Simulating the Gambler’s Ruin

April 14, 2013

(This article was first published on Statistical Research » R, and kindly contributed to R-bloggers)

The gambler’s ruin problem is one where a player has a probability p of winning  and probability q of losing. For example let’s take a skill game where the player x can beat player y with probability 0.6 by getting closer to target. The game play begins with player x being allotted 5 points and player y allotted 10 points. After each round a player’s points either decrease by one or increase by one we can determine the probability that player x will annihilate player y. The player that reaches 15 wins and the player that reach zero is annihilated. There is a wide range of application for this type of problem that goes being gambling.

Number of Time Until Annihilation

This is actually a fairly simple problem to solve on pencil and paper and to determine an exact probability. Without going into too much detail we can determine the probability of annihilation by \frac{1-\left(\frac{q}{p}\right)^i}{1-\left(\frac{q}{p}\right)^N}. In this example it works out to be \frac{1-\left(\frac{.4}{.6}\right)^5}{1-\left(\frac{.4}{.6}\right)^{10}} \approx 0.8703.

But this is a relatively boring approach and coding up an R script makes everything that much better. So here is a simulation of this same problem estimating that same probability plus it provides additional information on the distribution of how many times this game would have to be played.

gen.ruin = function(n, x.cnt, y.cnt, x.p){
x.cnt.c = x.cnt
y.cnt.c = y.cnt
x.rnd = rbinom(n, 1, p=x.p)
x.rnd[x.rnd==0] = -1
y.rnd = x.rnd*-1
x.cum.sum = cumsum(x.rnd)+x.cnt
y.cum.sum = cumsum(y.rnd)+y.cnt = cumsum(x.rnd)+x.cnt

if( any( which(>=x.cnt+y.cnt) ) | any( which(< =0) ) ){ = 1+min( which(>=x.cnt+y.cnt), which(< =0) )[] = 0



n.reps = 10000
ruin.sim = replicate(n.reps, gen.ruin(n=1000, x.cnt=5, y.cnt=10, x.p=.6))
ruin.sim[ruin.sim==0] = NA
hist( apply(ruin.sim==15 |, 2, which.max) , nclass=100, col='8', main="Distribution of Number of Turns",
xlab="Turn Number")
abline(v=mean(apply(ruin.sim==15 |, 2, which.max)), lwd=3, col='red')
abline(v=median(apply(ruin.sim==15 |, 2, which.max)), lwd=3, col='green')
x.annihilation = apply(ruin.sim==15, 2, which.max)
( prob.x.annilate = length(x.annihilation[x.annihilation!=1]) / n.reps )
state.cnt = ruin.sim
state.cnt[state.cnt!=15] = 0
state.cnt[state.cnt==15] = 1
mean.state = apply(ruin.sim, 1, mean, na.rm=T)
plot(mean.state, xlim=c(0,which.max(mean.state)), ylim=c(0,20), ylab="Points", xlab="Number of Plays", pch=16, cex=.5, col='green')
lines(mean.state, col='green')
points(15-mean.state, pch=16, cex=.5, col='blue')
lines(15-mean.state, col='blue')

Gambler's Ruin

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