Bankruptcy problem: Difference between revisions
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A '''bankruptcy problem''',<ref name=":0" /> also called a '''claims problem''',<ref name=":1">{{Cite journal|last=Thomson|first=William|date=2003-07-01|title=Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey|url=https://www.sciencedirect.com/science/article/abs/pii/S0165489602000707|journal=Mathematical Social Sciences|language=en|volume=45|issue=3|pages=249–297|doi=10.1016/S0165-4896(02)00070-7|issn=0165-4896}}</ref> is a problem of distributing a [[Fair division of a single homogeneous resource|homogeneous divisible good]] (such as money) among people with different [[Claim in bankruptcy|claims]]. The focus is on the case where the amount is insufficient to satisfy all the claims. |
A '''bankruptcy problem''',<ref name=":0" /> also called a '''claims problem''',<ref name=":1">{{Cite journal|last=Thomson|first=William|date=2003-07-01|title=Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey|url=https://www.sciencedirect.com/science/article/abs/pii/S0165489602000707|journal=Mathematical Social Sciences|language=en|volume=45|issue=3|pages=249–297|doi=10.1016/S0165-4896(02)00070-7|issn=0165-4896}}</ref> is a problem of distributing a [[Fair division of a single homogeneous resource|homogeneous divisible good]] (such as money) among people with different [[Claim in bankruptcy|claims]]. The focus is on the case where the amount is insufficient to satisfy all the claims. |
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The |
The canonical application is a [[Bankruptcy in the United States|bankrupt]] [[business|firm]] that is to be [[liquidation|liquidated]]. The firm owes different amounts of money to different [[creditors]], but the total worth of the company's assets is smaller than its total debt. The problem is how to divide the scarce existing money among the creditors. |
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Another application would be the division of an estate amongst several [[heir]]s, particularly when the estate cannot meet all the deceased's commitments. |
Another application would be the division of an estate amongst several [[heir]]s, particularly when the estate cannot meet all the deceased's commitments. |
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A third application<ref name=":1" /> is ''[[tax assessment]]''. One can consider the claimants as taxpayers, the claims as the incomes, and the endowment as the total after-tax income. Determining the allocation of total after-tax income is equivalent to determining the allocation of tax payments. |
A third application<ref name=":1" /> is ''[[tax assessment]]''. One can consider the claimants as taxpayers, the claims as the incomes, and the endowment as the total after-tax income. Determining the allocation of total after-tax income is equivalent to determining the allocation of tax payments. |
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== Definitions == |
== Definitions == |
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A ''division rule'' is a function that maps a problem instance <math>(c_1,\ldots,c_n,E)</math> to a vector <math>(x_1,\ldots,x_n)</math> such that <math>\sum_{i=1}^n x_i = E</math> and <math>0\leq x_i\leq c_i</math> for all ''i''. That is: each claimant receives at most its claim, and the sum of allocations is exactly the estate ''E''. |
A ''division rule'' is a function that maps a problem instance <math>(c_1,\ldots,c_n,E)</math> to a vector <math>(x_1,\ldots,x_n)</math> such that <math>\sum_{i=1}^n x_i = E</math> and <math>0\leq x_i\leq c_i</math> for all ''i''. That is: each claimant receives at most its claim, and the sum of allocations is exactly the estate ''E''. |
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=== Generalizations === |
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There are generalized variants in which the total claims might be smaller than the estate. In these generalized variants, <math>\sum_{i=1}^n c_i \geq E</math> is not assumed and <math>0\leq x_i\leq c_i</math> is not required. |
There are generalized variants in which the total claims might be smaller than the estate. In these generalized variants, <math>\sum_{i=1}^n c_i \geq E</math> is not assumed and <math>0\leq x_i\leq c_i</math> is not required. |
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Another generalization, inspired by realistic bankruptcy problems, is to add an exogeneous priority ordering among the claimants, that may be different even for claimants with identical claims. This problem is called a ''claims problem with priorities''. Another variant is called a ''claims problem with weights.'' |
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== Rules == |
== Rules == |
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There are |
There are various rules for solving bankruptcy problems in practice.<ref name=":0">{{Cite journal|title=Equal Awards vs. Equal Losses in Bankruptcy Problems|journal=SSRN|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2919582|last1=Alcalde|first1=José|date=2017-02-17|last2=Peris|first2=Josep E.|doi=10.2139/ssrn.2919582 |ssrn=2919582 |s2cid=158036131 }}</ref> |
