Have you ever received a chain letter? Suppose you did. Consider a
typical chain letter. It could be started by a person sending out two
letters with his name at the bottom of a 20 person list. You would be
asked to send a dollar to the person at the top of the list, remove
that person from the list, add your name to the bottom, and send two
copies of the revised letter to two other persons. Each time the
letter was received and the instructions followed, there would be twice
as many recipients. There would be approximately a thousand letters
after ten generations, a million after twenty, and a billion after
thirty (the number goes up by a factor of 1024 for each ten
generations). The person initiating the chain could expect the 19
friends he listed above himself to share over a million dollars (two
dollars for the first person, four dollars for the next, and so on).
He would expect to receive a cool million for himself.
Chain letters are illegal. They are like pyramid schemes, which are
also illegal. These scams involve large gains for the first few people
to become involved, and large losses for the many people who follow
them. Our government has decided that it’s bad for us to play such
games, although they have devised the Social Security system, which
works (or fails to work) in much the same way. Let’s Numinate about
the very first chain letter that must have been sent many years ago.
Assume that all recipients participated. Imagine that you were one of
them. The question is, how much could you have expected to receive if
you followed the instructions?
Let’s assume this was back when the world’s population was one billion
people. You first note that twenty names are on the list. You figure
how it grows and conclude that the game will be over after at most 30
generations. You can calculate that the most the person originating
the chain, or anyone else, could receive would be about $1 million. It
should be obvious to you that the later people into the game might
receive nothing. Their names can’t be promoted to the top, because
there are no new people to send letters to. The final few generations
could consist of letters to people who had already played the game, and
who knows how many times they would play it over. But, let’s go with
the simpler assumption that everyone plays exactly once. This means a
billion dollars are paid into the game ($1 for each person on earth).
How is the money paid out? The persons the originator listed above
himself will share about a million dollars among them. The originator
and the people in the nine generations that follow him will share the
remainder of the money (for a total of a billion dollars). There are
1023 such individuals. So, it appears that a few more than a thousand
people will share about a million dollars each. A thousand winners out
of a total population of a billion means that any person chosen at
random would have about a one in a million chance of being one of the
winners.
Are the true odds of being a winner one in a million? This is a
sequential phenomenon. It would seem that some additional information
might pertain. For example, it might make sense to assume that about
half the population received their letters before you did, and half
will receive their letters later. Depending on the vagaries of mail
delivery, that means you are in the 29th or 30th generation. In any
case, assuming you are anywhere close to the middle person receiving a
letter, you must be among one of the last two generations of the chain.
If so, you have no chance at all of being a winner (of course, zero is
not much less than one in a million).
You could make a more optimistic assumption: That your name was added
to the list about half way through the game, namely that you are in the
15th generation. But, again you are at least five generations too
late; only the originator and the nine generations that follow him will
receive a million dollars. It takes the first 20 generations to pay
the 20 people on the original list. The originator, himself, is paid
by the 20th generation. The 21st through 29th generations are sending
their dollars to the lucky people in the first nine generations
following the originator. The game begins dying off starting with the
30th generation, if it hasn’t already done so before that time, because
the 30th generation would require the entire population of the world to
send two billion more letters. Now, with each generation, every person
would begin receiving two, then four, then eight letters apiece. You
would think they would catch on at this point and drop out of the game.
No one after the 10th generation can
get promoted all the way to the top of the list. They simply have no
chance at all.
If you understand this analysis, there is a natural extension of its
logic. It has been called the Carter/Leslie doomsday argument. It
runs something like this. Here you are, a living human being. There
will ultimately have been some number of human beings before the mold
is broken and our species goes extinct. You may be among the first,
the last, or somewhere in the middle of the line. All things being
equal, it is reasonable to assume that you were born somewhere in the
middle. However, like the chain letter, there are more humans with
each generation. About ten percent of all humans ever born are alive
today. If humans were to continue reproducing at their current rate
for another hundred years, about half would have preceded those alive
today, and about half would follow us. If the rate could continue for
another millenium, you and I would be among the first 0.001 percent of
all humans to have lived. What are the odds? The Carter/Leslie
argument is that the odds are that you are more likely
near the middle. This can only be the case if the current rate of
human reproduction stops (or almost stops) in a hundred years or so.
The conclusion predicted is what they call “Doomsday Soon.” No reason
for Doomsday needs to be given. The case is purely mathematical.
Although this particular application is fairly new and not very
well-known, arguments like this are common in science. There are
basically three kinds of these arguments. The first, and most well
known, is Occam’s razor—the proposition that the simpler of two
competing explanations is the preferable (or more likely). The second
is that given a single data point (such as when you were born), assume
that it lies near any relevant mean (such as halfway in time from First
Man to Last Man). However, when faced with extremely unlikely
circumstances, the third option may be chosen. This was titled the
Anthropic Principle (by Carter of the Carter/Leslie, above). It
basically explains why we might find ourselves on a planet that might
be one in a thousand, orbiting a star that might again be one in a
thousand, perhaps in a galaxy that’s also one in a thousand, and in a
universe with laws of physics that are unlikely beyond estimation. The
reason is simple: Unlikely as they are, if all these things hadn’t
come to pass, we wouldn’t be here to observe them. Our being here
changes the odds according to the rules of conditional probability.
The doomsday argument is based on Bayes law. Suppose you were shown
two urns, and told that one urn contains 1000 balls and the other
contains only three balls. Furthermore, each urn contains exactly one
black ball. Now, someone reaches into one of the urns and draws out a
black ball. Your problem is to pick the urn that contains the fewer
balls. There are two urns, so it’s a 50/50 choice, isn’t it? Or, is
it more likely that the black ball was drawn from the urn that contains
only three balls? How much more likely? Given the truth of what you
know and have seen, Bayes law tells us there’s a 99.7% chance that the
urn from which the black ball was drawn has only two balls remaining in
it.
The doomsday argument asks you to pick the more likely scenario: You
were born among the first 0.001 per cent of human beings, or you were
born at about the 50% point. If you think the odds favor the second
hypothesis over the first, then you have to think that the odds also
favor Doomsday Soon, especially given the growth curve of world
population. In any case, it’s worth Numinating about, don’t you think?
Although protected by Copyright, the author grants
permission to reprint this article in a non-profit publication, or copy
it over the Internet, with its Title, Copyright, and this notice.
Notification to the author and courtesy copies of the publication would
be appreciated. For other publication, please contact the
author.