There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved because the work of buying items and getting others to buy items did not appeal to me. However, their justification for being in their industry was interesting: that it is meritocratic. Specifically, they said that people higher up in the recruitment tree did not necessarily make more money than those lower, due to a certain "differential formula" that distributes income earned among the various nodes.

Here is the general setup that I have been exploring recently: Let's assume that there is a rooted tree where a parent-child pair corresponds to the parent having recruited the child into the pyramid. Each node spends a certain amount of money on items that are on sale by the overarching company. For simplicity, we can assume that everyone spends the same amount of money. Then the company gives each person a certain non-zero amount of money back for spending, and here is the important part: each person also gets a certain non-zero amount of money corresponding to the spending of each of their descendants. The latter amount typically is decreasing (or at least non-increasing) as the distance between the receiving node and the descendant node increases.

I explored some constructions using geometric series. For example, supposing there is a tree of height $n-1$ (so the number ascendants of the lowest leaf is $n-1$), let $3^n -1$ be the amount of money that the company is willing to distribute among each person and their ascendants. Using the fact that $$3^n -1 = 2\cdot 3^{n-1}+2\cdot 3^{n-2}+\cdots + 2\cdot 3^2 + 2\cdot 3 + 2,$$ which has $n$ terms on the right, the lowest leaf could keep $2\cdot 3^{n-1}$, send $2\cdot 3^{n-2}$ to its parent, $2\cdot 3^{n-3}$ to the parent of the parent, and so on, until the root receives $2$. For a non-leaf node, the leftover tail end of the geometric series could be assigned to the root without compromising the fact that higher ascendants should not get more money than a lower ascendant from a descendant. I attempted to prove that that the root will always make more money than any non-root node, but it actually is not true. There are constructions where, under this geometric series model, nodes lower in the tree make more money than some its ascendants, including the root in some cases.

**Question:** Has any research been done into such "meritocratic" pyramid schemes? Any references to papers or results would be appreciated. Here are some more specific ideas in which I am interested:

- For the geometric series model or any other functions that distributes income in the general way described, is there a characterization of when a descendant makes more money than the root? How about that there exists an ascendant of this node which makes less money? Or maybe given a node and a specific ascendant, what is a characterization of when the descendant makes more money?
- I suppose one measure of the degree to which a pyramid scheme like this is meritocratic is that, if the number of descendants of each node is explicitly bounded and the height of the tree is known, we could ask for the probability that, in a descendant-ascendant pair, the descendant makes more money than the ascendant. We could also attempt to take the limit of this probability as the bound on the degree of the node goes to infinity, or as the height goes to infinity. Also, instead of just existence, we could weigh each case in the probability according to the number of such descendant-ascendant pairs.
- I am open to results regarding any other notions of merit.
- Another idea is that, instead of the ascendants of a node getting a fixed percentage all together, there is a fixed percentage for each individual ascendant, where the sum of the amounts that the ascendants get in total converges. I have not looked into this much because it would mean that nodes at lower level would have to give up a higher amount of money than upper nodes to ascendants, which would probably discourage people from joining the pyramid in real life.
- It would be interesting to know any general properties of distributing income from descendants to ascendants in a non-increasing non-zero way as one climbs up the tree, such as the amount of inequality in the final income distribution graph.

**Why I am interested:** I sit on the committee for the Problem of the Month at the University of Waterloo CEMC, and I am considering writing a problem based on this setup. Knowing existing results would be helpful.

mathematics of pyramid schemesinto Google turns up many other references. $\endgroup$