Bankruptcy Problems and Minimum Cost Flow Problems

Transcription

1 Bankrupcy Problem and Minimum Co Flow Problem Rodica Branzei Faculy of Compuer Science, "Alexandru Ioan Cuza" Univeriy, Iai, Romania Giulio Ferrari Univeriy of Genoa, Deparmen of Mahemaic, Genova, Ialy Vio Fragnelli Univeriy of Eaern Piedmon, Deparmen of Advanced Science and Technologie, Aleandria, Ialy Sef Tij CenER and Deparmen of Economeric and Operaion Reearch, Tilburg Univeriy, Tilburg, The Neherland Univeriy of Genoa, Deparmen of Mahemaic, Genova, Ialy AIROWINTER Corina (BL)

2 Bankrupcy Problem and Minimum Co Flow Problem 2 Ouline Join Projec Bankrupcy Problem and Relaed Flow Problem Two Generalized Bankrupcy Problem Bankrupcy Rule and Min Co Flow Problem Concluding Remark

3 Bankrupcy Problem and Minimum Co Flow Problem 3 Join Projec Projec whoe aciviie are carried ou by differen firm Game heoreic approach of raioning problem ariing from join projec managemen: Berganiño and Sánchez (2002): NTU game for allocaing ime Branzei, Ferrari, Fragnelli and Tij (2002): TU game for allocaing co Caro and Tejada (2004): allocaing ime depending on uiliy (ground operaion on aircraf) Branzei, Ferrari, Fragnelli and Tij (2002): Divide he co of he delay of a projec among he aciviie ha have a delay via bankrupcy approach Two new queion: - Wha o do if ome aciviie recover par of he delay? - Wha o do if an exra reward arie from an early erminaion of he projec?

4 Bankrupcy Problem and Minimum Co Flow Problem 4 Bankrupcy Problem and Relaed Flow Problem Claical bankrupcy problem (N,E,c) where N = {1,, n} E R + e of claiman eae o be divided among claiman c =(c 1,, c n ) R n + claim, wih 0 E C = c 1 + c c n A oluion i a real vecor x =(x 1,,x n ) i N x i = E;0 x i c i, i N Sandard flow problem Nework G(N, A) wih wo node, he ource wih no enering arc and he ink wih no ougoing arc; arc have minimal and maximal capaciy conrain A flow i a funcion x : A R + ha repec he capaciy conrain and uch ha j N x ij = j N x ji, i N \{, } A claical bankrupcy problem can be repreened a a andard flow problem E/E 0/c 1 0/c n Each feaible flow correpond o a oluion of he bankrupcy problem

5 Bankrupcy Problem and Minimum Co Flow Problem 5 Two Generalized Bankrupcy Problem Claiman and debor Generalized bankrupcy problem A - The bank ha an eae E (N c,n d,e,c,d) where N c = {1 c,,n c } e of claiman N d = {1 d,, n d } e of debor E R + eae c =(c 1c,, c nc ) R n c + claim, wih 0 E C = c 1c + + c nc d =(d 1d,, d nd ) R n d + deb E/E d 1d d nd c 1c c nc

6 Bankrupcy Problem and Minimum Co Flow Problem 6 The value of E, c and d idenify he following wo ubcae: A1 i N c c i E + i N d d i A2 E i N c c i <E+ i N d d i A1 (eay) (N,E,c ) wih N = N c, E = E + i N d d i, c i = c i, i N c E + / d i E + i Nd i N d d i 0/c 1c 0/c nc

7 Bankrupcy Problem and Minimum Co Flow Problem 7 A2i Allocae he quoa E = i N c c i E,N = N d,c i = d i, i N d / c i E c i E i Nc i N c 0/d 1d 0/d nd A2ii Allocae he aving E = E + i N d d i i N c c i,n = N d,c i = d i, i N d E + d i / c i E + d i i Nd i Nc i N d c i i N c 0/d 1d 0/d nd

