Necktie paradox

The necktie paradox is a puzzle or paradox within the subjectivistic interpretation of probability theory. It is a variation of the two envelope paradox.

Two men are each given a necktie by their wives as a Christmas present. Over drinks they start arguing over who has the more expensive necktie. They agree to have a wager over it. They will consult their wives and find out which necktie is the more expensive. The terms of the bet are that the man with the more expensive necktie has to give it to the other as a consolation prize.

The first man considers this: The probability of me winning or losing is 50:50. If I lose, then I lose the value of my necktie. If I win, then I win more than the value of my necktie. In other words, I can bet x and have a 50% chance of winning more than x. Therefore it is definitely in my interest to make the wager. The second man can consider the wager in exactly the same way; therefore, paradoxically, it is in both men's interest to wager their neckties. This is obviously not possible.

The solution is that there is no 50% chance of winning. If the first man's necktie is almost worthless, he has high probability of winning. If his necktie is very expensive, he has much lower chance of winning. So it is not in his interest to wager, the situation is indifferent.