How to calculate a negative feedback loop?

Question

Perhaps this could best be explained as a closed system between two people:

1) For every $1 person A receives, he will give 50% to person B and keep the rest.

2) For every 1$ person B receives, he will give 25% to person A, and keep the rest.

3) Now, person C hands person A $1. How do I calculate how much money person A and person B will end up with after they keep circularly giving each other a cut of the money they just received to seemingly infinity?

An excel document I made looks like this, where each line is a step in the cycle:

Given to A  | A's Total  | Given to B | B's Total
1             1            0            0
0             0.50         0.50         0.50
0.125         0.625        0            0.375
0             0.5625       0.0625        0.4375

And so on, until after 14 cycles the differences in totals between cycles diminish and we're left with A's total of 0.571429 and B's total of 0.428571

I can solve this problem with an Excel spreadsheet, but I assume there is a formula to for this sort of feedback problem.

One of my biggest problems finding a solution is that I don't know the correct terminology to describe the problem.

Answer 1

EDIT: I misread the question, as was pointed out in the comments. However, formalizing a question like this using recurrence relations is still often a sound strategy, you just have to model it correctly. If I could recommend a general strategy to questions like this, I'd start by writing out the first few terms by hand, try to find a pattern, and realize that pattern in some sequence that you can take the limit of.

I would solve this with recurrence relations.

If person A has been given $ x $ dollars, after one step they have $ \frac{x}{2} $ dollars, as they give half to person B. After the next step, they have $ \frac{x}{2} + \frac{x}{8} $, as person B gives a quarter of what they received to person A. Thus, if we model this with a recurrence relation, we'd have $ a_0 = 1 $, as person A is initially given one dollar, and $ a_{n+1} = \frac{5}{8} a_n $. Then the amount of money person A will have "after" this infinite process will be the limit of this sequence, which is 0. Can you figure out person B's situation from here?

Answer 2

This problem can be solved in a lot of ways but one cool way is through Markov chains. Consider the Markov chain with two states $${A, B}$$ where the probability for each state is the chance that any unit of money is in the possession of person A or B. We can define the transition matrix as

$$ P = \begin{bmatrix} .5 & .5 \\ .25 & .75 \end{bmatrix} $$ where the first row, second column corresponds to the rate money moves from A to B and the and the second row, first column corresponds to the rate money moves from B to A.

The initial money distribution is $$ M_0 = [1, 0] $$ and the limit as P goes to inf is $$ \lim_{n \to \inf} P^n = \begin{bmatrix} 1/3 & 2/3 \\ 1/3 & 2/3 \end{bmatrix} $$

so $$ \lim_{n \to \inf} M_n = M_0 * P_n = [1/3, 2/3] $$

so A ends up with \$0.333 and B ends up with \$0.66.