The two-envelope paradox | Tangente

In 1988, Sandy Zabell, a member of Northwestern University’s Department of Mathematics and Statistics in the United States, presented the exchange paradox (exchange paradox) in his book Symmetry and Its Discontent (see box).

The exchange paradox ------------------------

In the passage below, taken from Symmetry and Its Discontent (Cambridge University Press, 2005), Sandy Zabell sets out this intriguing argument (translated by the author of this article).

"As a final illustration of the seductive nature of symmetry arguments in probability, and as a challenge to the reader, I shall end with a little puzzle, which I shall call the exchange paradox :

A, B and C play the following game. C acts as referee and places an unspecified sum of money x in one envelope and the sum 2x in another. One envelope is then given to A and the other to B.

A opens his envelope and sees that it contains $10. He then reasons as follows: "There is a 50% chance that B’s envelope contains the smaller sum x (which would therefore be $5), and a 50% chance that B’s envelope contains the larger sum 2x (which would therefore be

20) . I f I s w i t c h e n v e l o p e s, m y e x p e c t e d p a y o f f w i l l b e (1 / 2) \times 20). If I switch envelopes, my expected payoff will be (1/2) ×

5 + (1/2) ×

20 =

12.50, which is $2.50 more than my current holdings. I should therefore try to switch envelopes."

When A offers to exchange envelopes, B readily agrees, since B has already reasoned in the same way.

It seems unreasonable that the exchange should benefit both of them, yet it seems difficult to fault either one’s logic. I shall resist the temptation to explain what I consider to be the resolution of the paradox, except to note that the entire problem rests on A’s seemingly innocuous symmetry assumption that B is equally likely to hold the envelope containing the larger or the smaller amount."

In this book, the mathematician offers his solution to the exchange paradox, arguing that the assumption that B is equally likely to hold the envelope containing either the larger or the smaller amount is untenable, and that switching envelopes is not rational. What do you think?

What matters at this stage is that the author concludes that it is unreasonable for the exchange to benefit both players. But that is not the issue... The question is whether it is unreasonable for each player to believe that switching is more advantageous than keeping their envelope, even though both know that the game is not win-win and that whatever one gains, the other will lose...

In 1992, Marilyn vos Savant, who wrote the Ask Marilyn ("Demandez à Marilyn") column for Parade magazine, received this question from a reader: "I am asked to choose between two envelopes and told that one contains twice as much money as the other. I find $100 in my envelope. Should I switch to increase my expected gain?" In this version, the opponent has disappeared and the referee plays a less conspicuous role.