It's a lottery
with a prize fund of $1 000 000. You can enter as many times as you want. There's only one catch
: the amount the winner gets is the the million dollars
, divided by the total number of entries.
This was a game introduced by Douglas Hofstadter in Scientific American in August of 1983, and described in Martin Gardner's Mathematical Games column. Scientific American put up the money, and anyone could enter; all you had to was send them a postcard containing the number of entries you would like to make.
When someone sends in a postcard with a number on it, they get that number of tickets with their name on them put into a hat. When all the entries are in, one ticket is removed from the hat at random, and the person whose name is on that ticket wins the value of the pot, which is $1 000 000 divided by the total number of tickets in the hat.
Simple as that.
Of course, it's not as simple as that. The more entries are sent in, the less the eventual winner would make. If the million readers of Scientific American all sent in a single entry, then the lucky winner would make $1, which doesn't sound like such a good deal. So what to do?
Well, Gardner had run his own version of the Prisoner's Dilemma game in the previous two issues, and had introduced his idea of 'superrationality'. That is, you assume that everyone else is thinking the way you are, and make your decision on that basis. For the prisoner's dilemma, it inevitably leads to cooperation. For the Luring Lottery, Gardner calculates that the optimal course of action on this basis is to roll a 6667 sided die (on the assumption of about 10 000 entries) and send in a single entry if your number comes up. You don't have a huge chance of winning, but if you do win you can expect to make about $520 000, which is enough to satisy the espresso needs of even the most extravagant Scientific American reader for some time.
With a smaller die you get more chance of getting something, but you win less; with a larger die the pot gets bigger but you stand less chance of getting anything. 6667 sides is the best.
Is this what everyone did? Is this what anyone did? I'm sure you can guess. The results were published in the September 1983 edition, and contained a table with part of the results of people sending in various numbers of entries:
1 000 000: 33
602 300 000 000 000 000 000 000 (Avogadro's number): 1
Googol (10100): 9
Googolplex (10googol): 14
The sad thing is that it's not really surprising. No doubt the 14 people who sent in a googolplex entries thought they were guaranteed the win (of, by this point, $0), and no doubt the people who sent in a single entry thought they were being perfectly fair (although as I mentioned above this is far from the case). It's possible some people did base their entry on a truly rational decision, and possible someone even scored their number on the 6667 sided die and sent in their entry, but we shall never know.
But there's more; the score chart is far from complete. Some people really, really wanted to win. There were postcards covered from top to bottom in tiny 9's, or exclamation marks signifying factorials, resulting in huge numbers, and the most extreme "exploited such powerful concepts of mathematical logic and set theory that to evaluate which one was the largest became a serious problem," and which the editors of Scientific American, mindful of their own sanity, didn't even attempt to decipher.
Needless to say, the very first of the huge entries (including the googols and googolplexes) reduced the value of the winnings to 0. And, since there was no reasonable way to evaluate the hugest numbers, there was no winner.
Is there a lesson to be learnt from all this? Probably; but not, I think, something we didn't already know.