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=== Proportional === |
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The '''[[Proportional rule (bankruptcy)|proportional rule]]''' divides the estate proportionally to each agent's claim. Formally, each claimant ''i'' receives <math>r \cdot c_i</math>, where ''r'' is a constant chosen such that <math>\sum_{i=1}^n r\cdot c_i = E</math>. We denote the outcome of the proportional rule by <math>PROP(c_1,\ldots,c_n ; E)</math>. |
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Examples with two claimants: |
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* <math>PROP(60,90; 100) = (40,60)</math>. That is: if the estate is worth 100 and the claims are 60 and 90, then <math>r = 2/3</math>, so the first claimant gets 40 and the second claimant gets 60. |
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* <math>PROP(50,100; 100) = (33.333,66.667)</math>, and similarly <math>PROP(40,80; 100) = (33.333,66.667)</math>. |
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Examples with three claimants: |
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*<math>PROP(100,200,300; 100) = (16.667, 33.333, 50)</math>. |
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*<math>PROP(100,200,300; 200) = (33.333, 66.667, 100)</math>. |
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*<math>PROP(100,200,300; 300) = (50, 100, 150)</math>. |
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There is a variant called '''truncated-claims proportional rule''', in which each claim larger than ''E'' is truncted to ''E'', and then the proportional rule is activated. That is, it equals <math>PROP(c_1',\ldots,c_n',E)</math>, where <math>c'_i := \min(c_i, E)</math>.<ref name=":1" /> The results are the same for the two-claimant problems, but for the three-claimant problems we get: |
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*<math>TPROP(100,200,300; 100) = (33.333, 33.333, 33.333)</math>, since all claims are truncated to 100; |
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*<math>TPROP(100,200,300; 200) = (40, 80, 80)</math>, since the claims vector is truncated to (100,200,200). |
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*<math>TPROP(100,200,300; 300) = (50, 100, 150)</math>, since here the claims are not truncated. |
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=== Adjusted proportional === |
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The '''adjusted proportional rule'''<ref name=":4">{{Cite journal|last=Curiel|first=I. J.|last2=Maschler|first2=M.|last3=Tijs|first3=S. H.|date=1987-09-01|title=Bankruptcy games|url=https://doi.org/10.1007/BF02109593|journal=Zeitschrift für Operations Research|language=en|volume=31|issue=5|pages=A143–A159|doi=10.1007/BF02109593|issn=1432-5217}}</ref> first gives, to each agent ''i'', his ''minimal right'', which is the amount not claimed by the other agents. Formally, <math>m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that <math>\sum_{i=1}^n c_i \geq E</math> implies <math>m_i \leq c_i</math>. |
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Then, it revises the claim of agent ''i'' to <math>c'_i := c_i - m_i</math>, and the estate to <math>E' := E - \sum_i m_i</math>. Note that that <math>E' \geq 0</math>. |
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Finally, it activates the truncated-claims proportional rule, that is, it returns <math>TPROP(c_1,\ldots,c_n,E') = PROP(c_1'',\ldots,c_n'',E')</math>, where <math>c''_i := \min(c'_i, E')</math>. |
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With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples: |
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* <math>APROP(60,90; 100) = (35,65)</math>. The minimal rights are <math>(m_1,m_2) = (10,40)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>; it is divided equally among the claimants. |
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* <math>APROP(50,100; 100) = (25,75)</math>. The minimal rights are <math>(m_1,m_2) = (0,50)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>. |
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*<math>APROP(40,80; 100) = (30,70)</math>. The minimal rights are <math>(m_1,m_2) = (20,60)</math>. The remaining claims are <math>(c_1',c_2') = (20,20)</math> and the remaining estate is <math>E'=20</math>. |
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With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are <math>(0,0,0)</math> and thus the outcome is equal to TPROP, for example, <math>APROP(100,200,300; 200) = TPROP(100,200,300; 200) = (20, 40, 40)</math>. |
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=== Constrained equal awards === |
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The '''[[constrained equal awards]]''' rule divides the estate equally among the agents, ensuring that nobody gets more than their claim. Formally, each claimant ''i'' receives <math>\min(c_i, r)</math>, where ''r'' is a constant chosen such that <math>\sum_{i=1}^n \min(c_i,r) = E</math>. We denote the outcome of this rule by <math>CEA(c_1,\ldots,c_n ; E)</math>. In the context of taxation, it is known as '''leveling tax'''.<ref name=":1" /> Examples with two claimants: |
