Three Holding a Cloak (Column 757)
With God’s help
Last Sabbath we were guests of our new in-laws, and a lively discussion developed about the case of three holding a cloak, and of course we also referred to Israel Aumann’s well-known article on the matter. In the past I wrote about this at length (book 13 in the Talmudic Logic series), and I also cited other articles that dealt with the issue. In this column I only want to touch on the difficulties with several of the different solutions that have been proposed.
Aumann’s Article
Many years ago I was hospitalized, and for some reason I thought there about the topic of those holding a cloak, and wondered whether there is a way to generalize the Talmud’s solution to more general problems, and whether there is a unique generalization. I examined several such possibilities, thought about several possible approaches, but did not find a general formulation of the solution, nor a unique one.
After some time I learned that Israel Aumann, the Nobel laureate in economics, had written an article on the topic of ‘three who put money into a common purse’ or ‘one who had three wives,’ which deals with dividing money among creditors (here you can read a more popular Hebrew version of the article). He compared this to the case of those holding a cloak, and proposed a general solution to the problem for any number of claimants and any distribution of claims. He proved both the existence of a solution and its uniqueness (well, not for nothing did I fail to win a Nobel Prize. Although in the above-mentioned book I too proved this, by another route). To the best of my knowledge, the algorithm presented in that article is now taught in game theory courses that deal with these topics (I spoke with a game theory lecturer who herself teaches it at Tel Aviv University). I should note that when I read the article, I had several questions about it (I corresponded a bit with Aumann on the matter), some concerning his interpretation of the Talmud and some concerning the justification for the algorithm itself. Here I will not deal with the Talmudic passage, but with the problem itself.
Two Holding a Cloak
The Mishnah at the opening of Bava Metzia deals with a case in which two people come to a religious court while both are holding a cloak, and each claims that he is the one who found it:
Two are holding a cloak: this one says, “I found it,” and that one says, “I found it.” This one says, “It is all mine,” and that one says, “It is all mine.” This one swears that he has no less than half of it, and that one swears that he has no less than half of it, and they divide it. If this one says, “It is all mine,” and that one says, “Half is mine,” the one who says, “It is all mine” swears that he has no less than three parts, and the one who says, “Half is mine” swears that he has no less than a quarter; this one takes three parts, and that one takes a quarter.
Two cases involving two claimants are presented here:
- If each claims the whole thing, the ruling is that they divide it (with an oath), that is, each gets 1/2 of the cloak.
- If one claims the whole and the second claims only half of it (apparently he admits that the other lifted it together with him), then the one claiming the whole receives 3/4 and the other receives 1/4.
The Tosefta at the beginning of Bava Metzia presents another case:
If one says, “It is all mine,” and the other says, “A third is mine,” the one who says, “It is all mine,” swears that he has no less than five sixths in it, and the one who says, “A third is mine,” swears that he has no less than one-sixth. The general principle is this: one swears only concerning half of what he claims.
That is, one claims the whole thing and the other claims a third; the division is 5/6 and 1/6.
It is clear that the principle defining the division is that there is a part of the cloak that is not in dispute, and that part goes to the one who claims the whole thing (since he is the only one who claims that part. There is no argument about it). As for the rest, there is a dispute in which each of the two claims all of it, and therefore they divide it (as in the first case in the Mishnah). Thus, in the first case in the Mishnah, where both claim the whole thing, there is no part that is undisputed between them, and so they divide everything. In the second case, where one claims the whole thing and the other claims half, there is half that is undisputed, and therefore the one who claims the whole thing receives it, and the remaining half is divided between them. As noted, the result is 3/4 and 1/4. This of course also explains the case in the Tosefta.
Notice that this method in fact covers all the possibilities involving two claimants. In the most general case we have two claimants, one claiming P1 and the other P2. If the sum of the two is less than 1, then each receives the part he claimed and the rest remains unclaimed, waiting for another claimant. The problem arises when the sum of the claims is greater than 1, because in such a case it is impossible to satisfy them both. Here a division is required, and it is carried out according to the algorithm described above.
One may ask what the ruling would be when one claims 3/4 and the other claims 1/2. This is a case in which the sum is greater than 1 and it is impossible to satisfy them both. On the other hand, it is not entirely clear how to apply the algorithm I described to such a case. The one claiming 1/2 admittedly concedes that 1/2 is not his, but the one claiming 3/4 concedes 1/4 to the other. Is that 1/4 not included in what the other concedes to him? The question is how to position the relation between their mutual admissions. It turns out that in such a situation the larger claimant receives 1/2 (because the other admits it to him), and the smaller claimant receives 1/4 (because the first admits it to him), and they divide the remaining quarter. So the larger one receives 5/8 and the smaller one 3/8.
