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How chain letters (don’t) work:
two demonstrations

I have posted two examples (here and here) of the particularly noxious weed known as the internet chain letter. These scams have been around since Hector was a pup (I once heard them satirized on a 1930s episode of the old Lum and Abner radio serial), but the internet has given them new life. First on Usenet news groups (whence they were once routinely ejected by automatic “cancelbots”), and later on e-mail, the chain letter has become a significant source of spam and an annoyance for net users, as well as a swindle for those who respond. Even if they involve an exchange of goods for money (i.e., if you buy bogus “reports” for your five bucks), chain letters are still mail fraud and illegal gambling by mail if they involve postal mail at any stage, and anyone who propagates them can be prosecuted.

The connection between these modern chain letters and spam is simple: chain-letter participants are inevitably required to use bulk mail (i.e., spam) to reach the largest number of people possible in order to (they think) generate the biggest return. Yet, as we shall see, this very greed simply hastens the collapse of the typical chain letter, and leaves its most recent victims holding the bag.

The typical chain letter is quite simple: you receive a message with (say) five names and addresses on it. You’re told to send some nominal sum (five or ten bucks) to each of the people on the list (possibly in exchange for worthless “goods”), and then you can delete the top name from the list, add your own name at the bottom, and forward the mail on to everyone you can (then, in theory, you will be one of the people getting the five bucks from everyone else).

The logic seems quite reasonable at first glance; after all, even if you don’t get the kind of response that the letter claims, you still ought to be able to make a few bucks, oughtn’t you? Under more careful scrutiny, however, this impeccable logic turns out to be very peccable indeed. Chain-letter victims, encouraged by the perpetrators, generally make false assumptions like these:

After some investigation and thought of my own, I’ve come up with a couple of ways to show why these and other assumptions are wrong, and why chain mails are inevitably doomed to fold up like a cheap suitcase. Take your pick: the mathematical model (complicated), or the thought experiment (easy).

By the way, the logic used here (particularly the notion of market saturation) can also be applied to many bogus MLM schemes, or other forms of “viral” or “network” marketing (like Herbalife) that are commonly touted in spam e-mail.

The mathematical model

We can use some simple (if staggering) math to get a peek at what’s going on with the typical chain letter.

The objective

Trying to precisely model a specific chain letter would quickly get pretty complicated; it would be hard enough to calculate exactly how much money any given person would make, even if you don’t try to quantify all the psychological vagaries (such as how many people would sign up more than once for the same letter, or whether the response rate would decline as the racket got old). For the moment, then, let’s just focus on a relatively simple objective: can the chain letter go on forever, or is it destined at some point to die?

The assumptions

To start with, we will make three basic assumptons:

Number of people in the world who are regularly online (and can receive chain letters)
1 billion
Proportion of the above who could and would actually respond to a chain letter
Number of new e-mails sent by each new participant in the chain (a relatively modest number, in spam terms)

My figure of one billion people online is pretty close in magnitude to most of the estimates I’ve seen, such as from the Global Reach site linked above. A one per cent response rate is just a guess, but as we will see it makes little difference either way to the ultimate destiny of the chain letter. The figure of 1,000 mails is purely arbitrary, but I figure that most people would be hard pressed to send more than 1,000 spams; in any case, we will also look at what happens if you increase this number.

I’m also going to assume that the chain letter will die (stop propagating) when everyone who might respond to it has been given at least one chance to do so; that is, when everyone on the internet gets at least one copy of the letter, and the total number of participants in the chain reaches the total number (our 1% figure) of people who would respond to it. I assume that beyond this point there is no more growth, and no further e-mail.

So what’s all this about limits, I hear some of you ask -- after all, we’ve got a pool of ten million people to work with, and it would take awhile to reach that mark, wouldn’t it? Well, just keep reading, that’s all I’ll say.

The calculations

Suppose someone makes up a chain letter and sends it out to 1,000 strangers via e-mail. According to my assumption above, 10 people (1%) will respond and become part of the gang. After this "zero’th generation," there are now 11 people in the chain (the originator plus the ten recruits), and 1,000 mails have been sent.

We could go on like this for a few more generations, except we can’t. By the end of the seventh generation, we’ve got a billion people in the pyramid, and we’ve sent an astonishing TEN BILLION (ten thousand million) e-mails! Unfortuantely, the gig is now up, because we’ve now run out of suckers (everyone who’d be interested in the scheme -- 1% of all the people online -- has already been contacted at least once). We’ve “saturated the market” as the economists say.