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* <math>CEA(60,90; 100) = CEA(50,100; 100) =(50,50)</math>; here <math>r=50</math>. |
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* <math>CEA(40,80; 100) = (40,60)</math>; here <math>r=60</math>. |
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Examples with three claimants: |
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*<math>CEA(50,100,150; 100) = (33.333, 33.333, 33.333)</math>; here <math>r=33.333</math>. |
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*<math>CEA(50,100,150; 200) = (50, 75, 75)</math>; here <math>r=75</math>. |
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*<math>CEA(50,100,150; 300) = (50, 100, 150)</math>; here <math>r=150</math>. |
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=== Piniles' rule === |
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The following rule is attributed<ref name=":1" /> to Piniles.<ref>{{Cite book|last=Piniles|first=Zvi Menahem|url=https://www.hebrewbooks.org/31823|title=Darkah Shel Torah (Hebrew)|publisher=Forester|year=1863|location=Wien}}</ref> |
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* If the sum of claims is larger than 2''E'', then it runs the CEA rule on half the claims, that is, it returns <math>CEA(c_1/2,\ldots,c_n/2; E)</math> . |
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* Otherwise, it gives each agent half its claim and then runs CEA on the remainder, that is, it returns <math>(c_1/2,\ldots,c_n/2) + CEA(c_1/2,\ldots,c_n/2; E-\sum_{j=1}^n c_j/2)</math> . |
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Examples with two claimants: |
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* <math>PINI(60,90; 100) = (42.5, 57.5)</math>. Initially the claimants get (30,45). The remaining claims are (30,45) and the remaining estate is 25, so it is divided equally. |
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* <math>PINI(50,100; 100) = (37.5, 62.5)</math>. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally. |
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* <math>PINI(50,100; 100) = (37.5, 62.5)</math>. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally. |
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Examples with three claimants: |
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*<math>PINI(100,200,300; 100) = (33.333, 33.333, 33.333)</math>. Here the sum of claims is more than twice the estate, so the outcome is <math>CEA(50,100,150; 100) = (33.333, 33.333, 33.333)</math>. |
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*<math>PINI(100,200,300; 200) = (20, 40, 40)</math>. Again the sum of claims is more than twice the estate, so the outcome is <math>CEA(50,100,150; 200) = (50, 75, 75)</math>. |
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*<math>PINI(100,200,300; 300) = (50, 100, 150)</math>. Again the sum of claims is more than twice the estate, so the outcome is <math>CEA(50,100,150; 300) = (50, 100, 150)</math>. |
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=== Constrained egalitarianism === |
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The '''constrained egalitarian''' rule<ref>{{Cite journal|last=Chun|first=Youngsub|last2=Schummer|first2=James|last3=Thomson|first3=William|date=1998|title=Constrained Egalitarianism: A New Solution for Claims Problems|url=https://urresearch.rochester.edu/institutionalPublicationPublicView.action?institutionalItemId=2175}}</ref> works as follows. |
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* If the sum of claims is larger than 2''E'', then it runs the CEA rule on half the claims, giving each claimant ''i'' <math>\min(c_i/2, r)</math>. |
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* Otherwise, it gives each agent i <math>\max(c_i/2, \min(c_i, r))</math>, |
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In both cases, ''r'' is a constant chosen such that the sum of allocations equals ''E''. |
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=== Constrained equal losses === |
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The '''[[constrained equal losses]] rule''' divides equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer. Formally, each claimant ''i'' receives <math>\max(0, c_i-r)</math>, where ''r'' is chosen such that <math>\sum_{i=1}^n \max(0, c_i-r) = E</math>. This rule was discussed by [[Maimonides]].<ref name=":2">{{Cite journal|last=Aumann|first=Robert J|last2=Maschler|first2=Michael|date=1985-08-01|title=Game theoretic analysis of a bankruptcy problem from the Talmud|url=https://www.sciencedirect.com/science/article/pii/0022053185901024|journal=Journal of Economic Theory|language=en|volume=36|issue=2|pages=195–213|doi=10.1016/0022-0531(85)90102-4|issn=0022-0531}}</ref> In the taxation context, it is known as [[Poll tax|'''poll tax''']]. Examples with two claimants: |
