The question mark paradox

This is how former CEZ CEO Martin Roman and the brilliant Czech politician Václav Klaus met while on vacation in Dubai. After a few bottles of fine straw wine, they start betting on who has the more expensive tie.

Enter

As we already know, after a few bottles of wine, Martin and Václav start betting on who has the more expensive tie. This tie was given to them by their wives for Christmas and they don't even know how much the tie costs. The exact wording of the bet is as follows: they ask their wives for the price, and whoever has the cheaper tie gets the other guy's tie. So if Václav has a cheaper tie than Martin, Václav gets Martin's tie. And vice versa. The first question is, is either of them at an advantage? Is it worth it for either of them to take the bet, or are the odds exactly the same for both?

Pseudo-solution

Maybe so. We will further assume that the price of ties is different. It doesn't change the solution, because if the price were the same, no one would get anyone's tie. Count me in: Wenceslas has a tie that costs X crowns. He knows nothing about Martin's tie, nor does he have any other clues, so the probability that he has the cheaper tie is 50 %, the probability that he has the more expensive tie is also 50 %. So if Wenceslas has the more expensive tie, he loses X crowns. But if he has the cheaper tie, he will gain more than X crowns. Both with probability 50 %: he has 50% chance of losing X crowns and 50% chance of gaining more than X crowns.

Thus, with the same probability, she can win more money (a more expensive tie) than she can lose. And it pays off! It looks like things are going very nicely for Václav, and he's better off taking the bet. He has more to gain than he has to lose.

But, as you can all guess, Martin can make the same calculation. Martin too has 50% a chance of losing his tie worth Y and 50% a chance of winning a tie worth more than Y.

Obviously, it is not possible that, given the same bet, both participants are at an advantage; something must be wrong.

Solution

First, let's break down who can actually win. Let's set any specific tie prices. Assume for simplicity that there are only two ties, priced at 3000 and 4000 crowns. Then, from Vaclav's point of view, the following cases can occur:

• Václav has a tie at 3000 and Martin has a tie at 4000 $\rightarrow$ Václav has obtained 4000.
• Václav has a tie for 4000 and Martin for 3000 $\rightarrow$ Václav has lost 4000.

(That they had ties for the same price we have ruled out, but it would not have been necessary - then neither of them would have won.)

As you can see, the game is always played for the same money, Václav can either gain 4000, or lose 4000. Of course, this is independent of the specific price, if you put the amounts 250 and 350 in there, it will always be played for 350 crowns. So it is not worth anyone's while to go (or not go) into the bet unless they have additional information.

If we had a total of three different ties, say for 2000, 3000 and 4000, we would get a bit more of those possibilities. First Wenceslas wins, then Wenceslas loses:

• Václav got a tie for 2000 and Martin got a tie for 3000 $\rightarrow$ Václav got 3000.
• Václav got a tie for 2000 and Martin got a tie for 4000 $\rightarrow$ Václav got 4000.
• Václav got a tie for 3000 and Martin got a tie for 4000 $\rightarrow$ Václav got 4000.
• Václav got a tie for 3000 and Martin got a tie for 2000 $\rightarrow$ Václav lost 3000.
• Václav got a tie for 4000 and Martin got a tie for 2000 $\rightarrow$ Václav lost 4000.
• Václav has a tie for 4000 and Martin for 3000 $\rightarrow$ Václav has lost 4000.

All possibilities are equally likely. As you can see, Václav can gain exactly the money he can lose at the same time. We would obtain similar tables for even more ties, it would always be true that Václav can lose the money that he can gain at the same time. And since all possibilities are equally likely, neither participant is at a disadvantage.

The main mistake Václav made was that he counted on a fixed price for his tie. He cannot do this because, as can be seen in the example with two ties, the price of his tie in the case of losing was different from the price of his tie in the case of winning. Similarly, in the next example with three ties, when Wenceslas won, the price of his tie was from the set {2000, 3000}, but when he lost, the price was from the set {3000, 4000}. Yet Wenceslas was still counting on a fixed amount.