Beating a Finite Automaton in the Big Match

We look at the Big Match game, a variation of the repeated Matching Pennies game where if the first player plays tails the game ends with the first player receiving the last round's payoff. We study this game when the second player js implemented as a finite automaton. We show several results including: • If the first player knows the number of states of the second player's automaton then he can achieve the maximum score with a deterministic polynomial-time algorithm. • If a deterministic first player does not know the number of states of the second player then he can not guarantee himself more than the minimum score. • If we allow player one to run in probabilistic polynomial-time then he still cannot achieve the maximum score but he can get arbitrarily close. • In a slight variation of the Big Match, the first player cannot have an even close to dominant strategy. 1 I n t r o d u c t i o n The king has a dilemma. Every year the crowned prince presents him with a sack of gold or a sack of stones. The king always loves to receive the gold but sees the stones as a slap in the face. Before the prince makes the presentation, the king has the option to order the guards to kill the prince. If the guards kill the prince, they will then tear open the sack to reveal the contents. If the sack contains stones then the king has justifiably eliminated the pesky prince. If, however, the sack contained gold then the masses will force the king from his throne for killing a prince bearing wonderful gifts. *Email: [email protected]. URL: http://www.cs.uchicago.edu/-fortnow. Partially supported by NSF grant CCR 92-53582. t Email: [email protected].