Tuesday, December 27, 2011

47: An Unusual Lottery

Today we're going to talk about a very strange type of lottery,
known as the "Luring Lottery", which originally appeared in one of
Douglas Hofstadter's Metamagical Themas articles in Scientific
American in the early 1980's.  Here's how it worked.  This lottery has
up to a 1 million dollar prize, and you can send as many entries as
you want.  In fact, to save time, you can just send a postcard with the
number of entries on it, and you have entered that many times.  But
there is a big catch:  the total prize will be 1 million dollars,
DIVIDED by the number of entries received.  So if two people each
enter once, the winning name drawn will win 500 thousand dollars.  But
if one million people each send in an entry, the winner will only get
one dollar.  So, what is the best way to play this lottery?
One naive strategy is that maybe you should just send in 1 entry:
that way, if everybody does it, everybody has an equal chance to win.
But suppose Scientific American has 10 thousand readers.  If they all
do that, the million dollar prize will become a mere hundred dollars,
not a very exciting victory for the winner.  Is there a better way?
Well, thinking about it, the best possible outcome is probably for
somebody to become a millionaire.  To do that, there would need to be
only one entry.  But if everyone cooperates, that can be arranged--
everyone can decide to run a random number generator on their computer
that generates a number between 1 and the total number of expected
participants, and they will send in an entry if and only if the number
picked was 1.  (Of course, there is a chance that a few people or no
people might enter due to random luck from the independent number
generators, but for the moment let's ignore that statistical
complication.)   Then one reader would win, just as in the everyone-
sends-an-entry scenario, but that reader would get a million dollars.
The monkey wrench in this idea is that pesky concept of human
nature.  Suppose, being a good guy, you decide to follow the rule we
just stated... and your computer generates the number 2.  "Pretty
close," you might rationalize, "and if I send in an entry, I have a
50-50 chance of winning 500 thousand dollars."   Of course you can see
the slippery slope here-- if one person does this, it's true, but then
if you reason that everyone else will think the same way, you need to
send in 2, or 3, or 100 entries in order to ensure you still have a
chance with all those cheaters around.  In the end, you need a lot of
willpower and moral stamina to remain a true cooperator, and help
achieve the best overall result, despite the fact that it may lead to
a locally less-than-optimal result for yourself.  If you're
philosophically inclined, you can think of a lot of less artificial
real-life situations where people face similar dilemmas.
I find the actual results of Hofstadter's lottery especially
amusing.  Many of his readers were cynical enough to believe that it
was nearly 100% certain that there would be enough non-cooperators to
reduce the effective prize to 0-- so a large number of them
re-interpreted the real prize to be that their winning name would
appear in Hofstadter's column.  Thus everyone competed to send in a
number so large that the other entries would be dwarfed, and they
would be almost certain to win.  There were nine googols, fourteen
googolplexes, postcards crammed with strange notations in tiny fonts,
and pages of forumulas and definitions.  In the end, even this elusive
name-in-print prize was lost, because the referees gave up on ever
being able to interpret the entries well enough to actually carry out
the drawing.  As for the prize money, needless to say, it would have
been so infinitesmally small that no bank would have been capable of
issuing it anyway.
And this has been your Math Mutation for today.

• Article on the Luring Lottery
• Metamagical Themas at Wikipedia
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