The Necktie Paradox

Two players, Alice and Bob, each receive a necktie as a gift. The prices of the two neckties are different positive values. Neither player knows either price. They agree to compare: whoever has the cheaper necktie wins the other's necktie. Alice reasons: 'If my necktie costs $x$, then I either gain a necktie worth more than $x$ or lose a necktie worth $x$. Since each is equally likely, my expected gain from the game is positive.' Bob makes the identical argument. Both conclude the game favors them — a contradiction since this is a zero-sum exchange. (a) Identify the precise flaw in Alice's reasoning. (b) Suppose the two necktie prices are drawn by a third party as $V$ and $2V$ where $V \sim \text{Uniform}(1, 100)$, assigned randomly to Alice and Bob. If Alice sees her necktie has price tag $x$, what is her expected gain from the game as a function of $x$? Show that the unconditional expected gain is zero.