The Mathematics Of A Pyramid Scheme.

  As math teachers we spend so much time explaining the basics of mathematics as determined by the curriculum but what if one day, we took a break and explained the mathematics behind various money making schemes such as the pyramid scheme or a Ponzi scheme. 

I've occasionally had students come in with letters or emails from lawyers or countries claiming they were due to inherit a ton of money or they won some sweepstakes they had not entered.  Those are easy to expose but the other money making schemes are harder but they still make a great topic in math classes.

It turns out the pyramid scheme area covers several types of scams including one that is not as common as it used to be.  The chain letter, the one where you'd get a letter from a friend or relative stating if you send this out to x number of other people, you'd get so much money in a certain amount of time.  Each level brings in twice as many as the previous level, if two people send back money.

Another more common one is the modified 8 ball model in which a person does not get paid until they have recruited enough people to have established several levels.  The idea is that you recruit two people, who recruit two more each who then recruit two more people each so there are three layers with 8 people involved.  At this point, their participation fee is given to the first person. So if the fee is $1000, you just made $8000.

Both schemes use a geometric progression to grow but there is a fine line between legal and illegal in that the people who join have to get merchandise equal to their investment. The actual math formula is well stated here but it boils down to about 88% of the people will loose their money because those past a certain level.

A new scheme is the two up model. The sales earnings from the first two people you recruited goes to the person who recruited you.  It isn't until the third person, you start making a profit.  This involves a geometric progression of three times rather than the two of the previous two.  Unfortunately, those at the very bottom usually do not make money.  In fact about 67% of the people involved in this one, do not make any money.

The formula used to calculate the above averages is the geometric series formula from calculus.

Now for the Ponzi Scheme. This is the people are more likely to be familiar with due to some of those shows that focus on con men who earn millions of dollars before getting caught. The most well known is Bernard L. Madoff who conned people out of over $50 million dollars but it was named after Carlo Ponzi the originator of the scheme.  In 1920, he collected almost $10 million from 10,550 people but only paid out about $8 million. 

The idea behind the Ponzi scheme is that you play upon the greed of people by offering extremely high rates of return which cannot be met so you take the money from later investors and give it to the earlier ones. The mathematics is rather complex but shown here in wonderful detail. 

Lets just say several calculus equations are involved in creating a mathematical model of the ponzi schemes.  The author of the paper, indicates that based on the zeros, it is possible to  determine what is going on with the fund.  If there are no positive zeros, the fund has a positive balance and  is solvent.  If there is one positive zero, it means the fund has collapsed and two positive zeros indicates the fund has become negative but will become positive later on with a bailout.

This explanation is a bit easier to follow but still involves a certain amount of calculus.

Let me know what you think.  I'd love to hear.  I find the mathematics of pyramid and Ponzi schemes fascinating.  Tomorrow, its the mathematics of regular pyramids.