# a bankruptcy problem in the talmud

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ABankruptcyproblemintheTalmud Idliketostartmylecturewithaquestion.Heresthequestion. A man dies, leaving debts totaling more than his estate. How should the estate be dividedamongthecreditors? Ok,letsmakethisproblemmorespecific. Letssay,amanowesof100,200and300,butdieswithinsufficientfunds/money topayeveryoneback.Thenhowshouldhisestatebedivided? AndIwillgiveyousomeoptions/alternatesforyoutochoose. Equaldivision(regardlessofthedebtsize). Proportionaldivision(accordingtothedebtsize). Itdepends(onsomeotherconditionssuchasthesumofmoneyandetc.). Illgiveyou15secondstomakeyoudecision Andwhothinkthefairdivisionisreasonable,pleaseraiseyourhands? Andwhothinktheproportionaldivisionismorereasonable,pleaseraiseyourhands? Thenwhothinkitdepends,suchasitdependsonthesizeofestate? Ihavetosaythattheremightnotbeonlyonecorrectanswer.Justlikesomeonesays thatthereisntasinglerightwaytoapproachanyproblems. For people who chose proportional division, and I find that most/all of you prefer proportional division that depends on the debt size. You may think its just kind of pay what you ordered method in restaurants where guests put money based on foodtheyordered. And I find that no one/only XXX chose equal division. But equal division is quite common particularly in some situations such as the heritage division among the youngchildreninafamily. Butinthistutorial,Idliketoshowyouanotherwaytodividetheestate,andgiveyou the principle behind the division method. An ancient code which was used almost 2000 years ago, called Babylonian Talmud, gives an explanation of the above bankruptcyproblemwhichisbasedonthesizeofestate.TheTalmudisarecordof discussions about Jewish laws and customs. The Talmud answer is not so obvious, and no one could give the rules/patterns behind it for all most 2 millenary. Until recent years, Robert Aumann just figures out the mathematical principles of the Talmudanswer.Andthatisourtopicinthistutorial.

TheNobelPrizeinEconomicsof2005wasawardedtoRobertJ.AumannandThomas C. Schelling "for having enhanced our understanding of conflict and cooperation throughgametheoryanalysis" Letsgobacktothequestionagain. A man owes debts of 100, 200, and 300, but dies with insufficient funds to pay everyone.Howshouldhisestatebedivided? The Talmud offers answers through three examples. The three cases are when the estatesizeis100,200and300. Inthefirstcasewhentheestatesizeis100,theTalmudawards331/3toeachparty. The division suggests a principle of an equal division. But strangely this is not the sameideausedintheothercases. 100:331/3,331/3,331/3 EqualDivision Inthethirdcaseof300,theTalmudoffersadivisionof50,100,and150.Themath here is a proportional division based on the size of the debt. In modern times, proportionaldivisionholdswideappealamonglawyersandeconomics.Atthispoint youmightaskwhythe300caseistreateddifferentlytothe100case. ProportionalDivision 300: 50, 100, 150 Ifthatquestionbothersyou,thengetreadyforanothersurpriseinthedivisionfor 200.Inthiscase,theestateissupposedtobedividedas50,75and75.Neitheran equaldivisionnoraproportionaldivision!Whyshouldthesecondandthirdcreditors begiventhesameamountofmoney?Andwheredothenumberscomefrom? 200: 50, 75, 75 ???Division BeforeIproceed,IdliketosummarizetheclaimandtheTalmudanswerasatable.In thetable,therowsareestatesize,thecolumnsareclaims,andtableentriesarethe divisionsize. Claims 100 200 300 Estate 100 331/3 331/3 331/3 size 300 50 100 150 200 50 75 75 In the 1980s, Professors Robert Aumann and Michael Maschler wrote a paper claimingtohavecrackedthemystery.TheydemonstratetheTalmudanswercanbe viewedasanapplicationofagametheoryprinciple.Whywasgametheoryused?It turns out the Talmud answer is the solution (the nucleolus) of a properly defined coalitionalgame.AumannandMaschlerexplaintheconceptinplainterms.Thatis equaldivisionofthecontested/disputed/controversialsum.

