The Mathematics of a Pyramid Scheme - The Scam Explained

There are many "get rich quick" pyramid schemes on the internet.  Some are illegal "pyramid schemes" (scams), and some are barely legal "business propositions" that bear a striking resemblance to an illegal pyramid scheme.

The Simplest Pyramid Scheme - A Chain Letter Scam
An example of the simplest form of a pyramid scheme would involve Person A sending a letter to 2 other people (Level B) requesting them to send $1 back, and to then send letters out to 2 other people (Level C) requesting a dollar.  So Person A and all people at Level B would theoretically make $10. This type of letter is known as a chain letter and is also often accompanied by superstitious claims that "breaking the chain" will bring bad luck. This is a pyramid scheme since the number of participants increases geometrically as shown below. A chain letter of this type is clearly illegal in the U.S.

A Modified Pyramid Scheme - The 8-Ball Model
In The 8-Ball Model, the person recruiting does not get paid at all until they have recruited 3 levels worth of new members.  Thus Person A at level 1 recruits 2 people at level 2, these 2 recruit 4 at level 3, and these 4 recruit 8 at level 4. When the 8 are recruited, Person A receives the "participation fee" for all 8 people of level 4. If the fee was $1000, then Person A would receive $8000. If 16 people are then recruited at level 5, then the 2 people at level 2 would each receive $8000, as shown below.  The appeal of this system over the simple chain letter is that fewer people need to be recruited, payout is higher, and there is incentive to help those in levels below you succeed. This type of scheme, if it does not deliver a product worth the participation fee, is illegal in the U.S. and is similar to that used by illegal scams called gifting clubs.

What Makes Many Pyramid Schemes Illegal?
An illegal pyramid scheme, by definition, is one that involves a geometric series of new members where "membership fees" do not pay for any product of value equal to the membership fee. So if one paid $1000 to join such a scheme yet received $1000 worth of goods or services, then the scheme would not be illegal  The problem with providing goods or services actually worth $1000 is that the full $1000 can not be given back to the top level "sponsors", but rather only a small percentage equal to the profit margin.  To both satisfy legality requirements and keep the full margin, pyramid schemes will often deliver a product with inflated value that involves virtually no cost.  An example of such a product would be ebooks.  A scheme could "sell" the new member 60 ebooks, each "valued" at $40 and thus provide a total value of $2400 for a reduced price of $1000.

Why Are Pyramid Schemes Such a Bad Thing?
In short, pyramid schemes that do not provide new members with a product that is fully worth the "membership fee" will result in a majority of all members losing their money.  If we look at the 8-Ball Scheme, there are twice as many members in each new level.  The bottom 3 levels always lose their money, and as shown below, this will  be at least 7/8 ≈ 88% of the total of all participants. Also, see this explanation of pyramid schemes.

A New and Popular Scheme - The 2-Up System
In the 2 up System, the "sales income" from the first 2 people you recruit goes to the person that recruited you. The sales income from the 3rd and each subsequent recruit you obtain goes to you, along with the first two "sales" of each of your recruits, as shown below for a system priced at $1000 per sale. This system is very popular because the income levels for the person at the top can grow exponentially. Also, there is tremendous incentive for each of your recruits to pursue that 3rd sale, since this it the one that has potential to create a lot of income.

The Problems With the 2 Up System
The 2-Up system, like the 8-Ball Scheme, involves a geometric series where each new level is three times as large  in number the previous level. If the "product" is not something worth the cost of joining and/or it a product that offers no prospects for repeated sales to the same person, we run out of people willing to buy into this “system” very quickly. So the recruits at the bottom level lose all their money. Mathematically, that will be more than 2/3 of all the people joining as shown below! Note also that the people on the second from the last level only break even, and may end up losing amounts equal to advertising costs associated with bringing in 3 recruits.

The formula used above is the geometric series formula commonly used in calculus.