Two Envelopes

I was just trying to refresh my knowledge of Bayesian probability theory (don’t ask) and I came across something I hadn’t seen before, the Two Envelopes Problem. I’m a fan of the classic game theory paradoxes, the most commonly known one being the Prisoner’s Dilemma.

The latter, as trite as it might seem after endless repetition, is still really a cornerstone of game theory, and the usual entree into discussions of how game theory can apply to economics and strategic financial decisions. For myself at least it was the first introduction to Nash Equillibrium, a state where no player can unilaterally benefit by changing strategy.

What’s so eye opening when you dive into Nash Equilibrium after a heavy indoctrination in efficient markets theory is the realization that a perfectly understandable system with clear rules, where each player is acting perfectly rationally, can produce an outcome that’s decidedly sub-optimal. It’s a dilemma indeed, which is why cops have been using it for centuries. No matter what strategy the prisoner chooses it’s nearly always optimal to confess, though clearly the optimal strategy for both prisoners is to keep their mouth shut.

Thoughtful and prudent market players acting rationally don’t always produce efficient (or Pareto-Optimal, to use the jargon) results. Interesting. We’ll have to get back to that one.

But back to the envelopes. To summarize, assume an actor is given two identical envelopes, each of them containing a sum of money. One envelope has twice as much as the other. The player selects one envelope and keeps whatever is in it, but as soon as they choose, and before they open the envelope, they are offered the option of switching.

Should they take the offer?

If the envelope in your hand has X amount of money, then the other envelope has either .5X or 2X, with a probability of one half. Now as any self-respecting gambler can, you should run an expected value calculation and thus determine that the value of switching will pay off at .5X half the time and 2X the other half, for an expected value of the switch that works out to 1.25X.

So switching, on average will yield a 25% better return than not switching. Of course once you’ve switched, and you have the other envelope in your hand, you start the problem over from the beginning. And now it makes sense to switch again, for exactly the same reason. And again, and again, and again.

Using math, we’ve proven that switching to the other envelope is always the better choice, no matter which envelope is in your hand. Sounds like the statistician’s version of proving that the grass is always greener.

If you’re waiting for the punch line, there isn’t one. That’s why it’s called a problem.