Yerushalmi Ketubot 10:4 ירושלמי כתובות י׳:ד׳

MISHNAH: If somebody was married to three women when he died, the ketubah of one was a mina⁵⁹100 zuz., of the other one 200, and of the other one 300. If the estate is only 100, they divide equally. If it was 200, the one who claims a mina takes 50, and those who claim 200 and 300 each take three gold denars⁶⁰1 gold denar was worth 25 silver denars. 3 gold denars equal 75 zuz, 6 gold denars 150 zuz.. If it was 300, the one who claims a mina takes 50, the one who claims 200 takes a mina⁵⁹100 zuz., and the one who claims 300 takes six gold denars⁶⁰1 gold denar was worth 25 silver denars. 3 gold denars equal 75 zuz, 6 gold denars 150 zuz.. Similarly, if three who invested together lost or gained they would split in this manner⁶¹This refers to the proportional distribution of the last case. Loss and gain of a stock company are distributed per share..⁶²This Mishnah poses difficult problems. First, it deals with a scenario that cannot arise since Mishnah 1 stated that a ketubah which precedes in time has to be satisfied from the estate before the later ketubah can be claimed. Therefore, the case of the Mishnah presupposes that the husband married three woman simultaneously. While this is possible in theory, it is excluded in practice. The problem is a practical one if applied to the case that the amount available in bankruptcy proceedings is not sufficient to cover all claims of equal rank.
At first glance, it seems that the Mishnah is inconsistent, applying three different rules to three different cases. Since the Mishnah must guide the judge to correctly apportion payment of claims c₁, c₂,c₃ if the available amount a < c₁+ c₂+ c₃, for all a between 0 and c₁+ c₂+ c₃, it is obvious that the Mishnah has to be explained by one single consistent formula or algorithm. In addition, the algorithm must be applicable to any number n of claims.
In the tenth Century, Saadya Gaon suceeded in reducing the number of different rules employed in the Mishnah to two (Oṣar Hageonim Ketubot p. 310). A complete discussion of the Geonic treatment of the Mishnah was given by I. Francus, שיטת הגאונים בפירוש משנה וברייתא, סיני פכו-פכז (תשס-תשסא) קצה-ריב (M. H. Katzenellenbogen Memorial Volume).
The first complete solution of the problem of finding a uniform rule for all three cases was given by R. J. Aumann and M. Maschler, Game Theoretic Analysis of a Bankruptcy Problem from the Talmud, J. Econom. Theory 36(1985) 195–213. In the framework of the theory of cooperative games, the authors show that the distribution of the Mishnah corresponds to the nucleolus strategy in game theory. Using only elementary mathematics, a single distribution formula covering all cases was given by M. Balinski in: Quelle équité, Pour la science 311 (September 2003) 82–87; What is Just? Am. Math. MONTHLY 112(2005), 502–511. While these authors succeed in explaining the Mishnah in itself, they do not explain the interpretation given to the Mishnah by the Talmudim. Nor do they explain why in both Talmudim the Mishnah is rejected out of hand. The explanation of the Babli is in language close to the Yerushalmi but translated into mathematical formulas the results are quite different; in both Talmudim the proposed explanation results in a procedure which is not monotone; i. e., it is possible that if the available amount a is increased, the payouts for some claimants are decreased. The solutions both of Aumann/Maschler and Balinski are monotone, increasing with a.
Balinski shows that the Mishnah can be understood in terms of a different distribution problem in Mishnah Baba meṣiʻa 1:1, as indicated by Alfasi (Ketubot 10, 51b in the Wilna ed.). Since this problem is also the basis of the arguments in both Talmudim, Balinski’s method is explained first. In Baba meṣiʻa there are two claimants for an amount of 1. The first claims it all, the second claims ½. Both claims are of the same status. Then the first is awarded ³/₄, the second ¹/₄. While the claims are in ratio 2:1, the awards are in ratio 3:1. The reason given is the rule ascribed to Symmachos, a student of R. Meïr, that “money in doubt is split evenly” [Babli Baba meṣiʻa 2b, Baba qama 35b; Yerushalmi Baba meṣiʻa 8:5 (11d 1. 18); cf. Chapter 2, Notes 9 ff.]. Since the second claimant asks only for ½, the other half is awarded to the first claimant. The first half is in doubt; therefore it is split evenly between the two parties, resulting in the awards mentioned.