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* <math>CEL(60,90; 100) = (35,65)</math>; here <math>r=25</math>. |
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*<math>CEL(50,100; 100) =(25,75)</math>; here <math>r=25</math> too. |
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* <math>CEL(40,80; 100) = (30,70)</math>; here <math>r=10</math>. |
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Examples with three claimants: |
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*<math>CEL(50,100,150; 100) = (0, 25, 75)</math>; here <math>r=75</math>. |
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*<math>CEL(50,100,150; 200) = (16.667, 66.666, 116.667)</math>; here <math>r=33.333</math>. |
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*<math>CEL(50,100,150; 300) = (50, 100, 150)</math>; here <math>r=0</math>. |
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=== Talmud rule === |
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This rule generalizes examples from the [[Babylonian Talmud]].<ref name=":2" /> |
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* If the sum of claims is larger than 2''E'', then it returns <math>CEA(c_1/2,\ldots,c_n/2; E)</math>; |
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* Otherwise, it returns <math>c/2 + CEL(c_1/2,\ldots,c_n/2; E-\sum_j (c_j/2))</math>. |
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The rule can also be explained constructively.<ref name=":1" /> Suppose ''E'' increases from 0 to the half-sum of the claims: the first units are divided equally, until each claimant receives <math>\min_i(c_i/2)</math>. Then, the claimant with the smallest <math>c_i</math> is put on hold, and the next units are divided equally among the remaining claimants until each of them up to the next-smallest <math>c_i</math>. Then, the claimant with the second-smallest '''<math>c_i</math>''' is put on hold too. This goes on until either the estate is fully divided, or each claimant gets <math>c_i/2</math>. If some estate remains, then the losses are divided in a symmetric way, starting with an estate equal to the sum of all claims, and decreasing down to half this sum. |
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Examples with two claimants: |
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* <math>TAL(50, 100; 100) = (25, 75)</math>; here CEL is used. |
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Examples with three claimants: |
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*<math>TAL(100,200,300; 100) = (33.333, 33.333, 33.333)</math>; here CEA is used. |
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*<math>TAL(100,200,300; 200) = (50, 75, 75)</math>; here CEA is used. |
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*<math>TAL(100,200,300; 300) = (50, 100, 150)</math>; here either CEA or CEL can be used (the result is the same). |
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The two-claimant version of the Talmud rule is called the [[contested garment rule]].<ref>{{Cite journal|last=Dagan|first=Nir|date=1996|title=New characterizations of old bankruptcy rules|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.319.3243|journal=Social Choice and Welfare}}</ref> |
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* The '''[[Proportional rule (bankruptcy)|proportional rule]]''' divides the estate proportionally to each agent's claim. Formally, each claimant ''i'' receives <math>r \cdot c_i</math>, where ''r'' is a constant chosen such that <math>\sum_{i=1}^n r\cdot c_i = E</math>. We denote the outcome of the proportional rule by <math>PROP(c_1,\ldots,c_n ; E)</math>. |
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=== Random arrival rule === |
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* There is a variant called '''truncated-claims proportional rule''', in which each claim larger than ''E'' is truncated to ''E'', and then the proportional rule is activated. That is, it equals <math>PROP(c_1',\ldots,c_n',E)</math>, where <math>c'_i := \min(c_i, E)</math>.<ref name=":1" /> |
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Suppose claimants arrive one by one. Each claimant receives all his claim, up to the available amount. The '''random arrival rule''' returns the average of resulting allocation vectors when the arrival order is chosen uniformly at random.<ref name=":5">{{Cite journal|last=O'Neill|first=Barry|date=1982-06-01|title=A problem of rights arbitration from the Talmud|url=https://www.sciencedirect.com/science/article/pii/0165489682900294|journal=Mathematical Social Sciences|language=en|volume=2|issue=4|pages=345–371|doi=10.1016/0165-4896(82)90029-4|issn=0165-4896}}</ref> Formally: |