Inthefollowing,IllshowyouallhowtheTalmudanswerdemonstratesit.Letsstart withasimplifiedversion:Howtoequallydividethecontestedsumbetween2people? Letslookatthefollowingexample: Suppose two people are arguing over a garment/cloth. One claims half belongs to himwhiletheotherclaimsthewholeishis.Ajudgeisaskedtodecidewhogetswhat. Whatwouldyoudo? Wecanhavevariousanswersonthisproblem.Onecouldproposeanevensplit(1/2, 1/2) or a proportional split (1/3, 2/3). But the Talmud offers a different answer, an answerthatturnsouttobeanequaldivisionofthecontestedsum(1/4,3/4). Howdoesthisprinciplework? Therearethreestages.First,decidewhatportionoftheclothisbeingdisputed.In thiscase,exactlyhalfofthegarmentisbeingclaimedbybothparties.Second,split thedisputeddivisionamongbothpartiesso1/4oftheclothisawardedtoeach.And third, give the remaining cloththe undisputed portionentirely to the person whoseclaimisnotdisputed. Thislogicyieldsasplitof1/4forthepersonclaiminghalfofthegarmentand3/4for thepersonclaimingthewhole. We can summarize the principle for any problems among two parties, using the following3steps: 1. Determinewhichportioniscontestedorclaimedbybothparties 2. Splitthecontestedportionequally 3. Assigntheuncontestedportiontothesolepersonclaimingit. Letsdosomeexercisestogetafeelforthisprinciple.Letsexaminehowtodivide estatesofvarioussizeswith2creditorsclaiming100and300. Claims 100 300 Estate 662/3 331/3 331/3 Size 125 50 75 200 50 150 Example1:(estate662/3) Iftheestateis662/3,thentheentireestateiscontested.Thesplitshouldbeevenat 331/3goingtoeachparty. Example2:(estate125)

Iftheestateis125,thenthefirst100iscontestedbybothpartiesanddividedevenly. Theremaining25isentirelyawardedtothe300claimant.Hence,thedivisionis50 and75. Example3:(estate200) Finally,iftheestateis200,thenagainthefirst100iscontestedbybothpartiesand divided evenly. The remaining 100 is entirely awarded to the 300claimant. Hence, thedivisionis50and150. Illgiveyousomeformsandyoucanworkingroupstocompletethem.Itsaboutthe divisionswhentheclaimsare(100,200)and(200,300).Iwillgiveyou2minutes. Claims Claims 100 200 200 300 Estate 662/3 331/3 331/3 Estate 662/3 331/3 331/3 Size Size 125 50 75 150 75 75 150 50 100 250 100 150 Compare this 3 tabular form to the big form on the white board. Can you find something? TheexplanationofTalmuddivision ToanalysistheTalmudsolution,letsgobacktotheTalmuddivisionfor3creditors.In the case of a 200 estate, the division was 50, 75, and 75 for parties that claimed debtsof100,200,and300.,justtakeanytwocreditorsandconsiderhowtheymight splitthetotalmoneyawardedtothem.Considerthepairofcreditorsclaiming100 and200.Togethertheyareawardedasumof125.Howdotheysplitit?Its50and 75.Itmatchestheworkwedidintheexercisesabove! Infact,thesameobservationcanbeseenwhenconsideringotherpairsofcreditors. Lookathowmuchthe100and300partiesaregetting.Togethertheyreceiveasum of125,andthisissplitas50and75.Again,thisanswerisconsistentwithanequal divisionofthecontestedsum. Finally,considerthetotalrewardtothe200and300parties.Inthiscase,thetotal sum of 150 is split as 75 to each. As the total sum is contested, this once again reflectsanequaldivisionofthecontestedsum. SotheprincipleequaldivisionofthecontestedsumisalsosuitabletotheTalmud solutions. Tosumup,thedivisionoftheestateamongthethreecreditorsissuch thatanytwoofthemdividethesumtheytogetherreceive,accordingtotheprinciple ofequaldivisionofthecontestedsum.