# Simulations Using Binomial Coefficients

In closed loop toy models with a fixed money supply, what are the downsides of calculating probable outcomes with a binomial coefficient?

For example, given a toy economy where trades yield a profit or loss of $t$, and all participants start with wealth $W$ as expressed by:

$f(x_{n+1})=x_n \pm t$

and

$f(x_0)=W$

I would predict the probability of one participant reaching a wealth of $0$ by iteration $q$ to be :

$$P = 1- \prod\limits_{n=1}^{q} {\left(1-\dbinom{n}{\frac{W+tn}{2t}}\frac{(n+1)\bmod 2}{2^n}\right)}$$

Are there better methods?

• You seem to be assuming even odds? I can't remember the details, but you can probably find something useful in a stochastic processes book. There are neat ways to model this as a Markov process. I think you are describing an N-player gambler's ruin, but it seems computationally difficult. – Pburg Nov 19 '14 at 5:12
• I am assuming even odds (though i've also used 'skill'), and have modeled as a Markov Chain, though the above equation is computationally less difficult with a good algorithm for computing the binomial as it cancels out the exponent. Essentially the outcome is always bounded, so it's easy to solve for large $W$ or large numbers of players. – Jason Nichols Nov 19 '14 at 5:18
• Also, this is more generalized than the gambler's ruin, in that the 'economy' can keep chugging along with or without a given member depending on scenario bounds. – Jason Nichols Nov 19 '14 at 5:19
• last comment, I swear, but I was looking for something like subversion.american.edu/aisaac/notes/… with some explanation of how Swan(2006) was different. – Jason Nichols Nov 19 '14 at 5:22