The graph below shows what happens over time to both the number of possible recruits available and to the number of mails sent. Note the sharp “knee points” during the sixth generation; this is pretty typical of the runaway growth invariably seen in geometrically-expanding processes.

What this graph tells us is that the chain letter rolls on pretty quietly in the first few generations, with lots of room for growth and relatively little spam load, but that at some magical point the growth suddently takes off and gobbles up the entire population of suckers in one gulp; the number of spam mails resulting from the racket also shoots up to truly astronomical levels (literally astronomical, as we will see below).

Conclusions, part one:
Why Mr. or Ms. Latecomer won’t be ordering that Bentley anytime soon

And here’s the crux of the biscuit (RIP Frank Zappa): the number of people you could recruit with a chain letter is very definitely limited. It may be very large, but it is limited. The later you are invited to participate, the more limited it will be. The inexorable geometrical expansion of the pyramid means that this limit is going to be reached, more likely sooner than later. The only question is how long the joy-ride will last before we run out of gas.

Here are some pertinent observations derived from the math:

And so, if you want to have any chance at all of making money with a chain letter, you’d better be the one who originates it (but cover your tracks, because the Postal Inspectors may come to call on you). By the time most of us get a chain letter, somewhere down in the n’th generation (after everybody’s already seen it), we have about as much chance of getting rich as we would have of winning a Nobel Peace Prize.

Conclusions, part two:
Chain letters and spam

Now, let’s look at the spam angle. As I noted above, using my original assumptions, the chain letter would put a load of ten billion e-mails on the network; that’s at least ten mails for everyone online today. If, on the other hand, we allow each new recuit to send 10,000 bulk mails, the total mail load jumps to a staggering ONE HUNDRED BILLION -- that’s one hundred copies of the same damn mail for every person now online (it’s also the estimated number of stars in the Milky Way galaxy). And, since the racket collapses in about half the time, nearly all of those hundred billion mails are a complete waste of time and resources for the senders as well as the recipients.


Be aware that this is only a model, and is only intended to show the general outlines of how the chain letter works. In real life:

Also, my notion of “generations” is pretty arbitrary (these things don’t happen in lock-step sequentiality). Nevertheless, you should be able to get the drift: chain letters DON’T WORK and are essentially schemes to try to move a lot of money from a big group of people to a much smaller group of people. You will generally not be included in that small group of lucky beneficiaries of the public largesse.

The thought experiment

With a nod to Professor Einstein, here’s my own non-mathematical thought experiment that may help you grasp some of the principles behind chain letters.

  1. Go to the department store and buy yourself a big bag of kitchen sponges. The total number of sponges you have represents the total number of people who would participate in chain letters. Take out one of the sponges and set it aside; soak the rest in a bucket of water (which here represents money).
  2. Take your dry sponge and set it in a bowl on the table. This sponge represents the originator of the chain mail. He’s dry because he doesn’t have any money, but this sponge has plans.
  3. Now, take one of the wet sponges and set it on top of the dry one. This sponge represents the first generation of chain-letter recipients. Note how, over time, gravity will draw the water from the top sponge to the bottom one. This is the money going from the recipient to the originator.
  4. Now, add a third sponge to represent the next generation of recipients; again, gravity will draw the water down towards the bottom sponge.
  5. You can continue to add more sponges to the stack, but you’ll soon run out (i.e., there’s nobody left to play the game); eventually, all of the sponges on top will be dry as a bone, while the ones at the bottom will be swimming in water (money).

That, folks, is how a chain letter works: the money flows upward toward the recipient, and keeps flowing until the supply is cut off (no more sponges). The last few sponges on the heap will get thoroughly dried out.

Where I come from, this is called a SWINDLE.

Closing remarks

The chain letter is like a sort of living creature: it must continually take in fresh meat (suckers) to keep going. Just as in any ecosystem, however, there’s never an unlimited supply of food for the chain letter, so eventually it dies.

Of course, as long as a living creature continues, it tends to distribute its new calories more-or-less equally; the heart gets some, the legs get their share, etc. This is not true of the chain letter, where all of the calories (money) go to the top guy. Imagine if everything you ate went to your brain, while the rest of your body starved!

I hope this has convinced you that the chain letter is a game for suckers, or that you will have some hard evidence to share with those who aren’t convinced. It seems to me that chain letters have been on the decline in recent months, so perhaps everyone else is getting the point too.

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Updated: Sun, 15 Jun 2008