The General Problem
We can now ask about the most general problem. N people are holding a cloak. Each of them claims Pi, where i ranges from 1 to N. In addition, the following must hold: 
, since only then does a problem arise. We are looking for the solution vector Q, each component Qi of which represents the amount that claimant i will receive. There are two vectors here of length N: one, the vector of claims, is the data of the problem; the second, the vector of the amounts they will receive, is the desired solution. We are essentially looking for an algorithm or formula that will take us from the claims vector P to the solution vector Q.
This of course requires a definition of what that solution must satisfy. It is a generalization of what we saw in the Mishnah, namely, to examine the admission of each claimant to the others and to make a division among all the others. But how is one to make this generalization? The natural move is to do it by induction. That is, to begin by finding a solution for three holding a cloak in the following way: from the standpoint of claimant 1, the other two are contending over the remaining part (which he admits to them), and likewise from the standpoints of claimants 2 and 3. We then have three possible results (each from the perspective of a different claimant). The question is what to do with the three results that are obtained here. How do we combine them in order to find the final and overall solution? Assuming we have solved this, we must then go back and find a solution for four holding a cloak, on the basis of four solutions for three holding a cloak (from the standpoint of claimant 1, the other three divide what he admits to them, and likewise for claimants 2–4). And in this way one can continue for any number N of claimants. In other words, we must understand how to divide a cloak among three claimants, and if we understand that, we can presumably generalize it to any number N of claimants and to any claims vector P.
Three Holding a Cloak: A Preliminary Discussion
Such a case does not appear in the words of the Sages, and several possible solutions can be suggested. For the sake of simplicity, let us begin with the case in which three are holding a cloak: one says ‘the whole thing is mine’ and the other two claim half of it. At first glance we might say that the division is (1/2,1/4,1/4), in proportion to the claims. But as we have seen, that is not the criterion. For example, in the case of the Mishnah where one claims the whole thing and the other claims half, if the criterion were division according to the proportion of the claims, we should have divided it 1/3 and 2/3, and that is not the solution determined by the Mishnah.
Beyond this, the logic of such a solution is very dubious. Think of a situation in which Reuven claims the whole thing and Shimon claims half. We saw that in such a case the Mishnah determines a division of 3/4 and 1/4. Now another claimant arrives, Levi, and he too demands half. If the solution were according to the proportion of the claims, namely (1/2,1/4,1/4), then it would turn out that Reuven, who previously received 3/4, must now transfer 1/4 to Levi, while Shimon gives him nothing. That is not plausible. Levi’s claim contends with the first two claimants, not only with Reuven.
In fact, what is implausible here is the assumption that underlies the division, and this is an important insight for understanding the problem. Levi, who claims half, is not claiming specifically the half that is not Shimon’s. Division according to proportions assumes that there are here two claimants of half, standing against the one who claims the whole thing, but that between the two smaller claimants there is no dispute. They are in effect forming a coalition and standing as a single claimant against Reuven. That is not the legal situation here. What we have here is a war of all against all, and each claimant is contending with the other two.
Of course, one could say that the two smaller claimants can form a coalition and win half, and then divide it between themselves. And perhaps the very fact that they could form a coalition means that they need not do so in practice. This is a migo (an argument from a stronger claim they could have made), and therefore they would be awarded half (although at least according to some of the medieval authorities (Rishonim), this appears to be a migo used to extract property, but I will not enter that discussion). Beyond this, it is not clear whether that result is actually good for them, that is, whether there really is a migo here. It may be that under the method of division without a coalition they ought to receive more than 1/4 each, and therefore the coalition does not help them. Beyond that, why not consider a coalition of Reuven and Levi against Shimon, or Reuven and Shimon against Levi? Perhaps that would help them even more, and they would prefer that coalition? In short, one must first calculate the outcome of division without coalitions, and only then might there be room to consider the possibility of different coalitions, assuming they really do improve the position of certain claimants.
Three Holding a Cloak: Maharil Diskin
In the responsa Torat Ohel Moshe of Maharil Diskin, sec. 5, he discusses the case of three holding a cloak, where the claims vector is: (1,1/2,1/2) = P. We are of course looking for the solution for the division vector: Q = (Q1,Q2,Q3).
At the first stage he proposes there the proportional solution and rejects it, as we saw above. He then proposes a solution that looks like a natural extension of the two cases in the Mishnah: Reuven will take the half that everyone admits belongs to him, and regarding the remaining half everyone (including Reuven) claims ‘the whole thing is mine,’ and therefore the three of them should divide it equally (each receiving 1/6). But he rejects that proposal as well, because it is not correct that Shimon and Levi agree that half belongs to Reuven. Shimon admits that the other half belongs jointly to Reuven and Levi, and Levi likewise admits that the other half belongs jointly to Reuven and Shimon. There is no agreed admission regarding the half that belongs to Reuven. Again, the insight is the same insight: each of the three is confronting the other two, and not only the one who claims the whole thing. This is everyone against everyone.
