There are a few statistical or game puzzles that intrigue me to the point that I continually return to them, mulling them over from different angles. One such puzzle (not really a puzzle so much as an illustration of a point) I got from Game Theory: A Nontechnical Introduction, by Morton Davis (one of my favorite books, by the way: recommended!). Davis uses an example to illustrate a point on arbitration. A paraphrase of the example:
**The International Experimental Commission on Game Theory and Bed-Wetting (a very prestigious organization that I just made up) is grossly over-funded and wishes to rid itself of excess cash. The Commission proposes to give you and Bill Gates ten million dollars, provided the two of you can agree on how to share it. If you can’t agree, you get nothing.
You say, “OK, Bill, fair’s fair, let’s split the ten million down the middle: five million for me, five million for you.”
Bill replies, “Hold on, little person. I am fabulously wealthy; you are a peon. Sure, I’d like the money, but it wouldn’t make a major change in my life like it would in yours, so I have more leverage. Therefore, I propose that I get nine million, and you get one million. If you don’t agree, I walk, and we both get nothing.”
What do you do? Do you stand on principle and demand a 50/50 split, or do you take the admittedly large amount of one million?**
The question could be extended, of course: Exactly where is your cut-off? Would you accept a 40/60 split? A 49.9/50.1 split? A 20/80 split? A 1/99 split? A 0.01/99.99 split? How about if the amount af money was substantially larger or smaller?
Me, I’d probably accept Bill’s offer, although I’d grouse a lot. I’d even go lower, but I’m not sure how much. $100,000 is a lot of money, but accepting a 1/99 split would rankle.
By the way, Davis points out that you can calculate the optimum split, provided you know the utility function for each person, i.e., how much a dollar means to you.