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* The '''adjusted proportional rule'''<ref name=":4">{{Cite journal|last1=Curiel|first1=I. J.|last2=Maschler|first2=M.|last3=Tijs|first3=S. H.|date=1987-09-01|title=Bankruptcy games|url=https://doi.org/10.1007/BF02109593|journal=Zeitschrift für Operations Research|language=en|volume=31|issue=5|pages=A143–A159|doi=10.1007/BF02109593|s2cid=206811949 |issn=1432-5217}}</ref> first gives, to each agent ''i'', his ''minimal right'', which is the amount not claimed by the other agents. Formally, <math>m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that <math>\sum_{i=1}^n c_i \geq E</math> implies <math>m_i \leq c_i</math>. Then, it revises the claim of agent ''i'' to <math>c'_i := c_i - m_i</math>, and the estate to <math>E' := E - \sum_i m_i</math>. Note that <math>E' \geq 0</math>. Finally, it activates the truncated-claims proportional rule, that is, it returns <math>TPROP(c_1,\ldots,c_n,E') = PROP(c_1'',\ldots,c_n'',E')</math>, where <math>c''_i := \min(c'_i, E')</math>. With two claimants, the revised claims are always equal, so the remainder is divided equally. With three or more claimants, the revised claims may be different. |
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* The '''[[constrained equal awards]]''' rule divides the estate equally among the agents, ensuring that nobody gets more than their claim. Formally, each claimant ''i'' receives <math>\min(c_i, r)</math>, where ''r'' is a constant chosen such that <math>\sum_{i=1}^n \min(c_i,r) = E</math>. We denote the outcome of this rule by <math>CEA(c_1,\ldots,c_n ; E)</math>. In the context of taxation, it is known as '''leveling tax'''.<ref name=":1" /> |
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* The '''[[constrained equal losses]] rule''' divides equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer. Formally, each claimant ''i'' receives <math>\max(0, c_i-r)</math>, where ''r'' is chosen such that <math>\sum_{i=1}^n \max(0, c_i-r) = E</math>. This rule was discussed by [[Maimonides]].<ref name=":2">{{Cite journal|last1=Aumann|first1=Robert J|last2=Maschler|first2=Michael|date=1985-08-01|title=Game theoretic analysis of a bankruptcy problem from the Talmud|url=https://dx.doi.org/10.1016/0022-0531%2885%2990102-4|journal=Journal of Economic Theory|language=en|volume=36|issue=2|pages=195–213|doi=10.1016/0022-0531(85)90102-4|issn=0022-0531}}</ref> In the taxation context, it is known as '''[[poll tax]]'''. |
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* The '''[[contested garment rule]]''' (also called the '''[[Talmud]] rule''') uses the CEA rule on half the claims if the estate is smaller than half the total claim; otherwise, it gives each claimant half their claims, and applies the CEL rule. Formally, if <math>2 E < \sum_{i=1}^n c_i </math> then <math>CG(c_1,\ldots,c_n; E) = CEA(c_1/2,\ldots,c_n/2; E)</math>; Otherwise, <math>CG(c_1,\ldots,c_n; E) = c/2 + CEL(c_1/2,\ldots,c_n/2; E-\sum_j (c_j/2))</math>. |
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* The following rule is attributed<ref name=":1" /> to '''Piniles.'''<ref>{{Cite book|last=Piniles|first=Zvi Menahem|url=https://www.hebrewbooks.org/31823|title=Darkah Shel Torah (Hebrew)|publisher=Forester|year=1863|location=Wien}}</ref> If the sum of claims is larger than 2''E'', then it applies the CEA rule on half the claims, that is, it returns <math>CEA(c_1/2,\ldots,c_n/2; E)</math> ; Otherwise, it gives each agent half its claim and then applies CEA on the remainder, that is, it returns <math>(c_1/2,\ldots,c_n/2) + CEA(c_1/2,\ldots,c_n/2; E-\sum_{j=1}^n c_j/2)</math> . |
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* The '''constrained egalitarian''' rule<ref>{{Cite journal|last1=Chun|first1=Youngsub|last2=Schummer|first2=James|last3=Thomson|first3=William|date=1998|title=Constrained Egalitarianism: A New Solution for Claims Problems|url=https://urresearch.rochester.edu/institutionalPublicationPublicView.action?institutionalItemId=2175}}</ref> works as follows. If the sum of claims is larger than 2''E'', then it runs the CEA rule on half the claims, giving each claimant ''i'' <math>\min(c_i/2, r)</math>. Otherwise, it gives each agent i <math>\max(c_i/2, \min(c_i, r))</math>, In both cases, ''r'' is a constant chosen such that the sum of allocations equals ''E''. |
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* The '''random arrival rule''' works as follows. Suppose claimants arrive one by one. Each claimant receives all his claim, up to the available amount. The rule returns the average of resulting allocation vectors when the arrival order is chosen uniformly at random.<ref name=":5">{{Cite journal|last=O'Neill|first=Barry|date=1982-06-01|title=A problem of rights arbitration from the Talmud|url=https://dx.doi.org/10.1016/0165-4896%2882%2990029-4|journal=Mathematical Social Sciences|language=en|volume=2|issue=4|pages=345–371|doi=10.1016/0165-4896(82)90029-4|issn=0165-4896|hdl=10419/220805|hdl-access=free}}</ref> Formally: |
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<math>RA(c_1,\ldots,c_n; E) = \frac{1}{n!} \sum_{\pi \in \text{permutations}} \min (c_i, \max(0, E-\sum_{\pi(j)<\pi(i)}c_j))</math>. |
<math>RA(c_1,\ldots,c_n; E) = \frac{1}{n!} \sum_{\pi \in \text{permutations}} \min (c_i, \max(0, E-\sum_{\pi(j)<\pi(i)}c_j))</math>. |
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It is possible to associate each bankruptcy problem with a [[cooperative bargaining]] problem, and use a bargaining rule to solve the bankruptcy problem. Then: |