Maharil therefore proposes the following algorithm:
Therefore, it appears, based on the principle of the Mishnah, that the claimant who says “it is all mine” takes one-half and one-third of a quarter of the cloak, while each of those who says “half is mine” takes half of a quarter and one-third of a quarter. Thus, the claimant who says “it is all mine” takes 14/24, and each of those who says “half is mine” takes 5/24.
His solution is that Reuven receives 14/24 of the cloak, and the other two receive 5/24 each. Immediately afterward he details how he arrived at this:
And the reason is that each of those who says “half is mine” concedes the other half belonging to the two of them—namely, of the one who says “it is all mine” and of the other who says “half is mine,” that is, a quarter to the claimant who says “it is all mine” and a quarter to the second claimant who says “half is mine.” It follows that the claimant who says “it is all mine” has a full admission from the two claimants who say “half is mine” regarding a quarter of the cloak, and no more; and thus he first takes a quarter without any litigation at all, solely by virtue of their admission.
As we saw above, Shimon and Levi each admit that there is a half that belongs to his counterpart and to Reuven. From the standpoint of each of them, that half should be divided equally between the two others (for both of them claim of it, ‘the whole thing is mine’). Therefore Shimon admits that from his perspective there is a quarter for Levi and a quarter for Reuven, and Levi admits that from his perspective there is a quarter for Shimon and a quarter for Reuven. If so, it turns out that they both admit that a quarter belongs to Reuven, and therefore first of all Reuven takes that quarter without dispute.
But regarding the quarter that Shimon admitted belongs to Levi, as well as the quarter that Levi admitted belongs to Shimon, Reuven does not agree. Therefore the division continues as follows:
And each claimant who says “half is mine” also has an admission from the other claimant who says “half is mine” concerning a quarter, so by strict law he should take a quarter on the basis of his fellow’s admission. However, the claimant who says “it is all mine” does not concede that quarter to him, for he claims the whole of it; therefore he disputes that quarter with him and takes from him half of that quarter. And likewise, from the second claimant who says “half is mine” he takes half of the quarter. Thus, the claimant who says “it is all mine” has the half, and each of those who says “half is mine” has half of a quarter, not by way of division, but rather as emerging from their admissions.
Reuven shares each such quarter with the one to whom the third party admitted it, and thus he receives two additional eighths, that is, another quarter. Likewise, each of the two others receives an eighth (his share in the quarter that was divided with Reuven). So for the time being Reuven has half, and each of the other two has an eighth. What remains of the cloak to be divided is a quarter. What is to be done with it?
He explains:
After that, concerning the fourth quarter, all of them claim the whole of it, and they divide it equally.
The remaining quarter is divided equally among all of them. Each receives an additional 1/12. If this is added to the sums that accumulated earlier, we get:
Thus the calculation is clear: the claimant who says “it is all mine” gets one-half and one-third of a quarter, and each of those who says “half is mine” gets half of a quarter and one-third of a quarter. Examine this carefully.
In conclusion, Reuven has 1/2 plus 1/12, and the two others have 1/8 plus 1/12. Thus we obtain the division: 14/24, 5/24, 5/24.
The extension of this solution to four holding a cloak, and also to additional claim vectors P, is far from simple. In my above-mentioned book I examined an extension by induction as I described above, as well as several other extensions (some of which were proposed in various articles). The conclusion is that it is difficult to find a correct solution in the sense that it is a unique extension from the cases of two and three holding a cloak.
A Technical Note
There is room to discuss how a case of claims of the whole, half, and half can arise at all. If the two half-claimants say that they lifted it together, then both are testifying that the claimant of the whole thing is a liar. If a half-claimant says that he lifted it with the claimant of the whole thing (while the latter claims that he lifted it alone), then the two of them are saying that the other smaller claimant is a liar. Perhaps it can be explained by saying that each of the smaller claimants argues that he lifted it with someone else, but does not remember with which of the other two. And the one who claims the whole thing argues that he lifted it alone (although here too it seems that the other two are testifying that he is a liar).
