The Winning Pair
Jane Street
You have a deck of 52 cards, and you keep taking pairs of cards out of the deck. If a pair of cards are both red, then you win that pair; if a pair of cards are both black, then I win that pair; if a pair of cards has one red and one black, then it's discarded. If, after going through the whole deck, you have more pairs than I do, then you win 1 dollar, and if I have more pairs than you or we have equal number of pairs do, I win 1 dollar. What is the value of this game in the long run?
Answer
The asymmetry in this game between you and me is that I win if we get an equal number of pairs. So when does this happen? Well imagine there are n pairs we draw that have one black and one red card - we discard these pairs. The remaining pairs are by definition those with both cards of the same color. After we've discarded n different-color pairs, there are 26-n remaining black cards and 26 remaining red pairs. Now we said the remaining pairs are all single colour, so the 26-n red cards are paired up to form (26-n)/2 pairs and 26-n black cards are paired up to form (26-n)/2 pairs. So we always end up with the same number of all-black and all-red pairs! This means I always win, so value of the game for me is 1 and for you it's 0.