Favorable red and black on the integers with a minimum wager

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)

First Committee Member

Victor Pestien - Committee Chair


We consider a problem where a gambler starts with an integer k dollars and is in a casino where the only game available is to stake an integer s dollars in order to gain s dollars. The probability p, of winning a single play of this game, is greater than ½. The casino requires that a minimum amount m be staked at each play, and the player is allowed to bet any integer amount up to k. The player may play the game repeatedly and has the objective of reaching the integer N before going broke. Our objective is to study optimal strategies for reaching the goal N.It was shown by Dubins and Savage that a similar game, played instead with the unit interval as the state space, has an optimal strategy, "bold play", when p < ½. Here "bold play" means to bet as much as we have or as much as is required to reach the goal. It was also shown by Ross (1974) that with an integer state space "timid play" is optimal when p > ½ and there is no minimum wager.We solve the problem completely for the case where the minimum wager is 2. We show that when N is even it is optimal to bet 3 at 3, 3 at N - 3, 2 at even integers and 2 or 3 at odd integers between 5 and N - 5. Furthermore, we show that there are no other optimal stakes. When N is odd we find that there are two optimal strategies depending on the value of p. We also solve the case where the minimum is 3 and the goal N is a multiple of 3. In cases where the goal N is a multiple of the minimum m, and m is a power of 2, we find optimal wagers at states which are not too close to N or to zero. In cases where N is a multiple of m, and m is any even number, we again find optimal wagers at states which are not too close to N or to zero. Finally we find an optimal strategy for the case where the minimum is 5 and N is a multiple of 5.



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