Maximizing Satisfaction
On Sabbath, a proposal came up for a solution that would maximize the satisfaction of the claimants. That is, we must formulate an expression for overall satisfaction and try to find a vector P that maximizes it. The assumption is of course that each claimant receives less than what he claimed, and the two quantities (the claim and the share of each claimant) are positive numbers between 0 and 1. The dissatisfaction of each claimant is expressed by the gap between the claim and the share he receives, but it makes sense to square it: 
.1 The larger the gap, the less satisfied he is. In summary, the total dissatisfaction is the sum over all the claimants:
In principle one must find P that yields a minimum of total dissatisfaction, under the constraint that the sum of the shares (the components of the vector P) is 1 and their values lie between 0 and 1.
In principle this can be done by Lagrange multipliers. But to simplify matters, let us first examine what is obtained in the case of two claimants. In this case we have one variable, p, which is the share claimant 1 receives, since the second receives 1-p. The total dissatisfaction in this case is:
The minimal value obtained here is:2
In the case of the Mishnah, we get p = 3/4, and therefore the vector P is (3/4,1/4), just as we saw in the Mishnah. In the case of the Tosefta as well (for claims of the whole and a third), the correct results are obtained: (5/6,1/6). In the case where nobody claims the whole thing, for example in the example we saw, (3/4,1/2), we get the following shares: (5/8,3/8). This too fits the calculation we made above. Apparently, then, we have obtained a surprising result, according to which the criterion presented in the Mishnah—namely, giving the part that is admitted and dividing the part that is disputed—fits perfectly with the solution that maximizes the satisfaction of the parties. But one must beware of hasty conclusions.
What happens in a situation of three claimants, as for example in the case of Maharil Diskin: (1,1/2,1/2)? Because of the symmetry between the two smaller claimants, it is clear that their shares too will be equal to each other. Therefore here too we have only one variable: p—the share of each of the smaller claimants. The share of the larger claimant is of course 1-2p.
The total dissatisfaction for this case is:
At the minimum value, one gets the following shares vector: (2/3,1/6,1/6).
This does not give us Maharil Diskin’s result. But perhaps he was mistaken, and this is the correct result? We saw above that it is proper to examine this case also in a way where initially there were two holding the cloak, with claims of the whole and half, and only afterward another one joined claiming half. The first two received (3/4,1/4), and if this result is correct then it follows that both of them give the next claimant the same fraction of their share: 1/12. But this is an implausible result. The larger claimant should have had to give him a larger fraction of his share than the fraction given by the smaller one.
The conclusion is that the criterion of minimizing dissatisfaction hit the correct result in the case of two holding a cloak, but only accidentally. It does not yield plausible results in the general case. I would add that this is rather to be expected, since the satisfaction of the claimants is irrelevant in this type of problem. The question here is who is right, or what it is proper to give each person, not that everyone should come out as satisfied as possible. In particular, given that in this case there are liars among them, it is even less plausible to demand maximum satisfaction. Later we will encounter a case for which satisfaction seems more relevant.
In the appendix, another possibility of dissatisfaction is examined, and the conclusion that dissatisfaction is probably not a good criterion for determining the shares vector is only reinforced.
Three Who Put Money into a Common Purse
Aumann, in the above-mentioned article, deals with a different Talmudic passage. The Mishnah in Ketubot 93a says:
If a man was married to three women and died, and the marriage contract of one was for one hundred zuz, of one for two hundred, and of one for three hundred, and there was only one hundred there, they divide it equally. If there were two hundred there, the one whose contract was for one hundred takes fifty, and the ones whose contracts were for two hundred and three hundred take three golden dinars each. If there were three hundred there, the one whose contract was for one hundred takes fifty, the one whose contract was for two hundred takes one hundred, and the one whose contract was for three hundred takes six golden dinars. And similarly, if three deposited money into a purse, whether it decreased or increased, they divide it in this manner.
The first clause of the Mishnah deals with a man who had three wives and died. Each had a different ketubah (marriage settlement): one for 100, one for 200, and one for 300. The estate does not suffice to cover all three ketubot, and therefore the question arises how to divide it among the three creditors. The division naturally depends on how much money was left after his death. The Mishnah discusses three examples, and taking into account that a gold dinar is worth 25 silver dinars, the division is as follows:
| Estate/claim | 100 | 200 | 300 |
|---|---|---|---|
| 100 |  |
|
|
| 200 | 50 | 75 | 75 |
| 300 | 50 | 100 | 150 |
In the latter clause the Mishnah adds that this is also the rule in the case of three who put money into a common purse (that is, invested different sums in a venture that lost money): they divide the losses equally. That is, if one invested 100, the second 200, and the third 300, then if 100 remains of the sum, they divide it equally; and if 200 or 300 remain, they divide it as in the last two rows of the table.
