# Simulating the Gambler’s Ruin

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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.

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 . In this example it works out to be .

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 ruin.data = cumsum(x.rnd)+x.cnt if( any( which(ruin.data>=x.cnt+y.cnt) ) | any( which(ruin.data< =0) ) ){ cut.data = 1+min( which(ruin.data>=x.cnt+y.cnt), which(ruin.data< =0) ) ruin.data[cut.data:length(ruin.data)] = 0 } return(ruin.data) } 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 | is.na(ruin.sim), 2, which.max) , nclass=100, col='8', main="Distribution of Number of Turns", xlab="Turn Number") abline(v=mean(apply(ruin.sim==15 | is.na(ruin.sim), 2, which.max)), lwd=3, col='red') abline(v=median(apply(ruin.sim==15 | is.na(ruin.sim), 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')

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