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How the Talmud Divides an Estate Among Creditors Steve Schecter North Carolina State University 1 2 The Talmud consists of: • Mishna (c. 200 AD), the first written compendium of Judaism’s Oral Law. • Gemara (c. 500 AD), a record of discussion by rabbis about the Mishna. First printed in Italy around 1520. Today’s printings: 60 tractates in 20 volumes occupying one meter of shelf space. 3 Tractate Page References to legal codes Modern glosses Tosefot (successors of Rashi) (12th-13th cent.) Chapter Number Chapter Name Mishna (~200) Gemara (~500) Rashi (11th cent.) Mishna Cross-references to Talmud Gemara Other medieval commentary Modern glosses 4 A Problem from the Talmud A man dies leaving • an estate of size e; • debts to Creditors 1, . . . , n of d1, . . . , dn; • e < d1 + · · · + dn. How much should each creditor get? A Mishna (Tractate Ketubot 93a): Assume d1 = 100, d2 = 200, d3 = 300. (1) If e = 100, each creditor gets 33 1/3. (2) If e = 200, creditor 1 gets 50, creditors 2 and 3 get 75 each. (3) If e = 300, creditor 1 gets 50, creditor 2 gets 100, creditor 3 gets 150. A literature stretching across 1500 years deals with the question: what algorithm is this Mishna describing? Of course, as in any legal system, the answer must be based on other Talmudic principles. 5 Ideas from the talmudic literature: (1) The Mishna is wrong. (This is the majority view.) (2) There are special circumstances that have not been explained. (3) There is an (unconvincing) rational explanation. (4) The text is corrupt. Alfasi (11th century): “My predecessors discussed this Mishna and its Gemara at length, and were unable to make sense of it.” What are some rational ways to divide an estate among creditors? An estate division problem is a pair (e, (d1, . . . , dn)) with the following properties: (1) 0 ≤ d1 ≤ d2 ≤ · · · ≤ dn. (2) Let d = d1 + . . . + dn. Then 0 < e < d . A division of the estate is (x1 , . . . , xn) with 0 ≤ xi for all i and x1 + · · · + xn = e. Some ideas: 6 Proportional Division. Assign to creditor i the amount (di/d)e. This method treats each dollar of debt as equally worthy. Mishna appears to use this idea when e = 300. Secular legal systems typically follow this idea. Equal Division of Gains. Assign to each creditor the amount e/n. This method treats each creditor as equally worthy. Mishna appears to use this idea when e = 100. Not sensible if d1 < e/n (i.e., not sensible for large estates). Constrained Equal Division of Gains. Give each creditor the same amount, but don’t give any creditor more than his claim. In other words, choose a number a such that min(d1, a) + min(d2, a) + . . . min(dn, a) = e. Then assign to creditor i the amount min(di , a). The number a exists and is unique because for fixed (d1, . . . , dn), the left-hand side is a function of a that maps the interval [0, d] onto itself and is strictly increasing on this interval. This rule was adopted by Maimonides (12th century). It is inconsistent with our Mishna (produces equal division in all our cases). Equal Division of Losses. Make each creditor take the same loss. The total loss to the creditors is d − e, so assign to creditor i the amount di − (d − e)/n. Not sensible if d1 < (d − e)/n (i.e., not sensible for small estates). Constrained Equal Division of Losses. Make each creditor take the same loss, but don’t make any creditor lose more than his claim. 7 References: R. J. Aumann and M. Maschler, “Game Theoretic Analysis of a Bankruptcy Problem from the Talmud,” J. Economic Theory 36 (1985), 195–213. M. M. Kaminski, “ ‘Hydraulic’ rationing,” Mathematical Social Sciences 40 (2000), 131–155. R. J. Aumann, “Game theory in the Talmud,” Research Bulletin Series on Jewish Law and Economics. 8 Some Other Mishnas Tractate Baba Metzia 2a: “Two hold a garment; one claims it all, the other claims half. Then the one is awarded three-fourths, the other one-fourth.” Explanation (Rashi, 11th century): The one who claims half concedes that half belongs to the other. Therefore only half is in dispute. It is split equally. Tractate Yevamot 38a: B dies childless. His widow marries his brother, C (as required by Deuteronomy 25:5-6). C already has two sons, c1 and c2, by his first wife. Eight months later the widow gives birth to a son, b, whose father is therefore doubtful. Next C dies. Finally, A, the father of B and C dies. Question: How is A’s estate to be divided among his grandchildren b, c1 , and c2 ? b says: Half goes to A’s son B and half to A’s son C. I am B’s only son, so I get his half. C’s half should be divided between c1 and c2 . c1 and c2 say: B had no children, and C had three sons. Therefore the entire estate goes to C, and then is divided equally among the the three grandchildren. The decision: c1 and c2 are treated as one claimant, b as another. The 1/2 of the estate that b concedes is not his goes to c1 and c2. The 1/3 of the estate that c1 and c2 concede is not theirs goes to b. The remainder of the estate, 1/6, is split equally: 1/12 to c1 and c2, 1/12 to b. Thus b gets 5/12 of the estate, and c1 and c2 get 7/12 to split. 