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

<p>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:</p>

<p>$f(x_{n+1})=x_n \pm t$</p>

<p>and</p>

<p>$f(x_0)=W$</p>

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

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

<p>Are there better methods?</p>

CC BY-SA 3.0
2014-11-19T05:12:00.347Z
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](http://en.wikipedia.org/wiki/Gambler's_ruin#N-player_ruin_problem), but it seems computationally difficult.
Pburg
CC BY-SA 3.0
2014-11-19T05:18:26.583Z
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
CC BY-SA 3.0
2014-11-19T05:19:37.810Z
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
CC BY-SA 3.0
2014-11-19T05:22:41.667Z
last comment, I swear, but I was looking for something like https://subversion.american.edu/aisaac/notes/gamblers_ruin.xhtml#swan-bruss-2006-japplprobab with some explanation of how Swan(2006) was different.
Jason Nichols
<mathematical-economics><simulations><algorithms>
Simulations Using Binomial Coefficients
Jason Nichols
# Simulations Using Binomial Coefficients

(Asked by: username_0 on 2014-11-19)

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?

**Tags:** mathematical-economics, simulations, algorithms

> **Comments:**
> **username_1**: 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](http://en.wikipedia.org/wiki/Gambler's_ruin#N-player_ruin_problem), but it seems computationally difficult.  
> **username_0**: 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.  
> **username_0**: 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.  
> **username_0**: last comment, I swear, but I was looking for something like https://subversion.american.edu/aisaac/notes/gamblers_ruin.xhtml#swan-bruss-2006-japplprobab with some explanation of how Swan(2006) was different.  

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## 0 answers

