Necktie paradox

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.

Statement of 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.

Resolution

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 one assumes for simplicity that the only possible necktie prices are $20 and $40, and that a man has equal chances of having a $20 or $40 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 $40 gain $40
$40 $20 lose $40
$40 $40 0

The first man has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $40, and a 25% chance of losing a necktie worth $40. Turning to the losing and winning scenarios: if the man loses $40, then it is true that he has lost the value of his necktie; and if he gains $40, 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.

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.

See also

References

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