Recently I was going through a box of some old correspondence and came across a handwritten letter from Douglas Hofstadter, dated September 1, 1983.
If that name sounds familiar, there’s a good reason. Hofstadter is a well-known professor of cognitive science and author of the Pulitzer Prize winning book Gödel, Escher, Bach: An Eternal Golden Braid, among other works.
Most of it was a form letter, and enclosed were a couple of articles about nuclear disarmament which he apparently sent to anyone who wrote to him for any reason. But there was a handwritten note at the bottom that was clearly a personal response to something I had written to him.
The handwritten portion said:
I liked following the train of your musings as you moved from a die to large numbers and then back down to 1. Curious sequence. I wish more people had even
considered die-throwing! Well, I hope the enclosed rings a responsive cord in you!
Now this presented a tantalizing puzzle. I had no recollection whatsoever of writing to Hofstadter. But I remembered that in the early 1980s Hofstadter had written a monthly column for Scientific American called Metamagical Themas, taking over the space that had long been occupied by Martin Gardner’s Mathematical Games. (You might notice that Hofstadter titled his column with an anagram of Gardner’s title – a very Hofstadter-esque thing to do.)
I guessed that I must have written to Hofstadter in response to one of his columns, probably from the summer of 1983. But I had no idea what the subject of the column had been or what I might have said about it. I certainly hadn’t saved any issues of Scientific American from that long ago. So I embarked on an internet search, and found that Hofstadter had published a collection of his columns as a book titled – what else – Metamagical Themas. I ordered a used copy, and a few days later it arrived.
Paging through the 800+ page book I eventually found the article that I’m pretty sure I was responding to. It’s titled “Dilemmas for Superrational Thinkers, Leading Up to a Luring Lottery.” Once I had the title, I found that the article is available in its entirety online, here.
It’s an investigation of the Prisoner’s Dilemma, a much-analyzed game that illustrates circumstances under which rational individuals might choose not to cooperate, even when it seems to be in their best interests to do so. In the classic game, there are two players. Each player can choose either to “cooperate” or “defect”, without knowledge of what the other player will do (or has done). The outcome is favorable for everyone if both players cooperate, and unfavorable for everyone if they both defect. But if one cooperates while the other defects, the defector comes out very well while the cooperator gets shafted.
It’s well known that if the game is repeated multiple times with the same players, the optimal strategy for each player to follow is “tit for tat” – that is, cooperate if the other player cooperated on the previous round, and defect if the other player defected.
But Hofstadter is most interested in what happens when the game is played only once, so the players have no history to refer to. He bends and extends the classic game in various ways, looking at what happens when there are larger numbers of players and the benefits and costs of cooperating and defecting are adjusted. This becomes a launching pad to examine what an assortment of purely rational thinkers ought to do when confronted with a situation that requires each player to guess what the other players might do in order to choose an action most likely to result in the outcome most favorable for him/herself.
I won’t try to replicate Hofstadter’s entire argument. But in essence, he proposes that if every player is purely rational, and if every player knows that every other player is purely rational, then you have a group of “superrational thinkers.” Knowing this, each player should figure out what the optimal course of action is and assume that every other player will come to the same conclusion. Therefore, the question becomes: assuming that all players make the same choice, which of the available choices maximizes the benefit for all players when all players choose it?
This is a very interesting, and very rational, conclusion to reach. But Hofstadter was surprised to find that many of his (very intelligent, very rational) friends did not reach the same conclusion.
Hofstadter ends his June 1983 column with this invitation:
This talk of holding back in the face of strong temptation brings me to the climax of this column: the announcement of a Luring Lottery open to all readers and nonreaders of Scientific American. The prize of this lottery is $1,000,000/N, where N is the number of entries submitted. Just think: If you are the only entrant (and if you submit only one entry), a cool million is yours! Perhaps, though, you doubt this will come about. It does seem a trifle iffy. If you’d like to increase your chances of winning, you are encouraged to send in multiple entries – no limit! Just send in one postcard per entry. If you send in 100 entries, you’ll have 100 times the chance of some poor slob who sends in just one. Come to think of it, why should you have to send in multiple entries separately? Just send one postcard with your name and address and a positive integer (telling how many entries you’re making) to:
c/o Scientific American
415 Madison Avenue
New York, N.Y. 10017
You will be given the same chance of winning as if you had sent in that number of postcards with ‘1’ written on them. Illegible, incoherent, ill-specified, or incomprehensible entries will be disqualified. Only entries received by midnight June 30, 1983 will be considered. Good luck to you (but certainly not to any-other reader of this column)!
And I’m pretty sure this (the Luring Lottery) is what I must have been responding to when I sent my musings to Douglas Hofstadter.
But – frustratingly – I still don’t know the substance of my own musings that led Hofstadter to send me that note.
Perhaps if I were part of a superrational group, we would all figure it out.
Postscript, April 7, 2015:
I emailed Douglas Hofstadter a link to this blog post, and he responded the next day:
Subject: Luring Lottery memories
From: Douglas Hofstadter
Date: Wed, 1 Apr 2015 14:52:42 -0400
To: David Woolley
Dear Mr. Woolley,
I read your blog and was very entertained by it. I also liked the letter that I wrote to my readers! And your description of the Prisoner’s Dilemma and of my columns concerning it was spot-on.
Now concerning your original letter to me, I surmise (in fact, I am almost sure) that in it, you must have told me that your very first thoughts concerning the Luring Lottery were that you would throw a die and if it came up (say) “6”, then you’d enter, and otherwise you would abstain. If all my other readers used this same strategy, then you figured you would have a small but non-trivial chance of winning a small but non-trivial amount of money. Then, however, you decided that you would like to increase your chances of winning some money, so you moved to (say) either a “5” or a “6” (a 1/3 chance of submitting instead of just 1/6), and from there, you slowly slipped up the slippery slope to the desire to submit a phenomenally huge number that would dwarf all other readers’ numbers, so that you’d be almost certain to win the Luring Lottery. However, since the amount you would win in that case would be infinitesimal, you then reconsidered once again. I surmise (in fact, I am almost sure) that you must have backed down the slippery slope and wound up entering the Luring Lottery exactly once (rather than entering it only with a probability of 1/6, as you had initially thought). Does this make sense?
Anyway, thanks for your nice memories. I’m glad you enjoyed my columns and my writing style. All the best to you!
It does make sense, and I’m inclined to think he has correctly summarized my thought process. It makes sense for the following reason: Assuming everyone who enters the lottery is rational, each of us would realize that sending a huge number of entries would make the prize almost worthless, so we would discard that idea. The way to keep the prize valuable is to keep the total number of entries as small as possible. The prize is most valuable if only one person enters, and that person enters only once. But that would require all the rest of us to forgo entering at all, which would reduce our probability of winning to zero. And we all want at least some chance of winning.
The conclusion all of us rational thinkers would come to is that it is best for each of us to enter exactly once, because that keeps us in the running while minimizing the number of entries and maximizing the value of the prize.
I still wish I had saved a copy of the letter I wrote him 32 years ago, but I think I can put this mystery to rest now.
Thank you, Professor Hofstadter.