In order to arrive at a mathematical formulation, assume that n claims c_i are advanced,
c₁< c₂ < … < c_n.
If several claims were identical they would be consolidated into a single claim and the award split evenly between the participants. For any amount a available for distribution, the payout for claimant i is p_i(a),
p₁(a)+ p₂(a)+ … + p_n(a)= a.
We also introduce the sum of the claims, S =c₁ + c₂+ … + c_n and M= S/2. For n = 2, Symmachos’s argument determines the payouts uniquely:
If a ≤ c₁, the entire sum is in dispute,
p₁(a)= p₂(a)= a/2.
Therefore, p₁(c₁)= p₂(c₁)= c₁/2.
If c₁ < a ≤ c₂, the amount a - c₁ belongs to claimant 2 and
p₁(a)= c₁/2,
p₂(a)= a-c₁+c₁/2= a-c₁/2,
and p₁(c₂) = c₁/2,p₂(c₂)= c₂-c₁/2.
If c₂< a, both claimants have equal rights to the excess over c₂,
p₁(a)= c₁/2+(a - c₂)/2
p₂(a) = c₂-c₁/2+(a-c₂)/2=(a+c₂-c₁)/2.
If a = c₂+c₁ = S, all claims are satisfied in full. Using the function min(x,y) = smaller of x or y, Balinski did express the formulas given above by
(1) p_i(a) = min(c_i/2, λ) where λ is chosen so that p₁(a)+ p₂(a)= a;i = 1,2.
An inspection of the formulas shows that the distribution is symmetric about the middle value of the claims:
(2) p_i(M-x)+ p_i(M+x)= c_i.
This means that the distribution for M ≤ a ≤ S is determined if it is determined for 0 ≤ a≤M. In the Mishnah here, it should be sufficient to indicate the distribution up to an estate of 300 if the sum of claims is 600. But the generalization of the procedure for n = 2 to n > 2 is not trivial. As Balinski points out, an arbitrary distribution function which yields the given values for a = c_i and satisfies (2) is compatible with the Mishnah. His own solution is to postulate (1) for i = 1, … n. Then (2) is satisfied and the distribution is monotone. For example,
{p₁(400),p₂(400),p₃(400)} = {100-p₁(200),200-p₂(200),300-p₃(200)} = {50,125,225}.
The problem with this solution, as with most other solutions proposed in the last 1000 years, is that the Talmudim take another view. The solution of the Babli is easily described as a recursive computational procedure. We assume that p₁(a),…,p_i-1(a) have been determined; leaving an amount a_i to be distributed. We determine p_i(a_i). If c_i ≥ a_i, p_j(a_i) = a_i/(n-i) for j = i, … n. Otherwise, all claims larger than c_i are consolidated into one claim C_i = c_i+1 + … + c_n and p_i(a_i) is the payout assigned to claimant #i in the two-person distribution problem for claims c_i, C_i and amount a_i. The fictitious second claimant then receives P_i(a_i) = a_i+1 and the process continues.
In the example of the Mishnah, c₁ = 100, c₂ = 200, c₃ = 300. For a = 100, by the first alternative p_j(100) = 100/3 for all j. For a = 200, c₁ = 100, C₁ = 400. Therefore, 100 is assigned to the second party and 100 is split evenly. This yields p₁(200) = 50, P₁(200) = 150. Since 200 > 150, the latter amount is split evenly, resulting in p₂(200) = p₃(200) = 75. For a = 300, one sees that p₁(300) = 50, P₁(300) = 250; p₂(300) = 100 = 200/2, p₃(300) = 50 + 200/2 = 150. This explains the numbers stated in the Mishnah, but what about a = 101? In that case, p₁(101) = 50, P₁(101) = 51; p₂(101) = p₃(101) = 51/2 = 25.5. This clearly is unacceptable; the procedure of the Mishnah has to be rejected.
