16 Sep 2014

A Simplistic Mathematical Model of a Cash Gifting System

Category: MathematicsAdmin @ 1:58 pm


A gifting system, in its purist form, sells no goods or services, but promises to make you money through the people you recruit to sign up.

Below is a simple mathematical model showing how, in principle, such a system cannot go on forever. There will always be people who lose money. On the other hand, there will also be people who make money (so why not you?) The model below is unrealistic because it does not concentrate money in the hands of the few the way real-world gifting systems operate. My model is the best one could ever hope for from a gifting system. It is almost fair. The ones in the real-world skew the rules towards the ones who start them and their very first recruits. I am hoping that by seeing how the best-case model works mathematically, you will have a better idea of why joining one may either lose you money or land you in jail.

See our blog article, “Cash Gifting Systems Exploit Math Illiteracy,” for more comments about real-world gifting programs.

The Model

  1. A single person starts a gifting system.
  2. This simplified system only has single dollar amount that everyone who joins will give. The amount does not matter for our model, so we can pretend it is always $1,000. (Real-world gifting programs confuse the matter by having different “gifting levels” and a set of rules about who can receive what value of gift according to the “levels” at which a person bought in.)
  3. The founding member may have advertising and other costs, but to keep it simple, let’s say he only needs to recruit 2 people in order to make money. At this point, 3 people are now involved, the one who started it and the 2 he recruited.
  4. Again for simplicity, we are going to pretend that no money flows up the chain past the ones currently recruiting, and the only people recruiting are the ones who just joined and haven’t made a profit yet. Once a person makes a profit, he stops recruiting. Obviously, this isn’t realistic. I am deliberately making the model go through the maximum number of possible iterations before we run out of people on the planet who can still join. Note that getting just 1 recruit only reimburses the money the recruiter gave. Profit is gained only when the second person joins.
  5. Our first 2 recruits each get 2 more recruits, for a total of 4 brand-new recruits. At this point (Iteration No. 3 in the table below), a total of 7 people have joined.
  6. The 4 newest members also each need to recruit 2 people, for a total of 8 more people, in order for everyone to make their profit. Now there are a total of 15 members in the system (Iteration No. 4).
  7. Each new person again needs to recruit 2 more people, and so it goes, on and on.
  8. Every person in the world wants to, and is able to, sign up for the system. (I hope it is obvious that in the real world, this won’t happen.)

How many iterations of recruitment until the entire world’s population has joined this club?

Between 32 and 33 (if you started with a single person). This is shown in detail in the table below.

Iteration No. New Recruits Total Members
Signed Up So Far
1 (The founder->) 1 1
2 2 3
3 4 7
4 8 15
5 16 31
6 32 63
7 64 127
8 128 255
9 256 511
10 512 1,023
11 1,024 2,047
12 2,048 4,095
13 4,096 8,191
14 8,192 16,383
15 16,384 32,767
16 32,768 65,535
17 65,536 131,071
18 131,072 262,143
19 262,144 524,287
20 524,288 1,048,575
21 1,048,576 2,097,151
22 2,097,152 4,194,303
23 4,194,304 8,388,607
24 8,388,608 16,777,215
25 16,777,216 33,554,431
26 33,554,432 67,108,863
27 67,108,864 134,217,727
28 134,217,728 268,435,455
29 268,435,456 536,870,911
30 536,870,912 1,073,741,823
31 1,073,741,824 2,147,483,647
32 2,147,483,648 4,294,967,295
33 4,294,967,296 8,589,934,591

Table 1: Iterations of Recruits


Before you finish the 33rd iteration, you have exceeded the world’s population. The US Census Bureau estimates that in 2015, the world’s population will be about 7,253,260,112 (just over 7 billion) people. Note that at the end of Iteration No. 32 in the table above, 4,294,967,295 (just over 4 billion) people have become members of the system, but 2,147,483,648 people (just over 2 billion, or more than ¼ of the population) still need to recruit two new members in order to get their money back and then make their own money.

