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The purpose of this article is to describe an application of mathematics to one controversial place in Talmud, as an example of relationship between religion and science. Some of the material is rather difficult; for this part, the hope is to target an audience with some experience in math and algorithms.

The concluding discussion may be of general interest to all readers; it is also related to the previous volume of Chidushei Torah@NDS.

THE CASE OF THREE WIDOWS

In Mishna Kethubot (10:4), there is a case of three widows, surviving the same husband. Each widow has a documented claim on her part of the property. The first one has a case for 100, the second one for 200, and the third one for 300. (All values are in some ancient units; one of course may substitute dollars or any other currency here.) . They would all obtain their claimed shares if the total value of the property amounted to 600 or more. The problem is that the deceased left much less. Mishna proposes the following distribution table.

Table 1

From the first glance, it appears that there is no general method applicable to all of the listed cases. The first line divides equally while the third one divides proportionally. The strangest case is the second line.

To find the rationale behind this decision is not easy. The case produced a lot of literature. In specialized publications, it is also called the bankruptcy problem (with shareholders standing for the widows and the bankrupt enterprise for the inheritance).

1. Proportional method. For this case, the table is a bit obvious. (Tosafot in Kethubot 93, “Rabbi omer”)

Table 2

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=100*	16 2/3	33 1/3	50
*Inheritance=200*	33 1/3	66 2/3	100
*Inheritance=300*	50	100	150

2. Rambam’method (Laws of lending and borrowing, 20:4): the whole is divided equally, insofar as it does not violate the requirement that nobody receive more her claim is. In other words, all unsatisfied claims are equal. To illustrate this method, we draw this table, which also includes larger values.

Table 3

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=100*	33 1/3	33 1/3	33 1/3
*Inheritance=200*	66 2/3	66 2/3	66 2/3
*Inheritance=300*	100	100	100
*Inheritance=400*	100	150	150
*Inheritance=500*	100	200	200
*Inheritance=550*	100	200	250

A similar rule is also found in Talmud for an auction case (Erakhin 27B).

1. Rabad method. All the three widows claim the first 100, so it must be divided into three equal parts. To the second 100, two widows of the three have rights, so it is divided into two equal parts. To the third 100, only the third widow is entitled, so she takes it. The table will look like this (for larger sums, the rule is not defined):

Table 4

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=100*	33 1/3	33 1/3	33 1/3
*Inheritance=200*	33 1/3	83 1/3	83 1/3
*Inheritance=300*	33 1/3	83 1/3	183 1/3

4-6. All the methods above can be inverted, in case the claimants choose to divide losses. Thus, theoretically, the amount of methods is doubled, although it is easy to see that for some rules, such as the proportional method, the inverted rule will result in the same distribution.

From all these tables, one can see that it is not easy to decide what the fairest method is. Another problem is that Mishna remains unexplained. It is stated explicitly by Tosafot on the same page (Kethubot 93, “Deamra la midin”: “The matter is not clear…”)

THE CASE OF CONTESTED GARMENT

This case is rather famous. Two men are holding a garment. The one claims that it is all his, the other that he has a half in it. Judicial decision: the former gets ¾ of its value, the latter ¼. (Baba Metzia 2A).

What is the reason for this decision?

The second contender admits that one half is not his, so it is awarded to the first contender without any argument. The other half is what is contested, with both men having equal rights to it. So, it is divided into equal parts of ¼.

Let us see another example.

A claims 40 out of 100, B - 70 out of 100. Similarly to the previous example, A admits that B is entitled to 60, and B concedes 30 to A. The remaining 10 is divided 5:5. So, A receives 35 and B 65.

It is fairly easy to obtain a general formula for this method.

