dreidel, dreidel, dreidel… oh wait, not that game
This blog post was a lot more difficult than the first mainly because a lot of pop culture references were already written about. So I decided to look for examples of game theory in religious texts. What I found was quite interesting.
In the article, the author discusses a situation of bankruptcy. A man dies and owes three different people three different sums of money. The players of the game are the size of the dead man’s estate, and the amounts of money he owes the three people. He has an estate worth 100, 200, or 300, and the people he owes are claiming 100, 200 and 300. This situation is laid out in the Babylonian Talmud, a record of discussions about Jewish laws and customs. According to Jewish custom, the matrix, with payoffs, looks like this:
When the estate size is 100, it is split evenly between the three people with claims. When the estate size is 300, the estate is split proportionally. When the estate is 200, there seems to be no pattern. People were perplexed by this inconsistency for thousands of years and some even contributed the weird payoff values to a mistake in the Jewish literature. In the 1980s, Professors Robert Aumann and Michael Maschler wrote a paper showing that they solved the mystery with the help of game theory. They found that the Talmud answer is the solution of a coalitional game. Simply speaking the game has payoffs made by equally dividing the contested sum.
In the linked article, the author goes through a few examples to show how this principle works. Basically, if you have two people, one desiring 3 dollars (A) and the other desiring 5 dollars (B), fighting over a certain amount, say 5 dollars, then the contested sum is 3 dollars because both parties want at least 3 dollars. Therefore, 3 dollars is split evenly between the two parties. A gets 1.5 and B gets 1.5. What to do with the other 2 dollars? The remainder is granted to the person that desired greater than the contested sum. So in the end, the 5 dollars will be split like this: A gets 1.5 and B gets 3.5.
This splitting of payoffs is consistent with that in the Talmud. When applying this to a game with six possible outcomes, you take the amount that any two people receive and see if that is consistent with the theory. For example, in the bankruptcy game, the people claiming 100 and 200 both receive 33.3 in the first situation. The total payoff for both is 66.6. since the first wants 100 and the second wants 200, 66.6 is the total contested amount. Thus, it is split evenly between the two. The same applies for this situation and the person claiming 300. In the situation where the estate size is 200, you go about the math the same way. The person who wants 100 gets 50 and the person who wants 200 gets 75. The total payoff is 125. The contested amount is 100, so that value is divided evenly and the remainder is granted to the person who wants 200. Thus the payoffs are 50 and 75 respectively. Doing the math for all situations shows that all payoffs are consistent with the theory. The author expands this argument, to include N players, with an algorithm.
In game theory, a coalition game is a game where groups of players can enforce cooperative behavior, making the game a competition between groups of players instead of between individual players. Even though we did not talk specifically about coalition game theory in class, I thought it was an interesting way to think about game theory in general. We spent a lot of time talking about game theory and it is my favorite topic of the class so far. I also thought that it was pretty neat to find game theory in a religious work. After many years, game theory provided an answer to what was thought of as a mistake to some.
Here is the link to the article: http://mindyourdecisions.com/blog/2008/06/10/how-game-theory-solved-a-religious-mystery/