Abstract:

The Big Match is a multi-stage two-player game. In each stage Player 1 hides one or two pebbles in his hand, and his opponent has to guess that number; Player 1 loses a point if Player 2 is correct, and otherwise he wins a point. As soon as Player 1 hides one pebble, the players cannot change their choices in any future stage.The undiscounted Big Match has been much-studied. Blackwell and Ferguson (1968) give an epsilon-optimalstrategy for Player 1 that hides, in each stage, one pebble with a probability that depends on the entire past history. Any strategy that depends on just the clock or just a finite memory is worthless (i.e., cannot guarantee strictly more than the least reward). The long-standing natural open problem has been whether every strategy that depends on just the clock and a nite memory is worthless.The present paper proves that there is such a strategy that is epsilon-optimal. In fact, we show that just two states of memory are sfficient.