The medieval authorities (Rishonim) there are very perplexed by the explanation of the Mishnah’s laws. It is not clear what criterion the Mishnah used to determine the shares that each claimant receives. In the Talmudic discussion there as well, the proposal of the Mishnah is rejected and it is stated that this is Rabbi Nathan’s approach, but in practical Jewish law they rule like Rabbi. At the end of the day, Rabbi Nathan’s approach and that of the Mishnah are unclear, and there is no explanation for them in the medieval authorities (Rishonim).
In his article, Aumann proposed an explanation for the Mishnah’s method. He surveys there additional Talmudic passages that deal with division, including the passage of two holding a cloak, and argues that if one examines the divisions in the table above, one sees that every two claimants divide the sum that the two of them receive according to the rule of two holding a cloak. Let us take the second row as an example, and look at the first two claimants, the 100 and the 200. One sees in the table that together they receive 125. One of them claims 100 and the other claims the entire 125. According to the principle of two holding a cloak, the one who claims 100 concedes 25 to the other, and so the latter receives it. The remaining 100 they divide between them, and thus each gets another 50. The division between them is therefore (75,50). We got exactly the result that appears in the table. The same applies to the last two claimants, and this time let us examine the third row. From the table one sees that together they receive 250, while one claims 200 and the other claims the whole thing. The one claiming 200 concedes 50 to the other, and the remaining 200 they divide equally. We thus obtain a shares vector of (150,100), exactly as in the table.
To be sure, this analysis assumes the results determined by the Mishnah and finds logic in them after the fact. But Aumann shows there that it is possible to find the solution from the requirement that in all cases the division be carried out such that every pair divides, by the mechanism of two holding a cloak, the share that they jointly receive. If one imposes this requirement on the solution, it turns out that for all the cases there is a solution that satisfies it, and it is unique. Aumann also finds that solution, and in this way gives a constructive proof of its uniqueness. His proof is elegant and astonishingly simple, and is based on a hydrostatic model. He presents there a general method for calculating the solution and also finds solutions for additional cases.
A Discussion of the Logic of the Solution
Aumann argues implicitly that this ought to be the solution also for the case of two or three people holding a cloak, since in his view putting money into a purse or dividing an estate are no different from the case of dividing a found object. Aumann’s solution fits all the cases of two holding a cloak, since that is how it is constructed. As for a case with more claimants, we have no binding Talmudic determination, and therefore it is hard to test it. In the Ketubot passage there are examples of three claimants, and Aumann’s solution of course fits them, as we have seen. But it is by no means clear whether he is right in thinking that the division of an estate should be carried out like the division of a found object. We saw above that there is a significant difference between the cases. In the case of dividing a found object, some of the claimants are liars, and the aim of the division is to get as close as possible to the truth. As I remarked, for that reason too it is not plausible there to demand from the solution maximum satisfaction. By contrast, in the division of an estate it is clear that everyone is telling the truth, and the problem is only how to divide the small estate among them in the fairest possible way.
According to Aumann’s assumption, the solution in the division of a found object is like that in the division of an estate. Let us test his solution on Maharil Diskin’s example. Consider a case in which the estate is 100, and the three claimants demand (100,50,50), that is, the whole thing, half, and half. If you look at the article you will see that Aumann’s solution for this case comes out to (50,25,25), that is, half, a quarter, and a quarter. But we saw above that this solution is implausible, because it turns out that the third claimant receives his entire share only from the large claimant. I explained there that this solution assumes that the two smaller claimants are arguing only with the larger claimant and not with one another (if they unite and claim the whole thing, then half goes to the large claimant and the other half is divided between them), but this is an obviously unreasonable assumption. It is clear that each claimant is arguing with both of his counterparts, not only with one of them.
The conclusion is that Aumann’s solution does not give us a plausible result for the division of a cloak. It is not correct to require that the solution divide between every two claimants in the form of two holding a cloak. I corresponded with Aumann on this matter, and he replied that his goal was only to explain the puzzling method of the Mishnah, not necessarily to arrive at the logical solution. This is indeed an explanation of the Mishnah’s puzzle, but as noted, the solution does not seem plausible. The matter is especially problematic when one takes into account that today Aumann’s solution is taught in game theory courses as the sensible form of division for the case of dividing an estate or dividing a found object. That is, they assume that Aumann is proposing it as the recommended and reasonable solution for such cases.
Maximizing Satisfaction in the Division of an Estate
In principle there is some logic in proposing a solution of maximal satisfaction for the case of dividing an estate or putting money into a purse. As noted, in such cases it makes sense to demand maximum satisfaction, since all the claimants really are right (there is no liar here). Yet from looking at the table it is quite clear that the first row (where all the claimants receive equal shares) does not satisfy that requirement.
