This puzzle, which appeared on the blog associated with the cute webcomic xkcd, would make a good exercise in a game-theory course:
Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.
Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?
A good start is to notice that the possibility that Alice may randomize is a red herring. You need a strategy with a greater than 50% chance of winning against any pair she may choose. If you can do this, it takes care of her randomized strategies for free. (A familiar argument to those who enjoy computing values of zero-sum games.)