# How game theory solved a 2,000 year old problem

The Talmud is one of the central texts of Judaism. It was written around 200 A.D. and yet contains a problem that no-one could understand until two game theory professors solved it.

The problem is known as the Three Wives problem. In the problem there are three wives each with a claim to their husband’s estate. The first wife had a claim to 100 dinars, the second to 200 dinars and the third to 300 dinars. When the estate is worth 600 dinars this is easy because each wife will get the amount they claim. But what happens when the estate is worth less than 600 dinars?

The Talmud doesn’t give a general rule but it gives the following examples. When the estate is valued at 100 dinars then it should be split equally between the three wives. When the estate is valued at 300 dinars then it should be split 50,100 and 150, which is in proportion to their claims.

These two answers don’t seem consistent, but the really confusing answer is for when the estate is worth 200 dinars. Then the split is 50,75,75.

So how can it be possible to come up with such seemingly different ways to split up at estate?

The key idea is that getting half your claim is a tipping point. Once a wife has reached half of their claim then they are deemed to have been reasonably compensated.

In the first case, where there is just 100 dinars to split, then it is split equally between the wives because there is not enough for anyone to even get to half their claim. The fact that the wives are claiming different amounts is not really relevant as there is not enough to go around anyway.

In the case of an estate worth 300 there is enough for each wife to get half her claim so this is what they get.

In the intermediate case of 200 the amounts are split equally until the wife with the lowest claim gets half her claim. This happens when each wife has been given 50 dinars. There is then still 50 dinars left in the estate and this is split equally between the two wives who have the higher claims, 25 each. This give the result that the wife with the lowest claim gets 50 and the other two get 75.

Seemingly inconsistent results can be achieved by using this method.

The problem was first solved by game theory professors Robert Aumann and Michael Maschler in 1985. You can find more about their solution here.

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