The explanation of the Yerushalmi (cf. Note 69) is not so clearly stated but it seems to be the following: The procedure also is recursive. If c_i ≥ a_i,p_j(a_i) = a_i/(n-i) for j = i, … n. Otherwise, one considers all two-person problems between claimants i and i+1,…, n. In praxi, this means that one has only to split between c_i and c_n. If c_i < a_i < c_n then p_i(a_i) = c_i/2. If c_n < a_i then write a_i = c_n + δ. Claims n and i overlap to the amount c_i - δ. This means that
p_i(a_i) = δ + (c_i - δ)/2 = (c_i + δ)/2.
For a ≤ M, the numbers obtained for Babli and Yerushalmi coincide; the Yerushalmi’s method also is not monotone. The results for a > M disagree. For example, for the data of Mishnah 4, the Yerushalmi gives
p₁(400)=100, p₂(400)=100, p₃(400)=200 while the Babli gives
p₁(400)=50, P₁(400)=350; p₂(400)=100, p₃(400)=250.

HALAKHAH: “If somebody was married to three women,” etc. Samuel said, sharpness of the judges⁶³Samuel seems to indicate that the court has to investigate and choose the method of distribution which seems most appropriate for the case before them, to whom to give real estate and to whom money or securities.
The translation of שוחדא is tentative. In the Babli, the word appears as שודא which either is derived from the Aramaic root שדי “to throw” (interpreted to mean that the court may “throw” the properties to the party it feels has the best claim) or it is identical with Galilean שוחדא because of the disappearance of ح in Babylonian speech. I am proposing to derive שוחד̇א from Arabic شحذ “to sharpen (a knife)”, Mishnaic Hebrew שחז.. As in the case of two mortgages written on one field, to whom the court wants to hand it over in possession they hand it over⁶⁴If the field is too small to be useful as property if divided up. The holder of the second mortgage (assuming both were written on the same day) has to be indemnified with money; this is Samuel’s position in Babli 94a.. The Mishnah disagrees with Samuel: “If the estate is only 100, they divide equally.⁶⁵Does this not indicate that all three women have to be treated exactly equally?” Was this said for the sharpness of the judges⁶⁶The Mishnah speaks only about the value, not about the mode of distribution.? And did we not state: If the heirs inherited an instrument of indebtedness, the male firstborn takes a double portion⁶⁷Since a documented claim is as good as collected, such a document is counted as ready money, not as future claim (cf. Note 50). In Babli Baba batra 124a and in Tosephta Bekhorot 6:17 the firstborn is given the right to refuse a double portion of secured future claims and then not pay double part of future secured claims against the estate.; and if an instrument of indebtedness was presented to them, the male firstborn pays a double portion. Was this said for the sharpness of the judges⁶⁸This is not a case that allows of judicial discretion.? Rebbi Abin in the name of Samuel: There is no difference, whether two mortgages were written on one field or one mortgage was written on two fields, on whom the court decide they hand it over in possession.

Samuel said, if they empower one another⁶⁹To deal separately with the co-wife claiming the smallest amount. In the Babli, 93a, Samuel explains that the second wife empowered the third to represent her also; cf. Note 62. Unless all parties agree to the procedure, the estate has to be distributed proportional to the claims. It is difficult to see why all parties should agree; the procedure of the Mishnah is eliminated from practical use.. If the third empowered the second to deal with the first. She said to her, is not a mina due to you⁷⁰If the amount available is larger than a mina, the first has to split the mina only with one co-wife.? Take 50 and leave!