This is not the place to propose a general solution. I will check the maximum-satisfaction solution for the claim vector (100,200,300), with an estate of 300. In fact, this is a case of an estate of 1, for which the claims are: (1,2/3,1/3). ChatGPT gives me the following result:
- For normalized satisfaction: (5/14,8/21,11/42). Here the solution does not even preserve the order of the claims, that is, the claimant of the whole thing receives less than the one who claims 2/3.
- For unnormalized satisfaction one gets: (2/3,1/3,0). The order is correct, but the solution is not plausible.
Summary
At bottom, there is some kind of puzzle here. We saw above Maharil Diskin’s proposal, which in terms of its logic sounds the most plausible, and I explained that it can be generalized by induction. This seems to me the most reasonable approach, even though it is difficult for me to offer an explicit expression for the solution for any number of claimants and any claim vector. The calculation has to be done step by step as the number of claimants increases.
I should note that in our above-mentioned book we survey and propose further solution methods, including coalition considerations (what follows if one takes into account the possibilities available to each player to form coalitions with different counterparts) and an application of the Shapley value. There we showed in chapter 14 that the Jerusalem Talmud, in the passage on division of the estate, indeed uses coalition considerations. From chapter 15 onward we examined the Shapley value for the various division problems, and I will not enter into all that here.
Appendix: Normalized Dissatisfaction
At first glance, it seems more reasonable to examine relative satisfaction, that is, satisfaction in percentages rather than absolute satisfaction. In this case, the dissatisfaction of claimant i is: 
. And the total dissatisfaction of all the claimants is:
In the case of two claimants we get here:
(where p is the share received by claimant 1). Let us differentiate and set the derivative equal to zero:
[This is a minimum because the second derivative is: 2(
) > 0]
The result we obtained is:
In the case of the Mishnah we get a shares vector of (3/5,2/5), and in the case of the Tosefta (the whole and a third) we get (7/10,3/10), results that do not fit the Mishnah and the Tosefta.
In Maharil Diskin’s case, with three claimants of the whole thing, half, and half, the division we obtained above is (2/3,1/6,1/6). I asked ChatGPT to check this, and that is indeed what it got. As for the normalized calculation, I did not calculate it myself, but instead gave it directly to our teacher ChatGPT to calculate. For some reason, what comes out there is an equal division: (1/3,1/3,1/3). Very implausible.
It seems that maximizing satisfaction is not the criterion relevant from the standpoint of the Talmud.
Link to the post in PDF format
- The squaring is done despite the fact that all the dissatisfaction values are positive. This is because if we sum the dissatisfaction values themselves, taking into account that the total of the shares is 1, we get that the sum of dissatisfaction is constant for every division: it always comes out to the total of the claims minus 1. Therefore different vectors P will not change anything in this respect, and it follows that by means of this tool we will not be able to find the optimal division P.↩︎
-
One differentiates the dissatisfaction and sets it equal to 0. The second derivative is positive, and therefore this is a minimum.↩︎
Discussion
Oman's method for two claimants comes out like the Mishnah: 3/4 and 1/4. The proportionality here is only incidental.
It seems there is one detail in the found object case that the rabbi is not addressing.
After all, not just any person from the marketplace can claim ownership of the cloak; only someone in possession of the cloak has some basis for his claim.
In addition, from the standpoint of the claim itself, there is really no monetary claim by one against the other, since neither one claims that the other “owes” him a cloak or half a cloak. In principle, then, the court should not have had to entertain their claims were it not for the fact that both are physically holding it, which forces the court to rule a division, as we see in the case of that boat: when they are not holding it, the court does not intervene and they apply “whoever is stronger prevails.”
Accordingly, in a case where one claims “it is all mine” and two others claim “half of it is mine,” if the two claiming half were to come to court, there would be no need for any rule of division at all, since if each claims half there is no problem and no need for a compromise.
If so, the two who claim half certainly become like a single claimant against the one who claims the whole, and then the ruling should be like the Mishnah: the one claiming the whole should receive 3/4, and those claiming half should receive the 1/4 and divide it.
And certainly in this respect two people holding is different from the estate case, for there there is no confrontation at all, and therefore there is no logic in combining the claims of two against the one.
What do you mean? Why are you assuming that those two are not opposed to one another? So what if without the third party they would have had no dispute. There is a third party, and therefore they do have a dispute. What if there were three people each claiming half? Would every two be considered one claimant? That makes no sense.
And even if they are both one claimant, the result is not 3/4 and 1/4 to be divided, but 1/2 and two quarters. Together they receive half, like one who claims "it is all mine."
I did not understand the point about derara demamona. What does it have to do with this?