Now, the following questions may naturally be asked:

1. May the garment case be generalized for more than two claimants?

2. Is there any connection between the widows’ case and garment case?

GENERALIZATION OF THE GARMENT CASE

Here we introduce the main heroes of the story – a pair of Israeli mathematicians, Robert Aumann and Michael Maschler. Both are experts in economics and game theory; the problem of the three widows is a typical sample problem from their field. They found many new interesting aspects of the subject matter from the points of view of both Torah and mathematics (R.Aumann, M.Maschler. Game Theoretic Analysis of a Bankruptcy Problem from the Talmud. J. of Economic Theory, 1984, vol. 35, no. 2, 195-213). Most of this text is based on the mentioned article, which is hereafter referred to as A&M.

A&M proposed the following criterion. Let us suppose that the estate is already distributed in some way between multiple contenders. Let us take any pair of the contenders; together, they are awarded a certain amount. Let us require that this amount be divided between the two as in the case of the contested garment. Then, the division will really be a generalization of the garment case.

A&M call a division that meets this requirement consistent. They prove that, for any total amount, any number of claimants and any set of claims, there is one unique consistent distribution. A&M further prove that, if the division is consistent, each subset of claimants will divide their common share in the same way. That is, the algorithm, if found, will give the same distribution for the initial problem and any of its subsets. They call the solutions with this property self-consistent. It will, furthermore, preserve the order, i.e. a larger claim will result in a larger award. Another nice feature is that the rule will yield the same division if the claimants choose to divide losses rather than the property value itself (the rule is “self-dual”).

Thus, the rule will be “good” – consistent, self-consistent, monotonous, self-dual and unique. We should, however, keep it in mind that so far we only know that this rule exists, which by itself does not enable us to find the distribution in practice.

Let us consider the division between the widows as an example. 200 will be divided by the proportional rule as 33?:66?:100, which means that widow-100 and widow-200 divide between them 100, either of them claiming =100. The garment principle will result in 50:50 and not 33?:66?. So, the proportional rule is not “garment consistent”.

Rambam’s rule awards each of the widows 66?, so the first two of them divide 133? with the claims of 100 and 200. If the rule were “garment consistent,” 133?-100=33? would be conceded to widow-200 as uncontested, and the rest would be divided equally, which would result in the shares of 50 and 83?. Therefore, Rambam’s rule is not consistent with the garment principle either.

It is easy to see that nor is Rabad’s rule garment consistent.

THE WIDOWS AND THE CONTESTED GARMENT

The second question raised above is whether there is a connection between the widows’ case and the contested garment case.

This may be argued both ways. On the one hand, there is a great deal of similarity between the cases; on the other hand, there are some differences: e.g. in the garment case, at least one of the claims is necessarily false. In the widows’ case, all their documents and claims may be perfectly valid, it is just that the inherited sum is not enough. Thus, the Halachic authorities were divided on this issue.

One of the experts who did see the connection was R. Hai Gaon. If we accept his view we will immediately find out that the distribution in Mishna is fair – because it turns out to be garment consistent!

Let us consider, for example, the second line of Table 1 – the division of 200:

Table 1(fragment)

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=200*	50	75	75

The first two widows together are awarded 125. Their claims are 100 and 200. The first widow cannot claim more than 100; so, the amount of 125-100=25 is given to widow-200 undisputed.

The remaining 100 is divided equally, so they must obtain 50 and 75, exactly as it is prescribed in this most difficult line of the table. Similarly, the other lines are also “garment compliant.”

So, according to R. Hai and the mathematicians, the table is smoothly explained – the distribution was made as in Bava Metzia for the contested garment.

HELP FROM JERUSALEM

Of course, we cannot be satisfied with the fact that the mathematicians found an abstract proof of the existence of an ideal division rule. We will seek this rule till it is found! And both math and Talmud help here.

In Yerushalmi, Shmuel says that the widows with larger claims can make a coalition against the others, if it is expedient for them. In math, it is called a coalitional game.

The two widows with the claims of 200 and 300

stick together into a coalition, claiming 500. Now, we have a contest of 500 and 100. These two sides divide the whole as in the garment case; then, the two members of the coalition divide between them in the same way.

