Economists’ Untold Secrets 013: Talmud I

Economists’ Untold Secrets 013: Talmud I


This video proves that the Talmud problems are bargaining problems and the Talmud rule is 50/50. Talmud is the ancient writing of Jewish teachings. Some people picked out from it two interesting sentences on dispute over money or property. This video considers the first one: the contested garment. The script says:Two hold a garment; one claims it all, the other claims half. Then the one is awarded 3/4, the other 1/4. Here are two contestants, for illustration only. Here are the results. But, how were they worked out? Define the whole garment as E or the estate, and the claims as C1 and C2 respectively. The Talmud rule, according to Aumann and Maschler, proceeds as follows: First, find the minimum right, which is the whole estate minus the opponent’s claim. It means something that the other has left over. So, E-C2 for person 1, and E-C1 for person 2. Next, find the disputed amount, which is the estate minus the two minimum rights. No. 3, Split the disputed amount equally between the contestants. Finally, Add the split to the minimum right. Sounds complicated? Here is the numeric example the minimum right for the first person is 1-1/2, as the second person claims half of the garment, so=1/2. while that for the second person is 1-1, as the first person claims the whole thing, so=0. And, the sum of the minimum rights is 1/2. The disputed amount is 1- 1/2, and therefore 1/2. The split of 1/2 is 1/4 , Finally, the first person’s award is: his minimum right, 1/2, + the split, 1/4, i.e., 3/4. and the second person gets 0+1/4 or 1/4. Still get lost? Here is my understanding of the Talmud Rule: The problem is to split an amount between two persons. The possible solutions can be represented by this line, with a slope of -1. You may call this the budget line. Point A means that Person I gets the whole amount, while Point B Person II gets the whole amount. If they share the amount, it will be some point on this line between A and B. The problem is to decide on which point. Person 1 claims the whole garment, so draw a vertical line passing through Point A. Person 2 claims half of the garment, so his claim line is a horizontal one halfway between 0 and Point B. Now, transform the claim into minimum right for Person 1, by swinging the C2 line to make it vertical with the pivot on the budget line. As person 1 claims the whole garment, the minimum right for Person 2 is zero. Now, the disputed amount is represented by the red line. To split the disputed amount equally, draw a 45 degree line from the right-angle of the smaller triangle to cut the red line. The result is 3/4 and 1/4 . I hope this is a clearer presentation than the previous one. Now, Person 2 may regret to have not claimed more. If he does, the minimum right for person 1
becomes smaller, and the vertical line shifts leftward. The disputed amount becomes bigger, but the solution is still one cut by the 45 degree line. Indeed, if person 2 also claims the whole garment, the solution will be 1/2 and 1/2. In retrospect, with reference to the disputed amount, Talmud dispute is nothing but a nash bargaining problem, and Talmud rule is to split the disputed amount equally. But, is “equal split” the only solution to the problem?

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