In the case where all three claim “half,” each of them necessarily claims that each of the other two has a quarter. For if one claims that he is a half-and-half partner with one of the other two, and that partner agrees with him, then we have one person extracting from two partners, and their status is like one who claims “it is all mine” against one who claims “half of it is mine,” since the force of partners is certainly no greater than that of a sole owner.
Therefore, since each claims “half is mine” and claims that the other two have a quarter each, it follows that all agree that each has a quarter in the cloak, and the entire dispute is only over the last quarter, which each one claims as “all mine.” That is the core of the quarrel, and clearly that quarter should be divided equally among the three.
This is not similar to the case where one claims “it is all mine” and two claim “half is mine,” where the agreement exists only between the two who claim half. For either way: if they claim that they are half-and-half partners between themselves, then they are considered like a single claimant asserting “it is all mine,” and that is not the case under discussion.
Rather, their claim must be, “half is mine and half belongs to the one claiming it is all mine,” and if so, both admit that the one claiming the whole owns half the cloak. It follows that he is certainly entitled to three quarters, just as if he were litigating against only one person claiming half, since they dispute with him only over the half-cloak that belongs to them, and over that half there is an internal dispute between them that they must divide.
A proof of this may be brought from the sugya of the doubtful heir and the yavam. It says there that when the doubtful heir and the sons of the yavam divide the grandfather’s estate, the doubtful heir claims that he is the son of the deceased, the brother of the yavam, and therefore inherits half; whereas the sons of the yavam claim that he is their brother, the son of the yavam, and therefore shares with them and takes one third. The rule is that he takes the third they admit to him, and they take the half he admits to them, and regarding the sixth over which they disagree, they divide it equally.
Now there, if each son of the yavam were considered a litigant in his own right, there would have been room to view this as one person claiming “half is mine” against two who claim “a third is mine.” But in practice they are treated as one side claiming two thirds, because there is no dispute among them.
What I prefaced regarding derara demamona was said only in order to distinguish this from the estate case, where one cannot view the two as a single claimant, for no agreement or joinder of claims exists between them. That is because there is no quarrel or dispute here at all about the right itself, as you noted; the whole question is only about the method of division, and there is no uncertainty or doubt about the essence of their claims.
It seems from the tone of the Maharil Diskin’s words that the main novelty of his view is that when Levi (“half is mine”) speaks about the other half as belonging to Shimon (“half is mine”) and Reuven (“it is all mine”), as an outsider to the dispute over that half, Levi sees it as Reuven’s quarter, and that is already embedded in his claim, which enables us to remove that quarter from the discussion entirely.
In other words, the novelty lies in the laws of admission, which lets us begin the algorithm from the parts that will remain in Reuven’s possession, rather than from a division among all of them according to their claims.
This is not so when money remains from an estate: even if 150 remain, the claim of the ketubah for one maneh relates to all 150, since this is a connection to repayment (as opposed to a claim of possession over a specific object), and one cannot begin the division algorithm with the outer portions over which there is supposedly no dispute.
Correction to the first section:
… then we have one person extracting from two partners, and their status is like one who claims “it is all mine” against one who claims “half is mine,” since the force of partners is certainly no greater than that of a sole owner.
It seems from your words (and from those of the Maharil Diskin) that it is obvious to you that the issue is exhausted by the relations among the different claims—that is, that “two are holding” is an ordinary case of monetary doubt (like two people claiming money without possession). But in truth that is not so, for the division in the case of “two are holding” depends on the fact that the two are in possession; since both are holding the cloak equally, the realization of the possessory standing of the different sides is through division. As all the Rishonim wrote there, it seems to me that this is an additional point that must be taken into account, no?
(I do not mean that this must necessarily produce a practical halakhic change in every case, for according to this, even in a situation where one says “it is all mine” and the other says “half is mine,” it ought to be that each takes half, since each is in possession of half, and the one claiming “it is all mine” is effectively claiming against his fellow’s half. Indeed, Tosafot HaRosh there noted this and wrote that regarding the half he admits, the other thereby becomes the possessor automatically—that possessory status is created only once there is a situation of monetary doubt. According to this, so too it should be in the other cases, but perhaps this has some significance regarding three who are holding a cloak, no?)
This is yeshiva nonsense. Possessory status has no significance where both sides are in possession, and the Gemara compares the Mishnah to Sumchus’s derara demamona and to the case of a maneh deposited with a third party, and also writes that in masrakh serukhei there should have been division were there uncertainty, and there are many more such places. Possession is a legal concept as against someone claiming money in your possession; it is not relevant to the discussion of “two are holding.” What the Rishonim mention about the “holding” is not connected to the law of possession, only to the fact that they are holding.
