Necktie paradox

Oliver and Harry meet on holiday in Dubai. After a few bottles of good straw wine they start to bet on which of them has the more expensive tie.

The problem

As we already know, after a few bottles of wine Oliver and Harry start to bet on which of them has the more expensive tie. They each got their tie for Christmas from their partners, and neither of them knows how much it cost. The exact wording of the bet is as follows: they ask their partners about the price, and whoever has the cheaper tie wins the other man’s tie. So if Harry has a cheaper tie than Oliver, Harry wins Oliver’s tie. And the other way round. The first question is whether either of them has an advantage. Is the bet worth taking for either of them, or are the chances exactly equal for both?

The pseudo-solution

Maybe it is. We will assume from now on that the ties have different prices. This changes nothing in the solution, because if the prices were the same, nobody would get anybody’s tie. Work it out with me: Harry has a tie that costs £X. He knows nothing about Oliver’s tie and has no other clues, so the probability that Oliver has the cheaper tie is 50%, and the probability that he has the more expensive tie is also 50%. So if Harry has the more expensive tie, he loses £X. But if he has the cheaper tie, he wins more than £X. Each has probability 50%: he has a 50% chance of losing £X and a 50% chance of winning more than £X.

With equal probabilities, he can win more money (the more expensive tie) than he can lose. And that is worth taking! It looks as if everything is going very nicely for Harry and that he should go for the bet. He can gain more than he can lose.

But, as you have probably guessed, Oliver can make exactly the same argument. Oliver also has a 50% chance of losing his tie, worth £Y, and a 50% chance of winning a tie worth more than £Y.

Obviously it cannot be that both players have an advantage in the same bet; something must be wrong.

The solution

First we write out what each of them can actually win. We substitute some specific tie prices. For simplicity, suppose there are only two ties, priced at £30 and £40. Then, from Harry’s point of view, these cases can occur:

• Harry has a tie worth £30 and Oliver one worth £40 → Harry wins £40.
• Harry has a tie worth £40 and Oliver one worth £30 → Harry loses £40.

(We ruled out the ties having the same price, but it would not be necessary – then neither of them would win anything.)

As you can see, the stake is always the same amount: Harry can either win £40 or lose £40. Of course this does not depend on the specific price. If you substitute the amounts £2.50 and £3.50, the stake will always be £3.50. So it is not worth anybody’s while to take the bet (or to refuse it) unless they have extra information.

If we had three different ties, say worth £20, £30 and £40, we would get a few more possibilities. First Harry’s wins, then Harry’s losses:

• Harry has a tie worth £20 and Oliver one worth £30 → Harry wins £30.
• Harry has a tie worth £20 and Oliver one worth £40 → Harry wins £40.
• Harry has a tie worth £30 and Oliver one worth £40 → Harry wins £40.
• Harry has a tie worth £30 and Oliver one worth £20 → Harry loses £30.
• Harry has a tie worth £40 and Oliver one worth £20 → Harry loses £40.
• Harry has a tie worth £40 and Oliver one worth £30 → Harry loses £40.

All the possibilities are equally probable. As you can see, Harry can win exactly the amounts of money that he can lose. We would get similar tables for even more ties, and it would always be true that Harry can lose the amounts that he can also win. And since all the possibilities are equally probable, neither player is at a disadvantage.

The main mistake Harry made was to count on a fixed price for his own tie. He cannot do that because, as you can see in the example with two ties, the price of his tie when he lost was different from the price of his tie when he won. Similarly, in the next example with three ties – when Harry won, the price of his tie was from the set {20, 30}, but when he lost, the price was from the set {30, 40}. Yet Harry kept counting on a fixed amount.