The necktie paradox is a puzzle or paradox within the subjectivistic interpretation of probability theory. It is a variation (and historically, the origin) 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 cheaper necktie. They agree to have a wager over it. They will consult their wives and find out which necktie is 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 as follows: winning and losing are equally likely. If I lose, then I lose the value of my necktie. But if I win, then I win more than the value of my necktie. Therefore the wager is to my advantage. The second man can consider the wager in exactly the same way; thus, paradoxically, it seems both men have the advantage in the bet. This is obviously not possible.

The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the value of my necktie"). If we assume for simplicity that the only possible necktie prices are \$20 and \$30, and that a man has equal chances of having a \$20 or \$30 necktie, then four outcomes (all equally likely) are possible:

Price of 1st man's tie Price of 2nd man's tie 1st man's gain/loss
\$20 \$20 0
\$20 \$30 gain \$30
\$30 \$20 lose \$30
\$30 \$30 0

We see that the first man has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth \$30, and a 25% chance of losing a necktie worth \$30. Turning to the losing and winning scenarios: if the man loses \$30, then it is true that he has lost the value of his necktie; and if he gains \$30, then it is true that he has gained more than the value of his necktie. The win and the loss are equally likely; but what we call the value of his necktie in the losing scenario is the same amount as what we call more than the value of his necktie in the winning scenario. Accordingly, neither man has the advantage in the wager.

In general, what goes wrong is that when the first man is imagining the scenario that his necktie is actually worth less than the other, his beliefs as to its value have to be revised (downwards) relatively to what they are a priori, without such additional information. Yet in the apparently logical reasoning leading him to take the wager, he is behaving as if his necktie is worth the same when it is worth less than the other, as when it is worth more than the other. Of course, the price his wife actually paid for it is fixed, and doesn't change if it is revealed which tie is worth more. The point is that this price, whatever it was, is unknown to him. It is his beliefs about the price which could not be the same if he was given further information as to which tie was worth more. And it is on the basis of his prior beliefs about the prices that he has to make his decision whether or not to accept the wager.

On a technical note, if the prices of the ties could in principle be arbitrarily large, then it is possible to have beliefs about their values, such that learning which was the larger would not cause any change to one's beliefs about the value of one's own tie. However, if one is 100% certain that neither tie can be worth more than, say \$100, then knowing which is worth the most changes one's expected value of both (one goes up, the other goes down).

This paradox is a rephrasing of the simplest case of the two envelopes problem, and the explanation of "what goes wrong" is essentially the same.

## References

• Brown, Aaron C. "Neckties, wallets, and money for nothing." Journal of Recreational Mathematics 27.2 (1995): 116–122.
• Maurice Kraitchik, Mathematical Recreations, George Allen & Unwin, London 1943
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