Gambler’s Ruin Math 394 1 The General Problem Suppose two players play consecutive games, where player #1 as probability p of winning any game, independently of the others, and player #2 has winning probability1−p =q. Suppose also that player#1 starts with a capital of c1, and player #2 with a capital of c2. Each time a player wins one unit In this article, we provide the general expressions for the classic 2-player gambler’s ruin problem, when the decision process relies on the occurrence of m successive and non-overlapping events. The Gambler’s ruin for fair games: We brie y discuss the case of fair game p= q. In that case the equation for the ruin’s probabilities x j simplify to x j+1 2x j + x j 1 which gives the quadratic equation 2 2 + 1 with only one root = 1. So we have only one solution x j = C. To nd a second solution after some head-scratching and we try the 1 Gambler’s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1−p respectively. Let R n denote the total fortune after the nth gamble. The gambler’s objective is to reach a total The bounded one-dimensional space of the gambler's ruin problem. the total noise coming from the m 0 = m , 1 partitions that are not competing directly is 2 = m 0 2 bb. Application - Gambler’s Ruin Problem Example: Consider once more the gambler who in each bet has probability pof winning one unit and probability of q= 1 pof losing one unit. Assuming that individual bets are independent, if the player starts with a fortune of iunits, what is the probability that he will reach a fortune of Nunits before going 1. Gambler’s Ruin Here is the code for the function function r = gambler(n) %The Gambler’s Ruin problem %The function returns a vector r %of length n+1. A gambler starts %with a stake of 0 dollars and tosses %a coin n times, winning one dollar for %each time heads is tosses, and losing %one dollar for tails. r(i) is the amount Thus the gambler’s ruin problem can be viewed as a special case of a first passage time problem: Compute the probability that a Markov chain, initially in state i, hits state j 1 before state j 2. 1 There are three communication classes: C 1 = {0}, C 2 = {1, . . . , N-1}, C 3 = {N}. C 1 and C 3 are recurrent whereas C 2 is transient. 4 the problem into the framework of the gambler’s ruin problem: p(a) = P i(N) where N= a+b, i= b. Thus p(a) = 8 <: 1 (q p)b 1 (q p)a+b;if p6= q b a+b; if p= q= 0:5. (7) Examples 1. Ellen bought a share of stock for $10, and it is believed that the stock price moves (day by day) as a simple random walk with p= 0:55. What is the probability that 1 Gambler’s Ruin Today we’re going to talk about one-dimensional random walks. In particular, we’re go- This problem is a classic example of a problem that involves a one-dimensional ran-dom walk. In such a random walk, there is some value - say the number of dollars we
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