(Red and Black; Dubins and Savage, 1976). An individual plays the following game of chance. If he begins the game with s dollars, he may bet any amount j I s dollars on each turn of the game. If he wins, which occurs with probability p, he receives his bet of J dollars plus an additional payoff of j dollars; while, if he loses, he receives 0. His objective is to maximize the probability that his fortune exceeds N. He may play the game repeatedly provided his fortune remains positive. When it reaches 0 he must stop playing
a. Formulate this as a positive bounded model.
b. Find an optimal strategy for playing the game when p 2 0.5.
c. Find a policy which maximizes the expected length of the game.
d. Suppose the only bet available is $1 and p = 0.5. Use the approach of Example 7.2.7 to find an optimal strategy for playing this game.
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