The necktie paradox
Asked at Jane Street
Two colleagues each receive a necktie as a gift and, not knowing the prices, agree to a wager: they will compare the two ties, and whoever is wearing the more expensive tie must give it to the other. Each reasons:
I might lose my tie, worth , or win the other's tie, which is more expensive than mine, so I gain more than . Either way my downside is capped at but my upside exceeds , so the bet favors me.
Both men reach this conclusion. But the bet is a zero-sum swap, so it cannot favor both.
Where exactly is the reasoning wrong?
Show a hint
The symbol (my tie's value) is being used as if it were a fixed number, but its meaning changes depending on which outcome you are in. Is the same in the "I win" branch and the "I lose" branch?
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.