It is possible to associate each bankruptcy problem with a [[cooperative bargaining]] problem, and use a bargaining rule to solve the bankruptcy problem. Then: |
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* The [[Nash Bargaining Solution|Nash bargaining solution]] corresponds to the [[constrained equal awards]] rule;<ref name=":3">{{Cite journal| |
* The [[Nash Bargaining Solution|Nash bargaining solution]] corresponds to the [[constrained equal awards]] rule;<ref name=":3">{{Cite journal|last1=Dagan|first1=Nir|last2=Volij|first2=Oscar|date=1993-11-01|title=The bankruptcy problem: a cooperative bargaining approach|url=https://dx.doi.org/10.1016/0165-4896%2893%2990024-D|journal=Mathematical Social Sciences|language=en|volume=26|issue=3|pages=287–297|doi=10.1016/0165-4896(93)90024-D|issn=0165-4896}}</ref> |
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*The lexicographic-egalitarian bargaining solution also corresponds to the constrained equal awards rule;<ref name=":1" /> |
*The lexicographic-egalitarian bargaining solution also corresponds to the constrained equal awards rule;<ref name=":1" /> |
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* The weighted [[Nash Bargaining Solution|Nash bargaining solution]], with weights equal to the claims, corresponds to the [[Proportional rule (bankruptcy)|proportional rule]];<ref name=":3" /> |
* The weighted [[Nash Bargaining Solution|Nash bargaining solution]], with weights equal to the claims, corresponds to the [[Proportional rule (bankruptcy)|proportional rule]];<ref name=":3" /> |
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* The [[Shapley value]] corresponds to the random-arrival rule;<ref name=":5" /> |
* The [[Shapley value]] corresponds to the random-arrival rule;<ref name=":5" /> |
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* The [[prenucleolus]] corresponds to the Talmud rule;<ref name=":2" /> |
* The [[prenucleolus]] corresponds to the Talmud rule;<ref name=":2" /> |
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* The Dutta-Ray solution |
* The Dutta-Ray solution corresponds to the constrained equal awards rule;<ref>{{Cite journal|last1=Dutta|first1=Bhaskar|last2=Ray|first2=Debraj|date=1989|title=A Concept of Egalitarianism Under Participation Constraints|url=https://www.jstor.org/stable/1911055|journal=Econometrica|volume=57|issue=3|pages=615–635|doi=10.2307/1911055|jstor=1911055 |issn=0012-9682}}</ref> |
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* The Tau-value solution |
* The Tau-value solution corresponds to the adjusted proportional rule.<ref name=":4" /> |
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An alternative way to associate a claims problem with a cooperative game<ref>{{Cite journal|last=Driessen|first=Theo|date=1995|title=An alternative game theoretic analysis of a bankruptcy problem from the Talmud: the case of the greedy bankruptcy game|url=https://research.utwente.nl/en/publications/an-alternative-game-theoretic-analysis-of-a-bankruptcy-problem-fr|language=English}}</ref> is its ''maximal right'' - the amount that this coalition can ensure itself if all other claimants drop their claims: <math>v(S) := \min\left(E, \sum_{j\in S}c_j\right)</math>. |
An alternative way to associate a claims problem with a cooperative game<ref>{{Cite journal|last=Driessen|first=Theo|date=1995|title=An alternative game theoretic analysis of a bankruptcy problem from the Talmud: the case of the greedy bankruptcy game|url=https://research.utwente.nl/en/publications/an-alternative-game-theoretic-analysis-of-a-bankruptcy-problem-fr|language=English}}</ref> is its ''maximal right'' - the amount that this coalition can ensure itself if all other claimants drop their claims: <math>v(S) := \min\left(E, \sum_{j\in S}c_j\right)</math>. |
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* '''Non-negativity''': each claimant should get a non-negative amount, <math>\forall i: x_i\geq 0 </math>. |
* '''Non-negativity''': each claimant should get a non-negative amount, <math>\forall i: x_i\geq 0 </math>. |
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*'''Claims-boundedness''': each claimant should get at most his claim, <math>\forall i: x_i\leq c_i </math>. |
*'''Claims-boundedness''': each claimant should get at most his claim, <math>\forall i: x_i\leq c_i </math>. |
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*'''Minimal-rights''': stronger than non-negativity: each claimant should get at least his minimal right, which is what's left if all other agents get their full claims: <math>\forall i: x_i\geq m_i, \text{ where } m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights. |
*'''Minimal-rights''': stronger than non-negativity: each claimant should get at least his minimal right, which is what's left if all other agents get their full claims: <math>\forall i: x_i\geq m_i, \text{ where } m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. |
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**Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights. |
