[David Sklansky]

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Here is a cooler problem, imo: show that the expected value of the # of people who get their hat back is 1, independent of n.

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Hmm, how about n(1/n)=1!
uA

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Obviously. And while I'm mad I just got to this thread so you posted it first, I'm quite happy that boris was nice enough to post a question that perfectly shows why clever amateur will sometimes beat not so clever pros.

Permit me to answer it in a way that everyone will understand.

There is 1000 players in a tournament redrawing for seats. The RIO is paying five thousand dollars to any player who gets his own seat. Each player has a one in a thousand chance of making a thousand dollars. Each player has an EV of $5.

I go around buying up everyone's EV for face value. It cost me five grand. Each purchase is a break even purchase for me. Thus the whole deal is a break even thing for me. Therefore my expected payoff from the Rio (which ranges from zero to 5mil) is the $5000 I paid. If the expected value of my prize is $5000, the expected number of matches is one. And of course this would work for any size tournament.