9 Neither Mishna treats a situation exactly analogous to an estate with creditors: there all claims are valid, in these two Mishnas both claims cannot be valid. Nevertheless, applied to an estate with two creditors, we get: Contested Garment Rule. Consider an estate division problem with two creditors: 0 ≤ d1 ≤ d2, d = d1 + d2, 0 < e < d . Creditor 2 concedes max(e − d2 , 0) to creditor 1. Creditor 1 concedes max(e − d1 , 0) to Creditor 2. The remainder of the estate, e − max(e − d1, 0) − max(e − d2, 0), is divided equally. Thus Creditor 1 receives 1 max(e − d2, 0) + (e − max(e − d1, 0) − max(e − d2, 0)). 2 Creditor 2 receives 1 max(e − d1, 0) + (e − max(e − d1, 0) − max(e − d2, 0)). 2 Note that if e is small (e ≤ d1), the estate is split equally. Is the Contested Garment Rule relevant to our Mishna? 10 Aumann and Maschler’s Observation Back to our Mishna: d1 = 100, d2 = 200, d3 = 300. (1) If e = 100, each creditor gets 33 1/3. (2) If e = 200, creditor 1 gets 50, creditors 2 and 3 get 75 each. (3) If e = 300, creditor 1 gets 50, creditor 2 gets 100, creditor 3 gets 150. Aumann and Maschler’s observation: Each of these divisions is consistent with the Contested Garment Rule in the following sense: If any two creditors use the Contested Garment Rule to split the amount they were jointly awarded, each will get the amount he was actually awarded. In an estate division problem (e, (d1, . . . , dn)), a division (x1, . . . , xn) of the estate is consistent with the Contested Garment Rule if, for each pair (i, j), (xi, x j ) is exactly the division produced by the Contested Garment Rule applied to an estate of size xi + x j with debts di and d j . Theorem (Aumann-Maschler). In any estate division problem, there is exactly one division of the estate that is consistent with the Contested Garment Rule. 11 Another Look at the Contested Garment Rule Contested Garment Rule. Consider an estate division problem with two creditors: 0 ≤ d1 ≤ d2, d = d1 + d2, 0 < e < d . Creditor 2 concedes max(e − d2 , 0) to creditor 1. Creditor 1 concedes max(e − d1 , 0) to Creditor 2. The remainder of the estate, e − max(e − d1, 0) − max(e − d2, 0), is divided equally. More detailed description: (1) If e ≤ d1, nothing is conceded, so everything is split: e Creditor 1: . 2 e Creditor 2: . 2 Each additional dollar of estate value produces an equal gain for each creditor. 12 (2) If d1 < e ≤ d2, e − d1 is conceded to Creditor 2, nothing is conceded to Creditor 1, and the remainder, d1, is split. Creditor 1: d1 . 2 d1 Creditor 2: (e − d1) + . 2 When e = d1, the estate is split, each Creditor has a gain of d1/2. Thereafter each additional dollar of estate value goes to Creditor 2. When e reaches d2, d Creditor 1 gets d1/2 and Creditor 2 gets d2 − 21 . Thus each Creditor has a loss of d1/2. Previously Creditor 2’s loss was larger. (3) If d2 < e ≤ d1 + d2, e − d1 is conceded to Creditor 2, e − d2 is conceded to Creditor 1, and the remainder, e − (e − d1) − (e − d2) = d1 + d2 − e, is split. d1 1 1 + (e − d2). 2 2 2 1 d1 1 Creditor 2: e − d1 + (d1 + d2 − e) = d2 − + (e − d2). 2 2 2 Creditor 1: e − d2 + (d1 + d2 − e) = In other words, the part of the estate above d2 is split equally. The two creditors’ losses remain equal. 13 Conclusion: The Contested Garment Rule linearly interpolates between Equal Division of Gains for e ≤ d1 and Equal Division of Losses for d2 ≤ e. At e = d1, both creditors gain d1/2; at e = d2, both creditors lose d1/2. “More Than Half is Like the Whole” The Contested Garment Rule is perhaps related to the Talmudic principle that “more than half is like the whole.” Example: Normally a lender has an automatic lien on a borrower’s real property. However, if the property is worth less than half the loan and the borrower defaults, the lender may not take the borrower’s property (Erakhin 23b). Rashi explains: since the property is grossly inadequate to repay the loan, the loan is presumed to have been made “on trust,” so the lender has no lien on the borrower’s property. In other words: if the property is worth less than half the loan, you cannot rely on the loan’s being repaid, so any repayment you do get is a gain relative to your expectation. If the property is worth more than half the loan, you expect the loan to be repaid, so any repayment you do not get is a loss relative to your expectation. 