Hypothetical Ethical Game Theory Puzzle/Question/Poll

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.

think about it this way:

Sure, Bills utility (the actual value of a dollar) is much, much, much, much, much lower than yours (meaning, of course, that a dollar is inherently worth more to you than him), but the social value of that money is much greater; ie: if Bill stiffs a common man out of millions, he looks like a complete ass because he is so disgustingly wealthy, as opposed to the case where Bill were to keep, say, Perot from getting a couple’a million = no big deal since they are both rich anyway and Perots utility is fairly low, too.

You could engineer such a public outcry at his blatant callus behavior and utter greed that he’d probably not only cede the entire sum to you, he might even stake you another ten mil. himself as a matter of goodwill (or charity).

I want half, send my check to:
Operating Excellence
Highway 74 East
Laurinburg, NC 28352

Thanks.

This seems to be a heavily modified version of the Prisoner’s Dilemma.
Is all game theory based on this one problem?

Ooh the dictator game. I’m not telling what I would do, but I’ve got a mad stare in my eyes.

What does “optimum” mean here? All bargaining solutions involving an agreement are Pareto optimal. I suspect what the author means is that a particular split can be said to be the best if the individuals’ utility functions are fully cardinal and interpersonally comparable. (This is implicit in thinksnow’s post) To put it mildly this is controversial.

The experimental evidence on this game is interesting. One major lesson seems to be: don’t play this game with an economist.

picmr