A gambler has $2. She is allowed to play a game of chance four times, and her go
ID: 3073219 • Letter: A
Question
A gambler has $2. She is allowed to play a game of chance four times, and her goal is
to maximize her probability of ending up with at least $6. If the gambler bets dollars
on a play of the game, then with probability 0.40 she wins the game and gains
dollars. With probability 0.60, she loses the game loses dollars. On any play of the
game, the gambler may not be more money than she has available. We assume that
bets of zero dollars (that is, not betting) are allowed. Determine a betting strategy
that will maximize the gambler’s probability of having at least $6 by the end of the
fourth game. Specifically, fill in the optimal bets and resulting outcomes for the
following diagram:
Explanation / Answer
We formulate Gambler's ruin as a stochastic dynamic program.
To derive the functional equation define b[t,s] a bet that attains f[t,s]. At the beginning of game 4
For t<4 the functional equation is
f[t,s]=max 0.4 f[t+1,s+b]+0.6 f[t+1,s-b]
where b ranges in 0,...,s; the aim is to find f[1,2].
Given the functional equation, an optimal betting policy can be obtained via forward recursion or backward recursion.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.