"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:
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)!"
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 devided 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" (now offline). A comprehensive review of this type of game can be found in "ULTIMATUM BARGAINING EXPERIMENTS: THE STATE OF THE ART" by J. NEIL BEARDEN.)
The reason this game is interesting is that it shows that humans are not always "rational optimizers" -- which the responses I've received have confirmed. We do seem to be concerned about "fairness" even if it costs us dearly! An article in Science, provides evidence that this is not true with our closest relatives in the animal kingdom, the chimpanzees. See "Chimpanzees Are Rational Maximizers in an Ultimatum Game".I would appreciate hearing from you on this puzzle. I am particularly interested in knowing whether you would accept the division proposed by Pat.
The "Centipede" game is an extension of the "Take it or Leave it" game. It is called "Centipede" due to the way the game looks when diagrammed -- not because of any relationship to centipedes by the game itself.
In this game, the players are allowed to play indefinitely with payoffs at each stage which are somewhat diabolical.
The game can, of course, be set up with any pay off but for our purposes we will assume $1. At each stage of the game, the house awards a payoff as follows: $1 each if the game is continued or $2 to the quitter and nothing to the opponent if one of the players quits. So a chart of the game looks like this:
0,0 1,1 2,2 3,3 4,4 ----->------->------->------->------->------ | | | | | V V V V V 2,0 1,3 4,2 3,5 6,4
The chart is intended to represent continued play by the arrow head pointing to the right and termination by the "V" pointing down. The first number represents your winnings and the second number represents the opponent's. Starting with the first and every other play, you get to chose. Your opponent chooses on the remainder.
It is assumed that a rational player will want to maximize her winnings (this can be arranged by properly structuring the game such as allowing repeated playings in a tournament with a large prize going to the person with the largest score). Further, you might want to assume that the players do not know each other and will remain so.
Given that, how would you play?
Now, consider a finite game with, for example, only 10 plays. Assuming you would have the ninth play, what would you do. Well of course, you would stop the game and be ahead. But your opponent, on the 8th play would know that, so he would quit there. And so on, back to the beginning of the game, meaning you would quit on the first play! But that doesn't make sense because you would both do better if you stayed longer.
If you would like to know more on the game and to even play it, check out "Centipede" by Ariel Rubinstein.
When you figure out this game and/or a good strategy, drop me a line.
This is similar to the Ultimatum game except now there are three people involved.
Three hapless characters, Bubba, Billy, and Bob, are managing to spend their last few dollars on a few more drinks before they get really serious about trying to find a job. While walking down the street looking for their second most popular bar, a car comes whizzing by and a roll of exactly 100 one hundred dollar bills is tossed out on the sidewalk. Close behind the speeding car is a cop car in hot pursuit. They pick up the bundle of cash and retire to the nearest bar to negotiate its distribution.
But just as they are about to work out a split, a cop comes in and says, "I saw that bundle of drug money tossed out of that car and I saw you guys pick it up. To heck with the high speed chase, I'm here to take the money."
There is much dispute, threats of lawsuits, etc., and a general reluctance by Bubba, Billy, and Bob to give up the money. Finally the cop says, "I tell you what -- I'll make you a deal. You guys decide, by unanimous agreement how to split up these 100 bills between the three of you and you can keep the money. Otherwise it is mine. Take your time on the negotiation but I ain't got all day -- eventually I will call for a vote between the three of you.
Bubba starts off the game by suggesting that he get 34 bills, Billy get 33, and Bob get 33. Bob says that is fine with him but Billy says, "Well, that sounds good, but I got a better idea. How about if we parcel out 40 to Bob, 40 to me, and 20 to Bubba. Bob likes that even better and suggests we vote right now. But Bubba says, "Wait a minute, Bob, you greedy bastard, I got a better proposal: How about if we give 45 to Billy, 40 to me and 15 to Bob."
Do you have any suggestions on how they can end the bargaining phase and move on to the vote?
The idea for this game came from David Friedman's online book, Price Theory: An Intermediate Text, Chapter 11.
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.
Geoff Wong has a series of Political Games on his web site. An interesting game at his site is the "$100 Game". Players write down a number between 1 and 1000. The player writing down the highest number wins. The winnings are determined by dividing the $100 by this highest number. Rarely does the 'house' have to pay out more than a few bucks!
Comments from you on these issues are very welcome. You may send email to firstname.lastname@example.org.
©1996, 1998, 1999, 2000, 2002Background courtesy of Roxy's Renditions.