Sunday, May 02, 2010

Two Envelopes

When I was an undergrad, a friend showed me a version of the following famous paradox.  Suppose somebody puts money into two envelopes, with one envelope getting thrice as much money as the other.  You pick an envelope at random and see that it has $90.  Now you have the option to exchange the $90 for the money in the other envelope.  Should you do it?  Well, not having much information, you figure it's about as likely that the other one has $30 as it does $270 and compute that in expectation you'll get 0.5($30)+0.5($270) = $150 if you swap.  This of course works for any amount x -- you can get about 1.67x in expectation by swapping.  You choose the envelope at random but always want to swap for the money in the other envelope -- this is the paradox! Does it trouble you?

It shouldn't -- because I cheated here.  I never said how the money was placed in the envelopes. For example, because there are infinitely many choices for possible amounts of money you can place in the envelopes, you can't choose from them uniformly at random.  And there are lots of intuitive distributions that won't work; perhaps the whole setting is impossible.

So let's be more careful.  Let's say the person placing the money chooses a positive integer y with probability 2^(-y) (this is an honest distribution) and then places 3^y dollars in one envelope and 3^(y+1) dollars in the other.  Now, when you open an envelope at random and see x dollars, you can figure out the probabilities of there being x/3 and 3x in the other.  If x = 3, you know you'll win by switching because the other envelope must have $9.  Otherwise it's not hard to see that you'll get x/3 with probability 2/3 and 3x with probability 1/3, giving you about 1.22x in expectation for switching -- not exactly the 1.67x, but still a gain.

Now again we get the same paradox: you choose an envelope at random but always win (in expectation) if you switch -- this even works if you don't look at the amount in the envelope!  Does this break probability or only intuition?

A note: various write-ups of this problem exist online, including one on Gowers's weblog from which I borrowed ideas for this entry.

Update (5/9/10): my following post is also about this paradox.


  1. Only intuition I think

    The reason here I think is the multiplicative difference in the payoff for switching. The expected value of the payoff is skewed by the fact that if you switch and win, you increase the payoff by 200%, but if you switch and lose, your payoff is reduced only 67%. This means that in the long run over many trials, you would expect to make more money by always switching than by not.

    If instead of a factor of 3x there were instead an absolute difference of, say, $30, there would be no benefit in the mean from switching.

  2. I think you're right, but it still seems paradoxical. You choose the envelope *randomly* between the two, but gain by switching.

    A new post on this problem coming soon...

  3. Bernard Kirzner, M.D.2:51 PM

    Wrong. Wrong. Wrong.

    I love you and your site, but on this one you are mixing up calculations about the difference between the two envelopes in absolute terms (270-30=240), in relative terms, and on average.

    If I switch every time or if I stay every time, my long term odds stay the same.

    On average over time I will hit somewhere near 30 half the time and 270 somewhere near half the time. On average I will make 270-30=$150...BUT THIS IS SO WHETHER I SWITCH OR STAY. BOTH CHOICES HAVE THE SAME LIKELIHOOD OF 1/3X OR 3X, I.E. 50%.

    The fatal flaw in this mathematics excercise is that figuring out the differences and averages mathematically HAS NOTHING TO DO WITH WHICH ENVELOPE HAS THE BIGGER OR SMALLER AMOUNT OF MONEY IN IT.

    It's a bait and switch, seemingly about the choice of envelopes, but really just a mathematical distraction with mathematical formulae from the equal choices of envelope switching (or holding.)

    Falsifiability means looking at what information would disprove the hypothesis. Having the same odds of succes by staying, disproves the wisdom, really the advantage, of switching. QED

  4. I'm sorry, Bernard Kirzner, M.D, but I do not understand your argument at all -- especially 1) when you say the difference in the amount of money in the envelopes has nothing to do with which has more and 2) when you say "270-30 = $150."

    Also, as I've noted, the $30 and $270 example isn't well defined. You should read to the bottom of my post, or to my next post.

    One last thought -- in my view, doing math formally is never a bait and switch. In fact, when you think about the problem sufficiently formally (see my next post), the paradox resolves.

  5. Anonymous3:04 PM

    You have to identify the random events first.

    I have two envelopes: red and blue and I randomly pick one envelope to put the higher amount in. This is the random event that happens first. (Let's say I picked blue.)

    Secondly, you pick one envelope at random, say, red. This is the second and *final* random event.

    There are *no more random events*. No matter how much you'd like to believe that "the other envelope has X% chance", it does not since money doesn't teleport between envelopes while you keep changing your mind. Everything's been already decided.