Classic Problems of Probability

Problem 25

Kraitchik's Neckties and Newcomb's Problem (1930, 1960)

Problem. Paul is presented with two envelopes, and is told one contains twice the amount of money than the other. Paul picks one of the envelopes. Before he can open it, he is offered the choice to swap it for the other. Paul decides to swap his envelope based on the following reasoning:

Suppose the envelope I initially chose contains D dollars. This envelope contains the smaller amount with probability 1/2. Then, by swapping, I will get 2D dollars and make a gain of D dollars. On the other hand, the envelope could contain the larger amount, again with probability 1/2. By swapping, this time I will get only D/2 dollars and make a loss of D/2. Hence swapping leads to a net expected gain of (1/2)(D) + (1/2)(-D/2) = D/4, and therefore I should swap.

Briefly state why Paul's reasoning cannot be correct.

Solution. If Paul swaps and applies the same reasoning to the other envelope, then he should swap back to the original envelope. By applying the reasoning again and again, he should therefore keep swapping forever. Therefore, Paul's reasoning cannot be correct.

25.1 Discussion

This little problem is usually attributed to the Belgian mathematician Maurice Kraitchik (1882–1957). It appears as the puzzle “The Paradox of the Neckties” in the 1930 book Les Mathématiques des Jeux ou Recréations Mathématiques¹ (Kraitchik, 1930, p. 253):

B and S each claim to possess the better necktie. They ask Z to arbitrate, the rules of ...