Suppose the sum is 200. The coalition first takes 100, the amount not contested by widow-100. The other hundred is divided 50:50. Now, widow-100 leaves the contest with 50. So Shmuel puts it: “You claim is 100? Take 50 and go!”. The coalition is left with 150, and both its members claim all of it, so it is divided 75:75.

This is how the most difficult part of Table 1 is explained. It can easily be shown that the entire Table 1 is explained this way.

However, in some other cases not mentioned in Mishna , the method causes problems. It may fail to keep the order of the awards consistent with the order of the claims.

Suppose the sum is 100. The coalition of 500 versus widow-100 will result in equal awards of 50. The coalition members divide 50 between them as 25:25. So, the final division will violate the natural order of the amounts awarded:

Table 5

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=100*	50	25	25

A more subtle problem arises with the sum equal to 500. Again, we have a coalition of claims whose total is 500 against a single claim of 100. Widow-100 cannot claim anything in excess of 100, so 500-100 = 400 goes to the coalition. The 100 is divided 50:50, and the coalition thus obtains 400+50=450 and divides it as in the garment case:

Widow-200 concedes 450-200=250 to widow-300.

Widow-300 concedes 450-300=150 to widow-200.

This leaves 450-250-150=50 as the disputed amount, which is divided equally, i.e.25:25. The final distribution is:

Table 6

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance=500*	50	175	275
*Losses*	50 =100-50	25 = 200-175	25=300-275

The problem is that the order of losses in this table is not natural – the greatest loss is with the smallest claimant. If we speak of investors instead of the widows, the one who invested the least loses the most, which is odd.

It turns out, and this is the crucial point of Aumann-Maschler’s article, that the coalition method with minor corrections yields the desired solution that is unique and consistent with the garment rule! The solution will also be the same for both division of property value and division of loss (“self-duality”), and it preserves the order.

The method of a coalition against the least significant claimant (LSC) must be applied always except two cases:

1) when the coalition members get less than the LSC;

2) when the LSC’s losses are greater than the losses of other claimants.

The final rule is algorithmic (by the way, Rambam’s rule is algorithmic too, as well as that of Rabad).

1. If there are only two claimants, apply the garment rule. STOP

2. Try to apply the coalition procedure:

3. If some of the higher claimants receive less than the LSC, divide the sum equally and STOP.

4. If the LSC’s losses are greater than those of some other claimant, divide the losses equally and STOP.

5. If the coalition procedure has succeeded the LSC obtains his part and leaves the contest. The amount of claimants decreases by 1.

6. GO TO 1.

These rules always result in a unique, monotonous, order preserving, garment consistent and self-dual distribution. The proof can be found in the article referred to above.

Now the table from the original Mishna is not only explained but can also be expanded.

Table 7

	*Widow-100*	*Widow-200*	*Widow-300*
*Inheritance =100*	33 1/3	33 1/3	33 1/3
*Inheritance=500*	66 2/3	166 2/3	266 2/3

In the first line, the coalition must avoid the distribution of Table 4, so all the claimants are awarded equally.

In the second line, the total loss is 100+200+300-500=100. It is distributed equally between all the claimants, so everybody receives his claim minus 33 1/3.

CONCLUSIONS

An ancient Talmudic problem of three widows is resolved quite satisfactory with the help of modern science and in the spirit of Talmud, yielding a superb example of possibilty of harmony between religion and science. We can also tentatively recognize this case in Mishna as a kind of scientific forsight, not fully understood even by later Sages.

On the other hand, many unsupported and outright dubios claims, that Sages had a supranatural knowledge of science, force us to execute an utmost restraint in field of relations between Sages, Science and Religion. Otherwise we will find ourselves in the position of three widows, not knowing how to divide their wealth.

AKNOWLEGEMENTS

The author wants to thank Zvi Bernstein for valuable discussion and references.

Levi Kitrossky

SAGES, SCIENCE, RELIGION AND THREE WIDOWS