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*'''Equal treatment of equals (ETE)''': two claimants with identical claims should get identical allocations: <math>c_i=c_j \implies x_i=x_j</math>. In generalized problems of ''claims with priorities'', equal treatment of equals is required to hold for agents in each priority class, but not for agents in different priority classes. |
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*'''Equal treatment of equal groups''': stronger than ETE: two subsets of claimants with the same total claim should get the same total allocation. |
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*[[Anonymity (social choice)|'''Anonymity''']]: stronger than ETE: if we permute the vector of claims, then the vector of allocations is permuted accordingly. |
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*'''Order-preservation''': stronger than ETE: agents with weakly-higher claims should get weakly-more and should lose weakly-more: <math>c_i \geq c_j \implies (x_i\geq x_j \text{ and } c_i-x_i\geq c_j-x_j) </math>. |
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*'''Group order preservation''': stronger than both group-ETE and order preservation: it requires order-preservation among every two subsets of agents. |
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== See also == |
== See also == |
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* [[Entitlement (fair division)]] |
* [[Entitlement (fair division)]] |
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* [[Proportional cake-cutting with different entitlements]] |
* [[Proportional cake-cutting with different entitlements]] |
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*[[Strategic bankruptcy problem]] |
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==References== |
==References== |
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{{game theory}} |
{{game theory}} |
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[[Category: |
[[Category:Bankruptcy theory|*]] |
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[[Category:Game theory]] |
[[Category:Game theory game classes]] |
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[[Category:Fair division]] |
[[Category:Fair division]] |
Latest revision as of 05:00, 15 January 2024
A bankruptcy problem,[1] also called a claims problem,[2] is a problem of distributing a homogeneous divisible good (such as money) among people with different claims. The focus is on the case where the amount is insufficient to satisfy all the claims.
The canonical application is a bankrupt firm that is to be liquidated. The firm owes different amounts of money to different creditors, but the total worth of the company's assets is smaller than its total debt. The problem is how to divide the scarce existing money among the creditors.
Another application would be the division of an estate amongst several heirs, particularly when the estate cannot meet all the deceased's commitments.
A third application[2] is tax assessment. One can consider the claimants as taxpayers, the claims as the incomes, and the endowment as the total after-tax income. Determining the allocation of total after-tax income is equivalent to determining the allocation of tax payments.
Definitions
[edit]The amount available to divide is denoted by (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by .
It is assumed that , that is, the total claims are (weakly) larger than the estate.
A division rule is a function that maps a problem instance to a vector such that and for all i. That is: each claimant receives at most its claim, and the sum of allocations is exactly the estate E.
Generalizations
[edit]There are generalized variants in which the total claims might be smaller than the estate. In these generalized variants, is not assumed and is not required.
Another generalization, inspired by realistic bankruptcy problems, is to add an exogeneous priority ordering among the claimants, that may be different even for claimants with identical claims. This problem is called a claims problem with priorities. Another variant is called a claims problem with weights.
Rules
[edit]There are various rules for solving bankruptcy problems in practice.[1]
- The proportional rule divides the estate proportionally to each agent's claim. Formally, each claimant i receives , where r is a constant chosen such that . We denote the outcome of the proportional rule by .
- There is a variant called truncated-claims proportional rule, in which each claim larger than E is truncated to E, and then the proportional rule is activated. That is, it equals , where .[2]
- The adjusted proportional rule[3] first gives, to each agent i, his minimal right, which is the amount not claimed by the other agents. Formally, . Note that implies . Then, it revises the claim of agent i to , and the estate to . Note that . Finally, it activates the truncated-claims proportional rule, that is, it returns , where . With two claimants, the revised claims are always equal, so the remainder is divided equally. With three or more claimants, the revised claims may be different.
