The paradox of the two envelopes | Field Notes

A lifetime ago – March 11, 2010 – I wrote in this space about a curious paradox I stumbled upon (“A Pair of Paradoxes”). It originated, as far as I know, in a 1953 book by Belgian mathematician Maurice Kraitchik titled Mathematical recreation. Paraphrasing from my column:

Around a coffee, Alice and Bart decide on a bet: each will open their wallet, and the one with the least money will receive the contents of the other’s wallet. Alice explains, “Since we are so well off, I have a 50/50 chance of winning or losing. I could lose what’s in my wallet, but I could gain both what’s in my wallet and what’s in Bart’s wallet. Good bet! ”Bart, of course, reasons in the same way. Since the situation is symmetrical, how come the odds seem to favor both?

I just watched a nice YouTube update of the Paradox, which led me to another video, and another …. It turns out people still have a hard time explaining it. I’ll try to save you a few hours of frenzy by summarizing “The Two-Envelope Paradox”, whose “wallet” roots are pretty clear:

You are shown two identical sealed envelopes, A and B, each containing money. One envelope contains 10 times more than the other. You choose envelope B, which contains X $ – you can open it if you want. Before leaving, you are given the opportunity to change, so you reason as follows: “Envelope A contains either 1/10 X or 10 times X, so clearly it is to my advantage to change. (For example, if envelope B contains $ 100, A contains either $ 10 or $ 100, so if you change, you win $ 900 or lose $ 90.) So you change. And now you have the option to go back. it is to your advantage to change and etc. You will be going back and forth forever.

WTF? How can the two envelopes be the best? I don’t know and probably neither can you. In fact, this paradox (and its variations) has baffled mathematicians and philosophers since the problem was first proposed. (Wikipedia and YouTube have plenty of references and threads, if you’ve got plenty of spare time.) Obviously, given the symmetry of the situation, there is something wrong with the reasoning above, but something thing is elusive, with different commentators focusing on different solutions. The best answer I’ve seen comes from the late philosopher / mathematician / magician / concert pianist Ray Smullyan, who has written some compelling books with titles like, What is the name of this book? and This book doesn’t need a title. Smullyan reaffirmed the problem, which I’ll also paraphrase:

Forget the odds. Let’s say the amounts in the envelopes are $ 10 and $ 100. If you choose the envelope containing $ 10, you will earn $ 90 by redeeming. If you choose the envelope with $ 100, you risk losing $ 90 by trading. Since you win or lose the same amount, there is no benefit to trading. (And no benefit not to trade.)

Me, I would exchange, taking example on the bet of Pascal. 17th century mathematician and theologian Blaise Pascal argued that one should try to believe in God, in case you risk winning more than losing i.e. unbelief could lead to eternal damnation. (You have nothing to lose by believing.) I would swap … just in case. Stupid, I know – the envelopes are the same, so there is no logical reason to do this. But Pascal had no logical reason to believe in Hell either.

So here’s the question: Would you like to trade?

Barry Evans (he / him, is still waiting for an invite to play the game.

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