Necktie paradox

From Wikipedia, the free encyclopedia

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 respective 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 the prize.

The first man reasons thusly: 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 seems both men have the advantage in the bet. This is obviously not possible.

The paradox is solved because the men's reasoning is flawed: each is considering his tie to be both the more expensive tie and the less expensive tie at the same time, while it can only be one or the other. The two outcomes of the game each should consider are:

• If I have the least expensive tie, and I make the bet, I win an expensive tie.
• If I have the most expensive tie, and I make the bet, I lose an expensive tie.

Thus each stands to either win or lose an expensive tie, each at 50% probability, so the game has no advantage to either man.