probabilistic dynamic programming

[mathsuser]

A gambler has 2 dollars. He is allowed to play a game four times, and his goal is to maximize his probability of ending with at least 6 dollars. If the gambler bets b dollars then, with probability 0.4, he wins the game and increases his capital position by b dollars; with probability 0.6, he loses the game and decreases his capital by b dollars. On any play of game gambler may not bet more than he has available. Determine the betting strategy that would maximize the gamblers probability of attaining at least 6 dollars by the end of fourth game. Assume that bets of zero dollars (not betting) are permissible.

By using probabilistic dynamic programming solve this.

Hint: Let f_t(d) be the probability by the end of game 4, the gambler will have at least 6 dollars, where d is the amount that he has at the beginning of game t which he can use for betting, and let b_t(d) be the amount that he bets to attain f_t(d).

max{0.4*f__{t+1}(d+b) + 0.6*f__{t+1}(d−b)}

But I don't understand how I can make a recursive relationship between f_t(d) and f_t+1 and solve this problem using hint.

 

[mathsuser]

A gambler has 2 dollars. He is allowed to play a game four times and
his goal is to maximize his probability of ending with at least 6
dollars . If the gambler bets b dollars then with probability 0.4
he wins the game and increases his capital position by b dollars;
with probability 0.6 he loses the game and decreases his capital by
b dollars. On any play of game gambler may not bet more than he has
available. Determine the betting strategy that would maximize the
gamblers probability of attaining at least 6 dollars by the end of
fourth game.Assume that bets of zero dollars(not betting) are
permissible.


By using probabilistic dynamic programming solve this.


Hint: let f_t(d) be the probability by the end of game 4, the gambler will have at least 6 dollars. $d$ is the amount that he has at the beginning of game t which he can use for betting and let b_t(d) be the amount that he bets to attain f_t(d).
max{0.4*f_{t+1}(d+b)+0.6*f_{t+1}(d−b)}


But I don't understand how I can make a recursive relationship between f_t(d)$ and f_t+1 and solve this problem using hint .

 

[Ishuda]

A gambler has 2 dollars. He is allowed to play a game four times and
his goal is to maximize his probability of ending with at least 6
dollars . If the gambler bets b dollars then with probability 0.4
he wins the game and increases his capital position by b dollars;
with probability 0.6 he loses the game and decreases his capital by
b dollars. On any play of game gambler may not bet more than he has
available. Determine the betting strategy that would maximize the
gamblers probability of attaining at least 6 dollars by the end of
fourth game.Assume that bets of zero dollars(not betting) are
permissible.


By using probabilistic dynamic programming solve this.


Hint: let f_t(d) be the probability by the end of game 4, the gambler will have at least 6 dollars. $d$ is the amount that he has at the beginning of game t which he can use for betting and let b_t(d) be the amount that he bets to attain f_t(d).
max{0.4*f_{t+1}(d+b)+0.6*f_{t+1}(d−b)}


But I don't understand how I can make a recursive relationship between f_t(d)$ and f_t+1 and solve this problem using hint .

Well, it's obvious that to maximize the expected amount of money at the end of the betting sequence, he should bet $0 each time. But given that you are going to bet, I would suspect something like a bet p*A where A is the amount remaining and p is a number between 0 (inclusive) and 1 (exclusive). Of course the simplest case is p is constant [=0.5?, related to the golden ratio? ???] for the four bets unless A gets to or exceeds $6 at which time p becomes 0.