The Bankruptcy . . . An Ancient Solution Examples Are Here, . . . Mystery Solved, . . . An Ancient Remaining Problem Bankruptcy Solution Analysis of the Problem Let Us Divide Equally, . . . Makes Economic Sense Which Points of the . . . No Matter What Our . . . Anh H. Ly 1 , Michael Zakharevich 2 Home Page Olga Kosheleva 3 , and Vladik Kreinovich 3 Title Page 1 Banking University of Ho Chi Minh City, 56 Hoang Dieu 2 ◭◭ ◮◮ Quan Thu Duc, Thu Duc, Ho Ch´ ı Minh City, Vietnam 2 SeeCure Systems, Inc., 1040 Continentals Way # 12 ◭ ◮ Belmont, California 94002, USA, michael@seecure360.com 3 University of Texas at El Paso, El Paso, Texas, USA, USA Page 1 of 37 olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 1. The Bankruptcy Problem: Reminder Examples Are Here, . . . • When a person or a company cannot pay all its obli- Mystery Solved, . . . gation: Remaining Problem Analysis of the Problem – a bankruptcy is declared, and Let Us Divide Equally, . . . – the available funds are distributed among the Which Points of the . . . claimants. No Matter What Our . . . • There is not enough money to give, to each claimant, Home Page what he/she is owed. Title Page • So, claimants will get less than what they are owed. ◭◭ ◮◮ • How much less? ◭ ◮ • What is a fair way to divide the available funds between Page 2 of 37 the claimants? Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 2. An Ancient Solution Examples Are Here, . . . • The bankruptcy problem is known for many millennia: Mystery Solved, . . . Remaining Problem – since money became available and Analysis of the Problem – people starting lending money to each other. Let Us Divide Equally, . . . • Solutions to this problem have also been proposed for Which Points of the . . . many millennia. No Matter What Our . . . Home Page • One such ancient solution is described in the Talmud, an ancient commentary on the Jewish Bible. Title Page ◭◭ ◮◮ • This solution is described in the Babylonian Talmud, in Ketubot 93a, Bava Metzia 2a, and Yevamot 38a. ◭ ◮ • This solution is actually about a more general problem Page 3 of 37 of several contracts which cannot be all fully fulfilled. Go Back • Like many ancient texts containing mathematics, the Full Screen Talmud does not contain an explicit algorithm. Close Quit
The Bankruptcy . . . An Ancient Solution 3. An Ancient Solution (cont-d) Examples Are Here, . . . • Instead, it contains four examples illustrating the main Mystery Solved, . . . idea. Remaining Problem Analysis of the Problem • In the first three examples, the three parties are owed Let Us Divide Equally, . . . the following amounts: Which Points of the . . . – the first person is owed d 1 = 100 monetary units, No Matter What Our . . . – the second person is owed d 2 = 200 monetary units, Home Page and Title Page – the third person is owed d 3 = 300 monetary units: ◭◭ ◮◮ d 1 = 100 , d 2 = 200 , d 3 = 300 . ◭ ◮ Page 4 of 37 Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 4. An Ancient Solution (cont-d) Examples Are Here, . . . • For three different available amounts E , the text de- Mystery Solved, . . . scribes the amounts e 1 , e 2 , and e 3 that each gets: Remaining Problem Analysis of the Problem d 1 = 100 d 2 = 200 d 3 = 300 Let Us Divide Equally, . . . E e 1 e 2 e 3 Which Points of the . . . 331 331 331 No Matter What Our . . . 100 3 3 3 Home Page 200 50 75 75 Title Page 300 50 100 150 ◭◭ ◮◮ ◭ ◮ • There is also a fourth example, formulated in a slightly different way – as dividing a disputed garment. Page 5 of 37 Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 5. An Ancient Solution (cont-d) Examples Are Here, . . . • In the bankruptcy terms, it can be described as follows: Mystery Solved, . . . the owed amounts are: d 1 = 50, d 2 = 100 . Remaining Problem Analysis of the Problem • The available amount E and the recommended division Let Us Divide Equally, . . . ( e 1 , e 2 ) are as follows: Which Points of the . . . No Matter What Our . . . d 1 = 50 d 2 = 100 Home Page E e 1 e 2 Title Page 100 25 75 ◭◭ ◮◮ ◭ ◮ Page 6 of 37 Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 6. Examples Are Here, But What is a General Examples Are Here, . . . Solution? Mystery