Gambler’s Ruin: A Review

All of us like to gamble, don’t we? We enjoy the thrill of taking risk with a prospect to gain more at the end. Most casinos are profiting huge amount just cashing on the gambling addiction of people, who just curse their fate for losing a shitload of money earned from a day’s hard work. But does it really depend on fate, or is it just a clever ruse to drain all of your pocket money? Let’s have a real hard look into this.

Why Study Gambler’s Ruin? 

Some of you might be wondering: “I don’t go to a casino. I don’t need to bother with this”. Well you are wrong. Since it’s inception, Gambler’s Ruin has found tremendous applications in several fields, including:

  • Credit Risk Managements
  • Growth Path and Survival Chances of a new Firm
  • Stock Market Analysis, and a lot of other areas.

Thus, basically, with appropriate modifications involved, we can apply the concept of Gambler’s Ruin in analyzing and predicting several important aspects of the modern world.

Scenario: Let’s just look at a simple scenario for now. Suppose a gambler walks into casino with n$ and starts to play a game where: Every turn a coin is tossed with a bet of 1$. If it’s Head, then he receives 1$, and if it’s tails, he loses 1$. (Of course the games in the casino are a little more complicated, but the essence is basically the same).

Let R_n denote the total fortune after the n^{th} gamble. The gambler’s objective is to reach a total fortune of N$, without first getting ruined (running out of money). If the gambler succeeds, then the gambler is said to win the game. In any case, the gambler stops playing after winning or getting ruined, whichever happens first.

P_N(n)= Probability that the gambler’s fortune reaches N$ before he goes bankrupt, given his/her initial starting was n$.

I am going to show the derivation of P_N(n) using Martingale Approach.

Case 1: Fair Game:

First consider the case when the probability of winning is 1/2. Though this is fairly the case in any real life scenario, I just want to cover it for the basic introduction of the method.

Let X_t =  Gambler’s fortune at time t.

Then X_t is a Martingale because E[X_{t+1}]=\frac{1}{2}(X_t-1)+\frac{1}{2}(X_t+1). If \tau, is the stopping time, the by Optimal Stopping Theorem, we can write: E[X_{\tau}]=E[X_0], which gives P_N(n)=\frac{n}{N}.

To find the expected number of plays, let Z_t denote the winnings of the gambler at time t, i.e. X_t=X_1+\sum_{i=1}^tZ_i.

Wald’s second equation gives:

E[\sum_{i=1}^\tau Z_t-\tau E[Z_t]]=E[\tau]Var[Z_\tau]

which combined with E[Z_\tau]=0 and Var[Z_\tau]=1, we have :


Case 2: Biased Game:

Let probability of Heads be p \neq \frac{1}{2}. Let \lambda= \frac{1-p}{p}, and f(x)=\lambda^x. Then \left \{ f(X_t) \right \} is a Martingale. By Optimal Stopping Theorem, f(X_\tau)=f(X_1).

Thus we can get P_N(n)=\frac{1-f(n)}{1-f(N)}=\frac{1-\lambda^n}{1-\lambda^N}.

Wald’s First equation is:E[\sum_{t=1}^\tau Z_t]=E[\tau]E[Z_t]

Using: E[Z_t]=\frac{1-\lambda}{1+\lambda}. Thus:                 E[\tau]=\frac{1+\lambda}{1-\lambda}[n-N\frac{1-\lambda^n}{1-\lambda^N}]

Discussion: In most gambling games p \lt \frac{1}{2} . Thus we can see that, with a finite starting amount, the probability of going bankrupt is always much higher than getting to the desired cap of N$. So, you must set the cap efficiently, keeping in mind the amount you are starting with.

Gambler’s Ruin In Three Dimensions:

Here three gamblers with initial fortunes a,b, and c, play a sequence of fair games until one of them is ruined. During the duration of the game, the total amount is fixed, i.e. a+b+c=v. Let’s find the probability that gambler 3 is ruined first.

This type of problem can be solved by modelling this as a Brownian motion in the plane of the equilateral triangle with barycentric coordinates (x, y, z) starting at the initial point, (a, b, c), and seek the probability that the Brownian motion first exits the triangle along the edge z = 0.

We can approach by using a mapping of the equilateral triangle onto a unit circle that maps (a,b,c) into the origin. Then we just need to find the proportion of the image of the edge z=0 on the circumference of the circle. For this, two steps are followed:

  • Step 1:  Map the triangle into the upper half plane
  • Step 2:  Map the upper half plane into the circle.

First part is done by using the inverse map of w=\frac{2}{B(1/2,1/3)}\int_{0}^{z} \frac{dt}{(1-t^2)^{\frac{2}{3}}}. The step 2 is done using the famous Mobius Transform.

Conclusion:  Thus the simple idea presented in the 2 dimensional case can be used in numerous real life applications, including but not limited to Credit Risk Management, Business Growth Analysis, 2 candidate general elections, etc. The extension of the method to n-player gambler ruin problem can be used in real life political election process. This field is providing the motivation behind vast number of publications in the scientific community even after almost 350 years of inception of the problem, when Blaise Pascal, Pierre Fermat and many others first started thinking about the problem.

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