Subfair red-and-black with a limit
Copyright policy )Subfair red-and-black with a limit
Suppose a gambler has an initial fortune in (0,1) and wishes to reach 1. It is known that, for a subfair red-and-black casino, the optimal strategy is always to bet min ( f , 1 − f ) \min (f,1 - f) whenever the gambler’s current fortune is f. Furthermore, the gambler should likewise play boldly if there is a house limit z which is the reciprocal of a positive integer; i.e., he should bet min ( f , 1 − f , z ) \min (f,1 - f,z) . We show that if 1 / ( n + 1 ) > z > 1 / n 1/(n + 1) > z > 1/n for some integer n ≧ 3 n \geqq 3 or if z is irrational and 1 3 > z > 1 2 \frac {1}{3} > z > \frac {1}{2} , then bold play is not necessarily optimal.
Decision theory for games
Decision theory for games
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