Two players, say A and B, play the following game: Both players have 100 marbles and may put anywhere between 1 and 100 marbles in the box each. This decision is not revealed to the other player. Then, they draw 2 marbles with replacement between trials. If the marble belongs to A, then assuming that A put a marbles in the box, A is paid 100−a monetary units from a third party. Similarly if the marble belongs to B, then assuming B but b marbles in the box, B is paid 100−b monetary units from a third party. Assume both players play optimally. Find the expected total payout of player A.