8 Bankrupcy Problem and Minimum Co Flow Problem 8 Example 1 Conider a iuaion in which a bank ha an eae E =10; he e of claiman i N c = {1 c, 2 c, 3 c } wih claim vecor c =(5, 6, 7) and he e of debor i N d = {1 d, 2 d } wih deb vecor d =(1, 2) Thi iuaion end in ubcae A1 and he correponding claical bankrupcy problem i (N,E,c ) wih N = {1 c, 2 c, 3 c },E =13,c =(5, 6, 7) If we conider an eae E =16we are in ubcae A2 and he correponding claical bankrupcy problem i (N,E,c ) wih N = {1 d, 2 d },E =2,c =(1, 2) according o he approach A2i or (N,E,c ) wih N = {1 d, 2 d },E =1,c =(1, 2) according o he approach A2ii

9 Bankrupcy Problem and Minimum Co Flow Problem 9 Generalized bankrupcy problem B - The bank ha a deb Ê (N c,n d, Ê,c,d) where N c = {1 c,,n c } e of claiman N d = {1 d,, n d } e of debor Ê R + deb of he bank c =(c 1c,, c nc ) R n c + claim d =(d 1d,, d nd ) R n d + deb, wih 0 Ê D = d 1 d + + d nd d 1d d nd Ê/Ê c 1c c nc

10 Bankrupcy Problem and Minimum Co Flow Problem 10 Again, he value of Ê, c and d idenify he following wo ubcae: B1 if Ê i N d d i < Ê + i N c c i B2 if i N d d i Ê + i N c c i B1 (eay) (N,E,c ) wih N = N c, E = i N d d i Ê, c i = c i, i N c / d i Ê d i Ê i Nd i N d 0/c 1c 0/c nc

11 Bankrupcy Problem and Minimum Co Flow Problem 11 B2i Allocae he quoa (N,E,c ) wih N = N d,e = Ê + i N c c i,c i = d i, i N d Ê + / c i Ê + i Nc i N c c i 0/d 1d 0/d nd B2ii Allocae he aving (N,E,c ) wih N = N d,e = i N d d i (Ê + i N c c i ),c i = d i, i N d d i Ê / c i d i Ê i Nd i Nc i N d i N c c i 0/d 1d 0/d nd

12 Bankrupcy Problem and Minimum Co Flow Problem 12 Example 2 Conider a iuaion in which a bank ha a deb Ê =12; he e of claiman i N c = {1 c, 2 c, 3 c } wih claim vecor c =(3, 4, 5) and he e of debor i N d = {1 d, 2 d } wih deb vecor d =(8, 9) Thi iuaion end in ubcae B1 and he correponding claical bankrupcy problem i (N,E,c ) wih N = {1 c, 2 c, 3 c },E =5,c =(3, 4, 5) If we conider a deb Ê =2we are in ubcae B2 and he correponding claical bankrupcy problem i (N,E,c ) wih N = {1 d, 2 d },E =14,c =(8, 9) according o he approach B2i or (N,E,c ) wih N = {1 d, 2 d },E =3,c =(8, 9) according o he approach B2ii

13 Bankrupcy Problem and Minimum Co Flow Problem 13 Bankrupcy Rule and Min Co Flow Problem A diviion rule i a funcion f which aign o any bankrupcy problem (N,E,c) a vecor f(n,e,c) R n uch ha: 0 f i (N,E,c) c i, for each i N f i (N,E, c) =E i N Well known diviion rule are he proporional rule (PROP), he conrained equalawardrule (CEA), he conrained equal lo rule (CEL),healmudicrule(TAL) and he adjued proporional rule (APROP)

14 Bankrupcy Problem and Minimum Co Flow Problem 14 PROP i (N,E,c) =( j N c j) 1 ci E, i N CEA i (N,E,c) =min{c i,α}, i N where α i he unique real number o ha i N CEA i(n,e, c) =E CEL i (N,E,c) =max{c i β,0}, i N where β i he unique real number o ha i N CEL i(n,e,c) =E { CEA i (N,E, 1 2 TAL i (N,E, c) = c) if E<1 2 j N c j c i CEA i (N,E, 1 2 c) if E 1 2 j N c,i N j where E = j N c j E The minimal righ of each player i N i m i (N,E,c) =max{0,e j N\{i} c j}, i N AP ROP i (N,E,c) =m i (N,E,c)+PROP i (N,E,c ),i N where E = E j N m j and c i = min{e,c i m i (N,E,c)}, i N A claical bankrupcy problem (N,E, c) and a diviion rule f can be relaed o a min co flow problem

