Pyramid schemes are usually explained by the scamster using a chart with a triangle. The person starting the pyramid is on top, with two people below him, 4 below them, and 8 at the bottom. The 8 don't fit side-to-side nearly as well as the upper participants, crowding 8 figures into a line only "4" wide. This is the reason they never show lower levels of the pyramids in these illustrations, the next line would have 16 in a line 5 wide, 32 in a line 6 wide, and 64 in a line 7 wide, a bit too crowded to see in the back rows.
If somebody asks about this, the clever huckster will say that a real pyramid is 3 dimensional and that fits all those extra people. Well, it could, but only to a point. Once you get 7 levels deep, crowding once again explodes. In fact, that's a good word for it, "Explodes."
This is an exponential growth curve from an online biology class. Note how it seems to take off and go almost straight up past a certain point. This is because the numbers in exponential growth do exactly that. The pyramid scheme's base "explodes" outwards quickly, becoming so big that there aren't enough suckers around to feed it.
It's hard to wrap your mind around exponential growth as something intuitive. To us, it appears that something simply appears and explodes in rapid growth. A good example is bread mold. You have a loaf of bread sitting on the counter in the bag for a week or more until one day it's covered with spots of mold. To your senses they suddenly appeared, but in actuality they were growing exponentially the whole time, until enough generations grew that you saw the "explosive" results as visible mold. The same it true of flowers and fresh growth in the springtime, after a spell of warm weather it's "bursting out all over," but actually the result of a growth ramp that was minimally perceivable until that point.
Let's take for example a simple scheme where each participant recruits two new members. Each new member gives $100 to the person who recruits them. If you join and successfully recruit 2 new members, you're $100 ahead on the deal. The numbers of people at each level are:
Level Number
1 1
2 2
3 4
4 8
5 16
6 32
7 64
8 128
9 256
10 512
11 1024
12 2048
13 4096
14 8192
15 16384
16 32768
17 65536
18 131072
19 262144
20 524288
So, to go 20 generations, all you need are a bit more than a million people willing to play the game, half a million of which are left holding the bag unless they can find another million suckers to hold it for them. This is because nowhere is any money somehow "made" in this scheme, it's simply transferred from the bottom up towards the top.
You will never see an actual scheme like this because 1) the person at the top who originated it doesn't get to really cash in and 2) the promised payout isn't big enough to attract a loads of people. Here's how to make your own pyramid scheme, make tons of money, and get thrown in jail when the Postal authorities find you (based on the one I studied in the early 70's).
To enter costs $100, which you give to the person who recruits you. You remove the name and address of the person at the top of the 10-name "pyramid list" and put your name & address at the bottom, making 3 copies which you sell/recruit for $100 each. You send $200 to the person who's name is at the top of your list (the one you removed for the copies) keep $100 yourself (making you even at this point) and wait for your name to come to the top of the list. In this case (the list being 10 long with 3 times the people every generation) if everyone recruited and was honest, you would get $11,809,800! Even if only 10% of that level is filled, the line goes, that's still more than a million dollars. You would be a chump not to join!
In reality, if "only 10% join" in every generation, the scheme dies immediately, you need 33% participation minimum just to keep it chugging along with a $200 total profit by the time you reach the top of the list.
So, you're talked into it, talk 3 friends into it, and send your $200 to the person at the top of the list. Another risk taker, much like yourself, waiting for his/her payoff, right? In reality, the 10 people at the top of the original list are all the same person. The other people get some fee for being a mail drop, but that's it (unless they're not honest and skim a few envelopes after realizing the scam), but the originator of the scam makes the money. Almost all the money, as it turns out.
When I modeled this it was for a town of 1 million people where 1% would be susceptible to this scam, but they become harder to find as the "low hanging fruit" is harvested. Even though 10,000 could enter, only about 1,000 do, or about 7 generations - ALL of which could send their money to the originator of the scam (well, if it fills evenly, which it won't). This assumes that the originator is honest and only sells 3 copies of the original list, I mean, what's to keep him/her from selling as many in slightly different markets at he/she can?
What it means is the suckers down the line see nothing and those at the bottom are left holding the bag. In the case I knew of, the friend who tried to talk me into this only sold one of his copies, not 3, and ended up not sending any money to the person at the top of the list. The person he sold it to actually sold all 3 of his, but none of his people sold any (and he was hounded about this for months). Of course the amount was only $15 instead of $100, but it was worth about the same, and enough to make someone mad enough to go to the postal authorities, who will pursue these schemes since they use the US mail (Online equivalents such as Paypal will cancel any account where there's even a hint of it being a pyramid scheme).
The other way of doing this is the 777 Prosperity way, where the originator "sells" something and gets a large chunk of each transaction along the way. The product is worthless, the sale comes from the promise of money for recruiting new members, and even if it "spirals" or is a "matrix" it still goes flat when new membership dries up leaving the last buyers holding the bag.