The Necktie Paradox

Tom and Michael are both given a cheap necktie by their respective wives for Christmas. At the office Christmas party they start arguing over which (who) is the cheapest. Having had a few drinks, they agree to have a silly bet. They will ask their wives how much their neckties cost. The guy with the more expensive necktie has to give it to the other as the prize.

Tom eagerly accepts the bet. He reasons that 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 this bet is ultimately to my advantage!”
Michael is eager to accept the bet as well... since he reasons in exactly the same way.

Either guy's reasoning seems sound, yet they cannot both have the advantage in the bet! Where is the fault?

Details: For a fair bet you'd expect both guys to have a 50% chance of winning it (100% together). In the extreme case where one guy would have a 100% chance of winning, it can only follow that the other guy has 0% chance, i.e. no chance at all. So that's why they can't both have the advantage (= a chance bigger than 50%).

Solution

Worry not, this paradox can be resolved. Let's make things a bit simpler and say that the only possible necktie prices are €5 and €10. Then four equally likely outcomes are possible:

• €5 vs. €5
• €5 vs. €10
• €10 vs. €5
• €10 vs. €10

We can ignore the outcomes where the bet ends in a tie (ha!).
Now, if Tom's necktie is the €5 one, then he will win €10. But if Tom's necktie is the €10 one, he will lose €10. The same goes for Michael! They will always win or lose the larger amount. In other words, no one has the advantage at all.

So where did the reasoning go wrong? Well, in their comparison they ignore the fact that their necktie must either be the cheaper or the more expensive one. What Tom and Michael call the ‘value of my necktie’ in the losing scenario, is the same amount as what they call ‘more than the value of my necktie’ in the winning scenario.

After asking their wives, however, they soon found this bet had no winners at all.