A friend asked comments about some websites he found, which promise to double your money deposit after you’ve joined them. The system works like this: you deposit N dollars to the site, and when new members join into the site, you can expect your payment to be doubled within some days.

The system in question seemed to be somekind of a variation of the snailmail pyramid schemes like chain letters, or a Ponzi scheme (named after Charles K. Ponzi), but in a different disguise. I decided to analyze the system a little bit as an intellectual diversion for that evening.

The following text is slightly edited and revised version of my original
mail to the friend.

In a pyramid scheme, those close to the top win a lot, in practice meaning those who start it. Pyramid schemes do not work well (if at all) for those who join it in a late phase. Also, the ones who join it early just get a small profit out of it. To reiterate, it works well for those who start it but eventually it’ll slow down and die off, because there simply just isn’t enough people in the world to support the growth of the pyramid.

A Ponzi scheme is a bit like that, but the returns to old investors are paid from the money given in by new investors. The system in question was something between a Ponzi scheme and a pyramid scheme: you need more and more new members to satisfy all the members who have joined, and at the same time you have a central controller who takes a percentage of all deposits.

Upon further investigation it was found that the term for such a system is a “straight-line matrix”, or an “elevator scheme”. An example of how the system works shall follow.

Suppose you ask people for x = 1 $, and promise to double that to 2x = 2 $, for each person who joins. Let’s say you take y = 0.5 = 50% of the deposit. The value of 50% is arbitrary, but since 50% is exactly one half, is easier to calculate for the purpose of this example.

Step 1: First person joins, deposits x. He wants 2x. You have yx = 0.5x, the overall pool of money is yx = 0.5x.

Step 2: Second person joins, deposits x. He wants 2x. You have now 2yx =0.5x + 0.5x = x, the pool of money is 2yx.

Step 3: Third person joins, deposits x. He wants 2x. You have now 3yx = 1.5x, the pool of money is 3yx = 1.5x.

Step 4: Fourth person joins, deposits x. He wants 2x. You have now 4yx = 2x, the pool of money is 4yx = 2x. Now you can pay the first person the promised 2x. The pool of money is now empty. You have 3 pending persons who have not received their payment. Total number of persons is four.

Step 5: You need again 2x / y = 2x / 0.5 = 4x money to pay the second person. Since each person pays x each, you will need 4x / x = 4 persons. Thus, four (4) persons need to join, each of who will pay x money, so that you can pay the second person the 2x. Then the money pool is empty again, and you have 2 + 4 = 6 persons pending who have not received their payment. At this point you will have 4yx + 4yx = 8yx = 2x + 2x = 4x. Total number of persons is eight.

Step 6: Again, you need 4 new persons so that you can pay the third person the doubed initial deposit. Then you will have 1 + 4 + 4 = 6 + 3 = 9 persons pending. You will have 3 * 4yx = 12yx = 2x + 2x + 2x = 6x. Total number of persons is 12. And so on.

See the pattern emerge?

Step z: You need 4 new persons to pay the ith person. Then you will have i * 4yx. You will have i * 4 – i = 3i persons pending. Total number of persons who got their money doubled will be i. Total number of persons processed is 4i.

Change the value of y and x appropriately and redo from step 1 if you want to play with the numbers.

So, how to get 1 million $…?

Recruit the queen of sciences and make an equation. 1000000 = i4yx. Remember, x was 1 dollar. y was 0.5=50%. i = 1000000 / 4 * 0.5 * 1 = 1000000 / 2 = 500000. So, to get 1 million $ to yourself, you need to recruit a total of 4i = 2 million people, of which 1.5 million will be waiting to get their money doubled. Then you take the money and *poof* disappear!

As long as you can keep new members coming in and keep a static deposit sum it’ll work for a while as a scam. Why static deposit? Well, if you promise money-doubling it’ll get difficult if someone puts in 10000 $ and the ones following that member just want to double their 1 $. The queue of people grows very long. How many members would you need in that case? That shall be left as an exercise to the reader!

Some further reading:

[1] Wikipedia: Pyramid Scheme. Link
[2] Cockeyed.com presents: Pyramid Schemes. Link.
[3] Wikipedia: Matrix Scheme. Link
[4] Pyramid Schemes, Ponzi Schemes, and Related Frauds. Link