Imagine that a stranger approaches you and hands you a sealed envelope with money inside. Before you can recover from the surprise, he offers to swap it for another envelope he carries, knowing that the new one might contain either twice as much money or half as much. What should you do? If you ever find yourself in this improbable situation, you will be facing the two-envelope problem, a curious statistical paradox worth knowing.
There are situations for which it is wise to be prepared. Setting aside how unlikely it is that someone would make you an offer, if you ever face a similar dilemma—say, in a game show—you would surely want to get the best possible deal. The two-envelope problem, one of those Machiavellian inventions that mathematicians and philosophers use to torment us, goes like this: we are asked to choose between two envelopes of money, being told that one contains twice as much as the other. Once we have chosen, we are given the option to swap. How should we act to maximize our gain? Is it better to keep the first choice, or is it better to change? That is what we will try to determine.
The Mathematical Argument
Suppose the amount in the envelope we first choose is A. That means the other envelope has a 50% probability of containing twice that amount (2A) and a 50% probability of containing half (A/2). Since both situations are equally likely, the “mathematical expectation” of the amount in the other box is:
0.5*2A + 0.5*A/2 = 1.25A
That is, if we swap envelopes, we obtain a 25% gain. Great, right? But before you run off to change envelopes, you should think a bit. Indeed, the above reasoning can be done exactly the same if you had chosen the other envelope, so maybe changing is not such a good idea after all. But where is the flaw?
A Concrete Example
Let's look at a concrete example. Suppose the chosen envelope contains 1000 euros. That means it is equally likely that the other contains 500 or 2000 euros. Therefore, if I swap the chosen envelope for the other, I either lose 500 or gain 1000. Since what I can gain is greater (twice, in fact) than what I can lose, there is no doubt that it is in my interest to change. But the paradox lies in the fact that the same argument can be applied to the other envelope. Or worse: once I have changed the envelope, I could use this argument again and again to keep swapping indefinitely. How is it possible that in both cases I can gain more than I lose if I swap?
The Flaw and the Resolution
In reality, the flaw occurs when thinking that the amount you will gain, if you win, is greater than the amount you will lose, if you lose. In truth, what you gain or lose is the same. If A is the amount in the first chosen envelope and the other contains either 2A or A/2, we can call B the difference between the two amounts, or equivalently, B is the smaller of the two amounts, or better yet, B = A.
If you win the exchange (swapping an envelope with A euros for one with 2A euros), you will gain A euros. Correct? And if you lose the exchange (swapping an envelope with 2A euros for one that only has A euros), you will lose A euros. This means that the amount you can gain or lose is the same and that there is no advantage in swapping envelopes. Since the probability of finding the larger amount is the same whether you swap or not, the paradox disappears. This means that if someone offers you an envelope with money, you can calmly take it and walk away without waiting for them to offer to swap it for another: the probability of winning or losing in the exchange is the same.
Wikipedia
https://old.neoteo.com/la-paradoja-de-la-eleccion/