Please, do not be angry. All the Rishonim at the beginning of Bava Metzia wrote this, and they even explained that in the case of the maneh with the third party it is considered as though both are in possession of it. The intent, however, is not possession in the ordinary sense. After all, if two people lift up a found object together, they do not acquire it, and in the laws of possession the possession has to be of a kind fit for acquisition. Here there is a more tenuous degree of possession.
It is a bit hard to explain here, but the idea in brief is that the possession here is something like what the Rishonim wrote in Bava Metzia (97b): that Sumchus, who holds that in the case of “two are holding a cloak” they divide, says so because it is considered as though both are in possession of it. This is not merely a compromise or peacemaking between the parties; rather, the money lies between them and each has some force in it, and the realization of the force of both sides is through division. When a person alone holds money, he has in it the force of a full possessor; when there is money that stands in a state of derara demamona, where both sides have some claim that it is theirs, then by its nature it stands between them and both have force in it. (And the Rabbis too agree with the core of the idea, as is explicit in Tosafot on Bava Metzia 117b; they only hold that actual seizure causes the money not to be in doubt at all.) In the case of “two are holding,” a state is created of “both are in possession of it” (on the level of possession that exists in monetary doubt) from the very fact that both are holding it, even without derara demamona.
To the rabbis Ish Ha’emet and Netanel: please continue enlightening us with interesting discussions and fruitful discourse.
Thank you for serving as a living example of the intellectual bankruptcy to which the yeshiva world has deteriorated.
It is evident that you are truly angry—at me and at the entire yeshiva world.
Well then, permit me to ask you a simple question: the Rishonim say that in every case of monetary doubt, “it is considered as though both are in possession of it,” and by this they explain why a person is not believed with a definite claim in a case of division, since it is considered like extracting from one in possession. So this is not mere rhetoric but a halakhic determination. How do you explain that?
If you have some explanation, then it applies equally to “two are holding a cloak”: “two are holding” is simply the situation of “both are in possession of it” when it occurs in practice and not only conceptually and theoretically. And if you have no explanation, then you are free to declare that the Rishonim too suffered from intellectual bankruptcy, together with the Amoraim and the Tannaim and the prophets. Fortunate are we that we have merited this.
More power to the rabbi for the interesting and thought-provoking column!
One question on the substance of the discussion about three people holding a cloak.
Do two of those holding it, each of whom claims half, in effect admit that they are partners with one another in the cloak, so that the whole of their give-and-take is only against the third, who claims “it is all mine”?
If so, one might say that their status is that of two litigants appearing together and claiming the whole against someone who claims “it is all mine,” so why not treat them as one person for this purpose?
And if you would say that they do not mean a partnership, but rather that each claims “my half is mine, and the other half belongs to the one claiming it is all mine,” then both admit that half the cloak belongs to the one claiming it is all mine. If so, the entire dispute concerns only half the cloak, over which all three claim ownership. In that case, by law it would seemingly be proper to divide that half equally among the three.
Have I erred in this understanding, or is there some additional factor here that must be taken into account?
The Gemara assumes that each one disputes with the other two. You are asking how such a case could happen? The two who claim half say that they lifted it together with someone else and do not know who. And the one claiming the whole says that he lifted it alone.
One could use Shapley values, so that each of the claimants is allowed to collect his claim in a certain order, and then one averages the sums each one received. For example, in the case of the Mishnah in Bava Metzia, either the one claiming “it is all mine” collects first and gets everything, and the other gets half, or the one claiming “half is mine” collects first and gets half, and then so does the other. The average of that division is three quarters to the one claiming “it is all mine” and a quarter to the one claiming half, exactly as ruled. According to this, when one claims “it is all mine” and two claim half, the division would be half to the one claiming the whole and a quarter to the other two, a division that also seems reasonable intuitively. What do you think?
I mentioned that in the book we also did a Shapley calculation for the problem.
But the result you mentioned is דווקא not reasonable, as I explained in the column.
1. It is not clear why, in the case of the cloak as well, a proportional division does not seem reasonable. The argument that only the larger claimant should give only from his own share to the third claimant holds only if the division from the outset follows the Mishnah. Had the division from the outset been proportional even for two claimants, the claim of irrationality regarding three would fall away. There has to be consistency in the method so that absurd divisions are not produced. And it seems to me that proportional division always yields a sensible and consistent division. So why, in your view, does the Gemara offer other methods?
2. In civil law, in the case of an estate that does not cover all the claims, doesn't the court divide proportionally to the claims? It seems to me that it does, because that is the mathematical logic everyone accepts. So on what is the claim based that there is no logic in proportional division? And again, why do the Gemara and also the Maharil Diskin not adopt that method?