15 Bankrupcy Problem and Minimum Co Flow Problem 15 Theorem 1 Le (N,E,c) be a claical bankrupcy problem The diviion rule PROP, CEA, CEL and TAL can be implemened via a min co flow problem Proof We only give for each rule a e of uiable co funcion PROP: k i (x i )= x2 i c i CEA: k i (x i )=x 2 i CEL: k i (x i )=(c i x i ) 2 For he Talmudic rule if E< 1 2 i N c i: TAL: k i (x i )=x 2 i,i N eing he maximal capaciy of he arc correponding o he claiman o 1 2 c i If E 1 2 i N c i: TAL: k i (x i )=(c i x i ) 2,i N eing he minimal capaciy of he arc correponding o he claiman o 1 2 c i

16 Bankrupcy Problem and Minimum Co Flow Problem 16 Remark 1 Le (N,E, c) be a claical bankrupcy problem The oluion APROP(N,E,c) can be obained via a min co flow problem wih e of co funcion: 0 if x i m i k i (x i )= (x i m i ) 2,i N c i m if x i >m i i

17 Bankrupcy Problem and Minimum Co Flow Problem 17 Concluding Remark Paricular choice of co funcion in a min co flow problem can lead o oher rule from he bankrupcy lieraure, or ugge new diviion according o differen fairne crieria, ailoring he amoun on each agen E/E d 1d d nd c 1c c nc Ê/Ê d 1d d nd c 1c c nc Bankrupcy approach aurae arc relaed o claiman or debor Min co flow approach may drive he oluion oward non-auraion of hee arc A muli-claim bankrupcy problem and/or an inerval bankrupcy iuaion can be repreened by an appropriae flow problem

18 Bankrupcy Problem and Minimum Co Flow Problem 17 Reference Aumann, RJ and M Machler (1985) Game Theoreic Analyi of a Bankrupcy Problem from he Talmud J Economic Theory 36, Berganiño, G and E Sánchez (2002) How o Diribue Co Aociaed wih a Delayed Projec Annal of Op Re 109, Borm, P, H Hamer and R Hendrickx (2001) Operaion Reearch Game: A Survey TOP 9, Branzei, R, G Ferrari, V Fragnelli and S Tij (2002) Two Approache o he Problem of Sharing Delay Co in Join Projec Annal of Op Re 109, Branzei, R, D Dimirov and S Tij (2003) Shapley-like Value for Inerval Bankrupcy Game Economic Bullein 3, 1-8 Branzei, R, G Ferrari, V Fragnelli and S Tij (2004) Join Projec Managemen wih Penalie and Compenaion Preprin, Diparimeno di Maemaica dell Univerià digenova519 Caro, J and J Tejada (2004) Una regla proporional a la duracione para el reparo de holgura en un problema PERT Working Paper Curiel, IJ, M Machler and SH Tij (1987) Bankrupcy Game Zeichrif für OR 31, Gondran, M and M Minoux (1984) Graph and Algorihm New York: Wiley Inercience Kaminki, MM (2000) Hydraulic Raioning MahSocSci40, O Neill, B (1982) A Problem of Righ Arbiraion from he Talmud MahSocSci2, Thomon, W (2003) Axiomaic and Game-heoreic Analyi of Bankrupcy and Taxaion Problem: A Survey MahSocSci45, Young, HP (1987) On Dividing an Amoun According o Individual Claim or Liabiliie Mah Op Re 12, Young, HP (1994) Co Allocaion In: RJ Aumann and S Har (Ed), Handbook of Game Theory, vol II, Amerdam: Norh Holland, Young, HP (1998) Diribuive Juice in Taxaion J Economic Theory 48,

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