Barry Hughes: Fair division in the Talmud

Written by Barry Hughes Wednesday, 01 August 2012 06:47
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Barry Hughes in his own words:


I love applying the principles of game theory to real life situations, especially in business, to find better solutions for everyone. I graduated from Cambridge University with a mathematics degree in 1993 and have since been the finance director of two of the largest newspapers in the UK and am now the chief executive of a not-for profit housing company based in the south-west of England.

My last post looked at the three wives problem from the Talmud. Today I will look at another problem of how to divide up goods also from the Talmud. This one has a very different answer, but there might be a good reason why.


The second problem is how to deal with splitting a father’s estate between four sons (the three wives problem looked at splitting a husband’s estate between three wives).


The detail of the calculation is at the end of the post but what is interesting is that the result is very different from the three wives problem. In that problem a method is used that ensures a more equal split of the estate. The wife that has the smallest claim still gets a significant part of the estate, whilst the wife with the largest claim does better but not by a huge amount.


This contrasts with the method used to split the estate between four sons. That method ensures that the son with the largest claim does very well, while the son with the smallest claim gets very little.


Perhaps there is a cultural explanation for this difference. It would have been unacceptable to leave a wife with almost nothing after her husband passed away, so the method that gives the most even split is used for giving an estate to widowed wives.


Sons would have been expected to support themselves and it makes more sense that the oldest son is given the bulk of the estate so he can continue to support the rest of the dependents while his brothers would be expected to make their own way in the world.


There are many ways to split up goods and one is not necessarily fairer than the other. What is seen as fair in one culture or situation may not be seen as fair in another.


The calculation

In the problem Jacob’s four sons, Reuben, Simeon, Levi and Judah, can prove that Jacob promised them they would each inherit part of the estate. Reuben had been promised the entire estate, Simeon half of the estate, Levi one third and Judah had been promised one quarter.


Clearly there is not enough estate to fulfil all these claims so we need a way to divide them up.


In this case the solution given is as follows for an estate worth 120 units:


One quarter of the estate is claimed by everyone. The estate is worth 120 units so one quarter of this is 30 units. This is split up equally between the four sons, each getting 7.5 units. Judah had the smallest claim and has now had his share of that claim.


Levi had the next smallest claim, a claim on one third of the estate. One third of 120 is 40, but the first 30 of this has already been split up between the four sons. The ten unit difference between 40 and 30 is split between the three remaining sons, each getting 3 1/3. Levi gets a total of 7.5 plus 3 1/3 = 10 5/6


There are now two sons remaining. Simeon claimed half the estate, which is 60 units, but 40 of this has already been divided up. The 20 difference between 60 and 40 is split equally between Simeon and Reuben, 10 each. This gives Simeon a total of 20 5/6.


The final half of the estate was only claimed by Reuben, so he gets all of it. This half is worth 60 units so he gets a total of 80 5/6 of the estate.

Judah gets 7 1/2 units; Levi gets 10 5/6 units; Simeon gets 20 5/6 units; Reuben gets 80 5/6 units

Last modified on Wednesday, 01 August 2012 07:00

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