Here’s the problem: The difference between the 7+ billion total people and the 4+ billion members already signed up is only 2,958,292,817 (almost 3 billion people). That is more than enough for all of the 2,147,483,648 newest recruits to get their money back from the same number of new people, but then there are only 810,809,169 people (not quite 811 million) left world-wide who have not yet given money. That means only 810,809,169 people out of the 2,147,483,648 will actually make a profit, and the other 1,336,674,479 (about 1-1/3 billion) people will only break even. Meanwhile, the 2,958,292,817 (almost 3 billion) people who signed up last are all going to lose every cent of money they give.

The total number of people who could not make any money, some of whom broke even (1,336,674,479) and some of whom lost all they gave (2,958,292,817), is 4,294,967,296 (a little over 4 billion people. If I made a math error along the way, please let me know.)

The best case scenario, if the world’s population happens to be evenly divisible by 2, is that half of the world’s population donates money to the other half. Half of the people are that much richer, and the other half are that much poorer. Since in actuality some people will recruit more than 2 people, there will be even fewer people left for those who don’t, so the new wealth will be concentrated in fewer than half the world’s population, and more than half may lose money (or at best, some will break even but most will lose money).

Note also that many gifting systems have a more complicated structure wherein the recruiter does not get to keep all of the money from his recruits, but is instead supposed to pass some of it up the chain, either as a percentage or because he did not “qualify” to be able to keep the new money yet. This is a wonderful strategy for the people at the top to get rich while the people near the bottom earn little and the people at the very bottom (or “end,” if you don’t want to use a pyramid model), lose all they contribute.

More Questions and Answers

When the entire population of the world has joined, how many people still have not been paid at all?

Based on an estimated world population for 2015 of 7,253,260,112, the total people who have lost money will be 2,958,292,817.

How many people, at minimum, will only be able to break even?

Based on an estimated world population of 7,253,260,112 and our simplistic model, the total people who break even without ever being able to make a profit will be 1,336,674,479.

Someone has suggested that people at the bottom could still make money because people who joined earlier might want to do it all over again. Since they are already members, why would they give money to new recruits now?

Generosity? Pity? Another reason? (Honestly, I have no idea.)

If everyone who made money gave money to people at the bottom of our pyramid so that they would be reimbursed, what would be the average net profit?

$0. Nada. Zilch. If that were to happen, everyone would end up with the money they started with. There would be no profit for anyone.

Closing Remarks

The bottom line is that a gifting system or pyramid scheme, whichever you prefer to call it, is simply a system for redistributing wealth without giving anyone a real good or a real service for the money a person has to “give” or “pay” to others. The perceived value is the opportunity to make money. For some, money will be gained. For others, money will be lost. It is not a system where everyone can make money. The sum of money made and money lost will always be zero. If you are fortunate (or devious) enough to make money using such a system, it is at the expense of someone else. Period.

If you nevertheless choose to use such a system, your best economic (but not moral) bet is to start your own, so that you will be at the top. If you don’t want to go to jail, though, be sure to plan on taking your money and getting out as quickly as possible. On your way out, be sure to take down your website, your ads, and your blog posts. Oh, and get a fake ID before you start and never use your real name.

You, too, can be a corrupt millionaire. You can even avoid paying taxes in the process. What more could you want? (Integrity? Who needs it!)

A Little Math Practice

A parting note about math, since this site is about education: Math is simply a descriptive tool with its own terminology. Like the computer, macramé and most other things, the more we play around with it, the more adept we become with it. So for those of you who want to, let’s play.

We can create a mathematical formula that fits the table above. Notice how the number of new recruits is always double the previous number. The number of new recruits is always a power of 2. Iteration No. 1 is 20, which is 1. Iteration No. 2 is 21, iteration No. 3 is 22, and so on. Thus, the power you raise 2 to is one less than the iteration number. If we set N = new recruits, and t = iteration number, then we can find the number of new recruits at each iteration like this:

N = 2(t-1).

(I used t instead of i because in math, i usually stands for the square root of negative 1.)

The equation covering number of people who have joined in total (3rd column) is only a little more complicated and I won’t get into it here (but you may contact me if you can’t figure it out). Notice how the total number of people who have joined happens to be just one less than the number of new recruits in the next iteration. There’s a hint there.

Tags: , ,

Comments are closed.