14 Interpreting the Contested Garment Rule in Glassware d1/2 d1/2 d2/2 d2/2 If an amount of liquid e is poured into this glassware, it will divide itself between the two creditors according to the Contested Garment Rule. 15 d1/2 d2/2 d2/2 d1/2 d1/2 d1/2 d1/2 d2/2 d2/2 d1/2 d2/2 d2/2 16 Proof of the Aumann-Maschler Theorem Given d1, . . . , dn, construct the following glassware: d1/2 d1/2 d2/2 dn/2 d2/2 dn/2 ... Pour in an amount e of liquid. It will divide itself among the n creditors in a way that is consistent with the Contested Garment Rule (since the glasses for each pair of creditors have the the same height of liquid). This division is unique: if we raise the height in one glass, we must raise the height in all, and the total amount of liquid will increase. 17 The Aumann-Maschler Theorem and Game Theory A cooperative game consists of (1) a set of players {1, . . . , n}. (2) a value V to be distributed among the players. Let S be a subset of {1, . . . , n} (a coalition). S can get for itself an amount v(S) no matter what. Assumptions: / = 0. (1) v(0) (2) v({1, . . . , n}) = V . (3) If S1 and S2 are disjoint, then v(S1) + v(S2) ≤ v(S1 ∪ S2). An allocation is a vector x = (x1, . . . , xn) such that all xi ≥ 0 and x1 + · · · + xn = V . Problem: Choose the allocation. Given an allocation x, the coalition S achieves the excess e(x, S) = ∑ j∈S x j − v(S). Idea: Coalitions with low excess will complain that they have been treated unfairly and won’t agree to the allocation. Choose x to minimize the complaints. 18 More precisely, given an allocation x, calculate all 2n − 2 excesses e(x, S). (We ignore the empty set and the set {1, . . . , n}.) Order them from smallest to largest to n form an excess vector e ∈ R2 −2. Given two excess vectors we can ask which precedes which in the lexicographic ordering. • Example: (1, 2, 4, 5) precedes (2, 2, 2, 7). • Example: (2, 2, 2, 7) precedes (2, 2, 3, 6). Definition. The nucleolus of a cooperative game is the allocation whose excess vector comes last in the lexicographic ordering. Theorem. Every cooperative game has a unique nucleolus. To find the nucleolus of a cooperative game, start with any allocation and adjust it to make one that follows it in the lexicographic ordering. 19 Example. A man dies leaving an estate of 200. There are three creditors with claims of 100, 200, and 300. Any coalition can guarantee itself whatever is left after those not in the coalition are paid in full. Start with proportional division and calculate excesses. S v(S) e(x, S) x = (33 13 , 66 23 , 100) {1} 0 x1 33 13 {2} 0 x2 66 23 {3} 0 x3 100 {1,2} 0 x1 + x2 100 {1,3} 0 x1 + x3 133 31 {2,3} 100 x2 + x3 − 100 66 32 The excess vector is (33 23 , 66 23 , 66 32 , . . .). To improve it in the lexicographic ordering we must take from Player 2 or Player 3 or both and give to Player 1. The most we can transfer before we start to go back down in the lexicographic ordering is 2 1 2 1 (66 − 33 ) = 16 . The excess vector will then be (50, 50, . . .). 2 3 3 3 We try reducing x2 and x3 by 8 13 each, and increasing x1 by 16 23 . 20 e(x, S) x = (33 13 , 66 32 , 100) x = (50, 58 13 , 91 32 ) 1 0 x1 33 31 50 2 0 x2 66 32 58 13 3 0 x3 100 91 23 {1,2} 0 x1 + x2 100 108 31 {1,3} 0 x1 + x3 133 13 141 32 50 {2,3} 100 x2 + x3 − 100 66 32 S v(S) The excess vector is now (50, 50, 58 13 , 91 32 , . . .). We know we can’t improve it by changing x1, and we can’t improve the start (50, 50, . . .). However, we can improve the third entry by taking from Player 3 and giving to Player 2. The most we can transfer before starting to go down in the lexicographic ordering is 21 (91 23 − 58 31 ) = 16 23 , so we do this. We’ve now done the best we can with the allocation to each individual, so the result is the nucleolus: x = (50, 75, 75). This is also the allocation the Talmud proposed for this problem. Theorem (Aumann-Maschler). In any estate division problem, the unique allocation that is consistent with the Contested Garment Rule is also the nucleolus of the associated cooperative game. 21 Final Remark In an estate division problem with n ≥ 3 and (d1, . . . , dn) fixed, there is typically no value of e for which the division that is consistent with the Contested Garment Rule is also the proportional division (I think!). Recall that our Mishna gave proportional division for one case. The Aumann-Maschler solution requires us to believe that this is a coincidence. Is this plausible?