope and keep
the money it contains. Having chosen an envelope at will, but before
inspecting it, you are given the chance to switch envelopes. Should you
switch?”
The problem has been around in various forms since 1953 and has been extensively discussed (see, for example
Gerville-Réache for a comprehensive analysis and set of references) although I was not aware of this until recently.
We actually gave this problem (using boxes instead of envelopes) as an exercise in the supplementary
material for
our Book, after
Prof John Barrow of University of
Cambridge first alerted us to it. The ‘standard solution’ (as in the
Monty Hall problem) says that you should always switch. This is based on
the following argument:
If the envelope
you choose contains $100 then there is an evens chance the other
envelope contains $50 and an evens chance it contains $200. If you do
not switch you have won $100. If you do switch you are just as likely to
decrease the amount you win as increase it. However, if you win the
amount increases by $100 and if you lose it only decreases by $50. So
your expected gain is positive (rather than neutral). Formally, if the
envelope contains S then the expected amount in the other envelope is
5/4 times X (i.e. 25% more).
In fact (as pointed out
by a reader Hugh Panton), the problem with the above argument is that it
equally applies to the ‘other envelope’ thereby suggesting we have a
genuine paradox. In fact, it turns out that the above argument only
really works if you actually open the first envelope (which was
explicitly not allowed in the problem statement) and discover it
contains S. As Gerville-Réache shows, if the first envelope is not
opened, the only probabilistic reasoning that does not use supplementary
information leads to estimating expectations as infinite amounts of
each envelope. Bayesian reasoning can be used to show that there is no
benefit in switching, but that is not what I want to describe here.
What
I found interesting is that I could not find - in any of the
discussions about the problem - a solution for the case where we assume
there is a
finite maximum prize, even if we allow that
maximum to be as large as we like. With this assumption it turns out
that we can prove (without dispute) that there is no benefit to be
gained if you stick or switch. See this short paper for the details:
Fenton N E, "Revisiting a Classic Probability Puzzle: the Two Envelopes Problem" 2018, DOI10.13140/RG.2.2.24641.04960
