## Pyramid of Sleaza

Mankind's love affair with the pyramid has endured for thousands of years.  From the ancient architecture of Egypt to the creepy-yet-fascinating Japanese watermelon industry, the pyramid is a shape that is recognized all over the world.

Yes, this is a real thing.

Our adoration of this geometric figure should not be unconditional, however, as not every association with pyramids is positive.  For example, consider the pyramid scheme, a ploy designed to make a small number of people rich at the expense of a much larger pool of victims.  At least, that's how pyramid schemes are typically described.  But how, exactly, do they work?

Imagine that you receive a letter in the mail with three names on it:

1. Alice

2. Bob

3. Carol

Along with the names are instructions, telling you to (1) send \$1 to Alice, Bob, and Carol, (2) remove the Alice from the list and add your name to the bottom, and (3) send the letter on to three other people:

1. Bob

2. Carol

How will this scheme play out?  Well, in a best-case scenario, each of your recipients will follow the instructions, send you (along with Bob and Carol) \$1, and pass the letter along.  And, if all goes well, each of those people will send you \$1 and pass the letter along again.  Assuming that everyone who gets a letter participates and nobody gets sent the letter twice, what sort of return on your investment can you expect?  Here's a diagram to help answer this question:

Let's say you're the purple dot, and you pass the letter along to the three blue dots.  You'll get \$3 from them (\$1 each), \$9 from the people in the next round (the green dots), and  \$27 from the people in the round after that (the orange dots).  After that, the money will stop flowing, since when the orange dots pass the letter along, your name will get cut from the top of the list.  In other words, the letters that the red dots receive will have the name of an orange dot, a green dot, and a blue dot.

Still, that's not a bad haul: \$3 + \$9 + \$27 = \$39, off of just a \$3 investment!  Anytime you make back 12 times what you spent, a celebration is in order.

But a healthy dose of skepticism is in order, too.  After all, if this is such a great idea, why doesn't everyone do it?  Well, part of the reason has to do with our finite population.  Already we've seen how quickly the number of recipients grows: from 1 to 3 to 9 to 27 to 81.  At each stage, if everyone passes the letter along to someone new, the number of people triples.  But there are only so many fresh sets of eyes to get in on the scheme.  In fact, after only 18 rounds, the letters will have reached 318 ≈ 387 million people, more than the population of the United States!  In other words, at some point we simply run out of new money, and the pyramid will collapse.  Of course, this is true even if we don't assume that everyone participates and the letters always get sent to new people.  Regardless of our assumptions, eventually we won't be able to find enough new investors.

When this pyramid does eventually collapse, who will shoulder the financial burden?  While the people at the top have already profited off of their initial investment, the people at the bottom need new investors to recoup their costs.  So ultimately, the folks at the bottom will hurt the most.

But how many people are we talking about, here?  Well, notice that at each stage the number of new recruits makes up a majority of the total number of participants.  For instance, if three people pass the letter along, the 9 new recruits will make up 9/12 = 75% of the participants.  If those 9 people pass the letter along, the new recruits will make up 27/39 ≈ 69% of the participants.  An so on.  In other words, if you randomly receive a solicitation for this investment opportunity, there's more than a 50% probability you're entering the scheme right before it collapses.

Because of this, it's probably best not to invest in a pyramid scheme.  That, and the fact that they're super illegal.

Teachers: Want to talk about pyramid schemes with your students?  Then check out our latest lesson, Pyramid of Sleaza.