Social Dilemma Games and Puzzles

By Leon Felkins, written 3/10/96

Revised on 3/10/97

• Hofstadter�s Games
  Douglas Hofstadter, in a series of essays on the Prisoner's Dilemma published in Scientific American, devoted one essay (Scientific American, June, 1983 and reprinted in his book, Metamagical Themas:) to the games based on Prisoner's Dilemma. At the end of the essay he makes the following offer to the readers:

  "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 without 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:

  Luring Lottery
  c/o Scientific American
  ...
  ...

  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 5:00 PM on June 30, 1983 will be considered. Good luck to you (but certainly not to any other reader of this column)!"

  
  Well, what do you think happened? It would have been in the best interests of a potential winner if everyone would have limited their entries to at most one or to not submit at all. If nearly all of us refrained from submitting, then at least someone would win a substantial amount of money (I understand Scientific American was actually a bit worried). Of course that didn't happen! No, he received many postcards with the largest number that the senders could write on a postcard! He received numbers like, 602,300,000,000,000,000,000,000, and "10 to the 10th to the 100th". Scientific American needn't have worried!

• "Take it or Leave it" (or The Ultimatum game)
  The following situation is described in the book, Bargaining Games, by J. Keith Murnighan:

  You and an acquaintance, "Pat", are walking down the street when you meet an older couple with a bag of money. The older couple makes the following offer: We wish to give the two of you $100,000 if you can decide how it should be decided between the two of you in the next 3 minutes. You say, "So, Pat, what do you say? How about fifty thousand dollars each". To your dismay, Pat answers, "Gee, I'm really sorry, but my mother needs an expensive operation. So, I'll take eighty thousand dollars and you can have twenty thousand. I won't settle for anything less!"

  What do you do? Insist on an even split and get nothing or take the $20,000 and be happy?

  (Another version of this game is called the "Ultimatum Game", in which two people play, with one getting to chose how the gift is to be divided and the other gets to decide whether to accept the division on nothing. A description can be found in one of Jon Elster's essays called "Doing our level best".)

  I would appreciate hearing from you on this puzzle. I am particularly interested in knowing whether you would accept the division proposed by Pat.

• Martin Shubik's Dollar Auction
  The following game is described in detail in the book, Prisoner's Dilemma, by William Poundstone: A dollar bill is offered to the highest bidder with the following rule of bidding: the highest bidder wins the dollar bill and both the highest bidder and the second-highest bidder both must pay what they bid. This actually has some relevance with respect to many competitive bids such as the government defence industry where making a bid is very expensive.

  This game is also described in J. Keith Murnighan's book, Bargaining Games, published by William Murrow and Co., Inc., Copyright 1992. This is an easy to read book that has far more discussion on the Social Dilemmas than the title would indicate. I have a copy of Chapter 9 which describes the Dollar Auction, included here by permission of the author. Note that this is copyrighted and is not to be duplicated without permission.

  When this game is played, a dollar bill is often auctioned for as high as 3 or 4 dollars! Try it at your next party.

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