The Mathematics of Cons – Pyramid Selling
Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!
The most basic version of the fraud starts with an individual making the following proposition, “pay me $1000 to join the club, all you then need to do is recruit 6 more people to the club (paying $1000 each) and you will have made a $5000 profit.”
There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM). These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell. However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.
The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:
You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up. Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members. For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money – and these levels are usually filled by those in on the scam.
Ponzi schemes (like that run by Bernie Madoff) use a similar method. A conman takes money from investors promising (say) 10% annual returns. Lots of investors sign up. The conman then is able to use the lump sum investments to pay the 10% annual returns. This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.
This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.
If you like this topic you might also like:
Benford’s Law – Using Maths to Catch Fraudsters – the surprising mathematical law that helps catch criminals.
Amanda Knox and Bad Maths in Courts – when misunderstanding mathematics can have huge consequences .
Essential resources for IB students:
1) Exploration Guides and Paper 3 Resources
I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
IB Maths Resources from Intermathematics 