In my stochastic processes class, Prof Mike Steele assigned a homework problem to calculate the ruin probabilities for playing a game where you with 1 dollar with probability p and lose 1 dollar with probability 1-p. The probability of winning is not specified, so it can be a biased game. Ruin probabilities are defined to be the probability that in a game you win 10 before losing 10, win 25 before losing 25, and win 50 before losing 50, etc. In total, I found three distinct methods to calculate.
This is a particularly great example to illustrate how to solve a problem using three fundamentally different methods: the first is theoretical calculation, second is simulation to obtain asymptotic values, and third is numerical linear algebra (matrix algorithm) which also gives exact values.
Method 1: First Step Analysis and Direct Computation of Ruin Probabilities
Let h(x) be the probability of winning $n before losing stake of x dollars.
First step analysis gives us a system of three equations: h(0) = 0; h(n) = 1; h(x) = p*h(x+1) + (1-p)*h(x-1).
How to solve this system of equations? We need the "one" trick and the telescoping sequence.
The trick is: (p + (1-p)) * h(x) = h(x) = p*h(x+1) + (1-p)*h(x-1) => p*(h(x+1) - h(x)) = (1-p)*(h(x) - h(x-1)) => h(x+1) - h(x) = (1-p)/p * (h(x)-h(x-1))
Denote h(1) - h(0) = c, which is unknown yet, we have a telescoping sequence: h(1) - h(0) = c; h(2) - h(1) = (1-p)/p * c; h(3) - h(2) = ((1-p)/p)^2 * c ... h(n) - h(n-1) = ((1-p)/p)^(n-1) * c.
Now, add up the telescoping sequence and use the initial conditions, we get: 1 = h(n) = c*(1+ ((1-p)/p) + ((1-p)/p)^2 + ... + ((1-p)/p)^(n-1)) => c = (1 - (1-p)/p) / (1 - ((1-p)/p)^N-1). So h(x) = c * (((1-p)/p) ^ x - 1) / ((1-p)/p)-1) = (((1-p)/p) ^ x - 1) / (((1-p)/p)^N - 1)
Method 2: Monte Carlo Simulation of Ruin Probabilities
The idea is to simulate sample paths from initial stake of x dollars and stop when it either hits 0 or targeted wealth of n.
We can specify the number of trials and get the percentage of trials which eventually hit 0 and which eventuallyhit n. This is important - in fact, I think the essence of Monte Carlo method is to have a huge number of trials to maintain accuracy, and to get a percentage of the number of successful trials in the total number of trials.
In each step of a trial, we need a Bernoulli random variable (as in a coin flip) to increment x by 1 with probability p and -1 with probability 1-p.
In Python this becomes:
from numpy import random import numpy as np def MC(x,a,p): end_wealth = a init_wealth = x list = [] for k in range(0, 1000000): while x!= end_wealth and x!= 0: if np.random.binomial(1,p,1) == 1: x += 1 else: x -= 1 if x == a: list.append(1) else: list.append(0) x = init_wealth print float(sum(list))/len(list) MC(10,20,0.4932) MC(25,50,0.4932) MC(50,100,0.4932)
You can see the result of this simulation by plugging in p = 0.4932 = (18/37)*.5 + .5*.5 = 0.4932, which is the probability of winning the European Roulette with prisoner's rule. As the number of trials get bigger and bigger, the result gets closer and closer to the theoretical value calculated under Method 1.
Method 3: Tridiagonal System
According to wiki, a tridiagonal system has the form of a_i * x_i-1 + b_i * x_i + c_i * x_i+1 = d_i where i's are indices.
It is clear that the ruin problem exactly satisfies this form, i.e. h(x) := probability of winning n starting from i, h(x) = (1-p)*h(x-1) + p*h(x+1) => -(1-p)*h(x-1) + h(x) -p*h(x+1) = 0, h(0) = 0, h(n) = 1.
And therefore, for the tridiagonal matrix, the main diagonal consists of 1's, and the upper diagonal consists of -(1-p)'s, and the lower diagonal consists of -p's.
In Python this becomes:
import numpy as np from scipy import sparse from scipy.sparse.linalg import spsolve n = 100 p = 0.4932 q = 1-p d_main = np.ones(n+1) d_super = -p * d_main d_super[1] = 0 d_sub = -q * d_main d_sub[n-1] = 0 data = [d_sub, d_main, d_super] print data A = sparse.spdiags(data, [-1,0,1], n+1, n+1, format='csc') b = np.zeros(n+1) b[n] = 1 x = spsolve(A, b) print x