Solved, . . . • In many other ancient mathematical texts, where the Remaining Problem general algorithm is very clear from the examples. Analysis of the Problem Let Us Divide Equally, . . . • However, in this particular case, the general algorithm Which Points of the . . . was unknown until 1985. No Matter What Our . . . • Actually, many researchers came up with algorithms Home Page that: Title Page – explained some of these examples, ◭◭ ◮◮ – while claiming that the original ancient text must ◭ ◮ have contained some mistakes. Page 7 of 37 Go Back Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 7. Mystery Solved, Algorithm Is Reconstructed Examples Are Here, . . . • This problem intrigued Robert Aumann, later the No- Mystery Solved, . . . bel Prize winner in Economics (2005). Remaining Problem Analysis of the Problem • He came up with a reasonable general algorithm that Let Us Divide Equally, . . . explains the ancient solution. Which Points of the . . . • To explain this algorithm, we need to first start with No Matter What Our . . . the the case of two claimants. Home Page • Without losing generality, let us assume that the first Title Page claimant has a smaller claim d 1 ≤ d 2 . ◭◭ ◮◮ • The first case is when the overall amount E is small – ◭ ◮ smaller that d 1 . Page 8 of 37 • Then, the amount E is distributed equally between the Go Back claimants, so that each gets e 1 = e 2 = E 2 . Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 8. Mystery Solved (cont-d) Examples Are Here, . . . • When the available amount E is between d 1 and d 2 , Mystery Solved, . . . i.e., when d 1 ≤ E ≤ d 2 , then: Remaining Problem – the first claimant receives e 1 = d 1 Analysis of the Problem 2 , and Let Us Divide Equally, . . . – the second claimant receives the remaining amount Which Points of the . . . e 2 = E − e 1 . No Matter What Our . . . • This policy continues until we reach the amount E = Home Page d 2 , at which moment: Title Page – the first claimant receives the amount d 1 = d 1 ◭◭ ◮◮ 2 and ◭ ◮ – the second claimant receives e 2 = d 2 − d 1 2 . Page 9 of 37 • At this moment, after receiving the money, both Go Back claimants lose the same amount of money: Full Screen d 1 − e 1 = d 2 − e 2 = d 1 2 . Close Quit
The Bankruptcy . . . An Ancient Solution 9. Mystery Solved (cont-d) Examples Are Here, . . . • The third case is when E larger than d 2 (but smaller Mystery Solved, . . . than the overall amount of debt d 1 + d 2 ). Remaining Problem Analysis of the Problem • Then, the money is distributed in such a way that the Let Us Divide Equally, . . . losses remain equal, i.e., that Which Points of the . . . d 1 − e 1 = d 2 − e 2 and e 1 + e 2 = E. No Matter What Our . . . Home Page • From these two conditions, we get: Title Page e 1 = E + d 1 − d 2 e 2 = E − d 1 + d 2 , . ◭◭ ◮◮ 2 2 ◭ ◮ • The division between three (or more) claimants is then explained as the one for which: Page 10 of 37 Go Back – for every two claimants, – the amounts given to them are distributed accord- Full Screen ing to the above algorithm. Close Quit
The Bankruptcy . . . An Ancient Solution 10. Mystery Solved (cont-d) Examples Are Here, . . . • This can be easily checked if we select, Mystery Solved, . . . Remaining Problem – for each pair ( i, j ) Analysis of the Problem – only the overall amount E ij = e i + e j allocated to Let Us Divide Equally, . . . claimants from this pair. Which Points of the . . . • As a result, for the pairs (1 , 2), (2 , 3), and (1 , 3), we No Matter What Our . . . get the following tables: Home Page Title Page d 1 = 100 d 2 = 200 ◭◭ ◮◮ E 12 e 1 e 2 ◭ ◮ 662 331 331 3 3 3 Page 11 of 37 125 50 75 Go Back 150 50 100 Full Screen Close Quit
The Bankruptcy . . . An Ancient Solution 11. Mystery Solved (cont-d) Examples Are Here, . . . d 2 = 200 d 3 = 300 Mystery Solved, . . . Remaining Problem E 23 e 2 e 3 662 331 331 Analysis of the Problem 3 3 3 Let Us Divide Equally, . . . 150 75 75 Which Points of the . . . 250 100 150 No Matter What Our . . . Home Page d 1 = 100 d 3 = 300 Title Page E 13 e 1 e 3 ◭◭ ◮◮ 662 331 100 ◭ ◮ 3 3 125 50 75 Page 12 of 37 200 50 150 Go Back Full Screen Close Quit
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