“Similarly, if three who invested together lost or gained they would split in this manner.” ⁷³This paragraph has a differently worded parallel in Baba qama 4:1 (by a different editorial team) which explains the somewhat cryptic wording here:
תַּמָּן תְּנִינָן. הָאַחֲרוֹן נוֹטֵל מְנָה וְשֶׁלְּפָנָיו חֲמִשִּׁים זוּז וּשְׁנַיִם הָרִאשׁוֹנִים דִּינָרֵי זָהָב. רִבִּי שְׁמוּאֵל בְּשֵׁם רִבִּי זְעֵירָא, וְכֵן לְשָׂכָר. אָמַר רַבִּי יוֹסֵי. הָדָא דְּרִבִּי זֵירָא פְּלִיגָא עַל דְּרַבִּי לָעָזָר. אָמַר רַבִּי מַנִּי. קַשְׁיָתָהּ קוֹמֵי רְבִּי יוּדָן. אָמַר לִי. לֹא מוֹדֶה רְבִּי לָעָזָר שֶׁאִם הִתְנוּ בֵּינֵיהֶן שֶׁזֶּה נוֹטֵל לְפִי כִּיסוֹ וְזֶה נוֹטֵל לְפִי כִּיסוֹ. שְׁוָרִים כְּמֻתָּנִים הֵן. חָזַר וְאָמְרָה קוֹמֵי רְבִּי יוֹסֵי. אָמַר לֵיהּ. בְּפִירוּשׁ פְּלִיגֵי. רִבִּי לָעָזָר אָמַר. סְתַמָּן חוֹלְקִין בְּשָׁוֶה. רְבִּי זְעֵירָא אָמַר, סְתַמָּן זֶה נוֹטֵל לְפִי כִּיסוֹ וְזֶה נוֹטֵל לְפִי כִּיסוֹ.
וְכֵן שְׁלֹשָׁה שֶׁהִיטִּילוּ לַכִּיס פָּחֲתוּ אוֹ הוֹתִירוּ כָּךְ הֵן חוֹלְקִין. אָמַר רִבִּי בּוּן. נִרְאִין דְּבָרִים אִם נָֽטְלוּ מַרְגָּלִית. דְּיָכוֹל מֵימַר לֵיהּ אִילּוּלֵי עֲשַׂרְתָּא דֵינָרַיי לֹא הֲוִיתָה מְזַבִּין כְּלוּן. אֲבָל דָּבָר שֶׁדַּרְכּוֹ לֵחָלֵק מֵבִיאִין לָאֶמְצָע וְחוֹלְקִין. אָמַר רִבִּי לָֽעְזָר וַאֲפִילוּ דָּבָר שֶׁדַּרְכּוֹ לֵחָלֵק. דְּיָכִיל מֵימַר לֵיהּ אַתְּ פְּרַגְמָטַּיָּא דִידָךְ סַגִּין וְאַתְּ מַנְעָא מַזְבִּנְתָּא. אֲנָא פְּרַגְמָטַּיָּא דִידִי קָלִיל וַאֲנַא הֲפַךְ וּמִתְהַפֵּךְ בְּדִידִי וּמַטִּי בָךְ. עַד כְּדוֹן בְּשֶׁהָייְתָה פְּרַגְמָטַּייוֹ נְתוּנָה כָאן. תָֽיְתָה פְּרַגְמָטַּייוֹ נְתוּנָה בְּרוֹמֵי. דְּיָכִיל מֵימַר לֵיהּ. עַד דְּאַתְּ סְלִיק לְרוֹמֵי אֲנַא הֲפַךְ וּמִתְהַפֵּךְ בְּדִידִי וּמַטִּי בָךְ.
"Similarly, if three who invested together lost or gained they would split in this manner." Rebbi Abun said, the statement looks reasonable if they bought a precious stone because he can say to him, without my ten denars you could not have bought anything. But anything that usually is split {smaller units that can be bought with less capital) one adds together and splits (proportionally to the capital invested). Rebbi Eleazar says, even things that usually are split [are divided evenly], because he can say to him, you have a lot of merchandise and you have difficulty selling it. 1 have little merchandise and turn it over rapidly and make as much as you do. So far if his merchandise was here. What if his merchandise was in Rome? He can say to him, by the time you went to Rome, I turn mine over rapidly and make as much as you do.
