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. Blackwell and Ferguson (1968) give an ε-optimal strategy for Player 1 that hides, in each stage, one pebble with a probability that depends on the entire past history. Any strategy that depends just on the clock or on a finite memory is worthless. The long-standing natural open problem has been whether every strategy that depends just on the clock and a finite memory is worthless. We prove that there is such a strategy that is ε-optimal. In fact, we show that just two states of memory are sufficient.
[1]
D. Blackwell,et al.
THE BIG MATCH
,
1968,
Classics in Game Theory.
[2]
S. Sorin.
A First Course on Zero Sum Repeated Games
,
2002
.
[3]
Kristoffer Arnsfelt Hansen,et al.
The Big Match in Small Space - (Extended Abstract)
,
2016,
SAGT.
[4]
Frank Thuijsman,et al.
Repeated Games with Absorbing States
,
2003
.
[5]
L. Shapley,et al.
Stochastic Games*
,
1953,
Proceedings of the National Academy of Sciences.
[6]
Dean Gillette,et al.
9. STOCHASTIC GAMES WITH ZERO STOP PROBABILITIES
,
1958
.