The Wallet Paradox Revisited

Put your wallet on the table next to mine. The game is this: The person whose wallet has less money wins all the money in the other person’s wallet. Do you want to play? You might think along these lines: “I don’t know how much money that other wallet has, and I’m not even sure how much is in mine. If I have more money, then I’ll lose it, but if I have less, I’ll win the larger amount. I have no idea what the odds are, but since I stand to win more money than I can lose, it seems like a good game.” Upon further thought, you realize that both players are probably thinking the same thing! Can both be correct? How can a game favor both players? It can’t! In any two-person, zero-sum game (where one person wins what the other person loses), it is not possible for the game to be advantageous to both players. Believing that the wallet game favors both players is a paradox, one discussed by Martin Gardner [3]. A variation of this game was originally posed by Kraitchik [1] where the person with the greater amount in her wallet gives the difference to the other. What if you and I decide to play this game day after day? We will need to establish a few more rules, because it would not be a very interesting game if neither of us ever carried any money. Since we do not want to mandate a minimum amount that must be carried, we agree that on average (and in the long run) we will carry the same amount of money. How should you decide how much to carry each day? Kraitchik shows that if the amount of money each player carries is uniformly (discretely) distributed between 0 and some large x (he uses the total amount of money that has been minted to date), then the game is fair (each player’s expected payoff is zero). Gardner notes that this does not explain the source of the paradox. Merryfield, Viet, and Watson [2] argue that the source of the apparent paradox is that the players do not take into account the probabilities of winning and losing. They argue that if the amounts of money in the players’ wallets are determined by independent, identically distributed random variables, then the game is also fair. Hence, the game is fair when the players are required to use the same distributions.