There, we have stated: “If [the ox] gored an ox worth 200, the last one takes 100, the one before him 50, and the two first ones a gold denar.” Rebbi Samuel in the name of Rebbi Ze‘ira: The same holds for earnings. Rebbi Yose said, the statement of Rebbi Ze‘ira disagrees with Rebbi Eleazar. Rebbi Mani said, I asked this before Rebbi Yudan. He said to me: does Rebbi Eleazar not agree if they contracted between themselves that each can take according to his contribution? Oxen are as if contracted. He turned around and said this before Rebbi Yose, who answered him, they disagree explicitly: Rebbi Eleazar said, if nothing was said, they split evenly; Rebbi Ze‘ira said, if nothing was said, each takes according to his contribution. {This paragraph is a direct quote from Ketubot since the quote from “there” refers to here, Baba qama 4:1}. Rebbi Eleazar said: You say that in case the tetradrachma lost or added value. But for earnings or losses they split evenly⁷⁴Only results of financial operations are split per share. Results of personal effort by the shareholders are split evenly; they are socii pro aequa parte of Justinian’s legislation.. That is difficult! One gave 100 denars and the other gave ten, and you say so? The colleagues say, he may say to him, by my contribution the merchandise rose. So far, if they ran after little merchandise⁷⁵Expensive items, as explained in the parallel text.. If there was much merchandise? Rebbi Abun bar Ḥiyya said, he may say to him, by the time you sold one piece, I am selling ten. That is, at a nearby place. If it was far away? Rebbi Hila said, he may say to him, by the time you go and return once, I am going and returning ten times. There⁷⁶Mishnah Baba qama 4:1. Ex. 21:35 decrees that if an ox kills another ox, the owner of the attacking animal and the owner of the victim become co-owners of both the living and the dead animal. If the attacking ox attacks another one before it can be slaughtered, its co-owners now become co-owners with the owner of the second victim. The case quoted in the Mishnah is about an ox worth 200 zuz which attacks three oxen, each of which was worth 200 zuz but whose carcasses are not worth anything after the attack. Then the owner of the ox which was killed last takes half the combined value of the attacker and its victim as prescribed in the verse, 100 zuz. The owner of the second ox has a claim of 50% of the value of the combined value of the attacker and its victim. But since only 100 zuz remain of the value of the attacker and the victim is not worth anything, the second owner gets only 50 zuz. By the same argument, the owner of the first ox gets 25 zuz, the same amount as the owner of the attacker retains., we have stated: “If [the ox] gored an ox worth 200, the last one takes 100, the one before him 50, and the two first ones a gold denar.” Rebbi Samuel in the name of Rebbi [Ze‘ira]: The same holds for earnings⁷⁷R. Ze‘ira interprets the Mishnah in Baba qama that in the company created by the attacking ox, the owner of the third victim contributed 200, the owner of the second 100, the owners of the first and of the attacker 50 each, and 50% of the value was lost as prescribed in the verse. Therefore, the Mishnah precribes proportional appropriations for a case which is not a financial operation; the Mishnah in Ketubot can be interpreted that in a company, all gains and losses have to be distributed in proportion to the capital invested.. Rebbi Yose said, the statement of Rebbi Ze‘ira disagrees with Rebbi Eleazar. Rebbi Mana said before Rebbi Yudan: Does Rebbi Eleazar not agree if they contracted between themselves that each can take according to his contribution⁷⁸Since in money matters, biblical law is not prescriptive.? Oxen are as if contracted. He turned around and said, they disagree explicitly: Rebbi Eleazar said, if nothing was said, they split evenly; Rebbi Ze‘ira said, if nothing was said, each takes according to his contribution⁷⁹Since R. Ze‘ira is the later and higher authority, the Yerushalmi decides that all distributions must be proportional. The Babli, 93b, quotes the opinion attributed here to R. Eleazar in the name of Samuel as interpreted by Rav Hamnuna, it clearly decides that all distributions must be split equally..