- The constrained equal awards rule divides the estate equally among the agents, ensuring that nobody gets more than their claim. Formally, each claimant i receives , where r is a constant chosen such that . We denote the outcome of this rule by . In the context of taxation, it is known as leveling tax.[2]
- The constrained equal losses rule divides equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer. Formally, each claimant i receives , where r is chosen such that . This rule was discussed by Maimonides.[4] In the taxation context, it is known as poll tax.
- The contested garment rule (also called the Talmud rule) uses the CEA rule on half the claims if the estate is smaller than half the total claim; otherwise, it gives each claimant half their claims, and applies the CEL rule. Formally, if then ; Otherwise, .
- The following rule is attributed[2] to Piniles.[5] If the sum of claims is larger than 2E, then it applies the CEA rule on half the claims, that is, it returns ; Otherwise, it gives each agent half its claim and then applies CEA on the remainder, that is, it returns .
- The constrained egalitarian rule[6] works as follows. If the sum of claims is larger than 2E, then it runs the CEA rule on half the claims, giving each claimant i . Otherwise, it gives each agent i , In both cases, r is a constant chosen such that the sum of allocations equals E.
- The random arrival rule works as follows. Suppose claimants arrive one by one. Each claimant receives all his claim, up to the available amount. The rule returns the average of resulting allocation vectors when the arrival order is chosen uniformly at random.[7] Formally:
.
Bankruptcy rules and cooperative games
[edit]Bargaining games
[edit]It is possible to associate each bankruptcy problem with a cooperative bargaining problem, and use a bargaining rule to solve the bankruptcy problem. Then:
- The Nash bargaining solution corresponds to the constrained equal awards rule;[8]
- The lexicographic-egalitarian bargaining solution also corresponds to the constrained equal awards rule;[2]
- The weighted Nash bargaining solution, with weights equal to the claims, corresponds to the proportional rule;[8]
- The Kalai-Smorodinsky bargaining solution corresponds to the truncated-claims proportional rule;[8]
- The extended-equal-losses bargaining solution corresponds to the truncated-claims constrained-equal-losses rule.[2]
Coalitional games
[edit]It is possible to associate each bankruptcy problem with a cooperative game in which the value of each coalition is its minimal right - the amount that this coalition can ensure itself if all other claimants get their full claim (that is, the amount this coalition can get without going to court). Formally, the value of each subset S of claimants is . The resulting game is convex,[4] so its core is non-empty. One can use a solution concept for cooperative games, to solve the corresponding bankruptcy problem. Every division rule that depends only on the truncated claims corresponds to a cooperative-game solution. In particular:
- The Shapley value corresponds to the random-arrival rule;[7]
- The prenucleolus corresponds to the Talmud rule;[4]
- The Dutta-Ray solution corresponds to the constrained equal awards rule;[9]
- The Tau-value solution corresponds to the adjusted proportional rule.[3]
An alternative way to associate a claims problem with a cooperative game[10] is its maximal right - the amount that this coalition can ensure itself if all other claimants drop their claims: .
Properties of division rules
[edit]In most settings, division rules are often required to satisfy the following basic properties:[2]
- Feasibility: the sum of allocations is at most the total estate, .
- Efficiency: stronger than feasibility: the sum of allocations equals the total estate, .
- Non-negativity: each claimant should get a non-negative amount, .
- Claims-boundedness: each claimant should get at most his claim, .
- Minimal-rights: stronger than non-negativity: each claimant should get at least his minimal right, which is what's left if all other agents get their full claims: .
- Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights.
- Equal treatment of equals (ETE): two claimants with identical claims should get identical allocations: . In generalized problems of claims with priorities, equal treatment of equals is required to hold for agents in each priority class, but not for agents in different priority classes.
- Equal treatment of equal groups: stronger than ETE: two subsets of claimants with the same total claim should get the same total allocation.
- Anonymity: stronger than ETE: if we permute the vector of claims, then the vector of allocations is permuted accordingly.
- Order-preservation: stronger than ETE: agents with weakly-higher claims should get weakly-more and should lose weakly-more: .
- Group order preservation: stronger than both group-ETE and order preservation: it requires order-preservation among every two subsets of agents.
See also
[edit]- Entitlement (fair division)
- Proportional cake-cutting with different entitlements
- Strategic bankruptcy problem
References
[edit]- ^ a b Alcalde, José; Peris, Josep E. (2017-02-17). "Equal Awards vs. Equal Losses in Bankruptcy Problems". SSRN. doi:10.2139/ssrn.2919582. S2CID 158036131. SSRN 2919582.
- ^ a b c d e f g h Thomson, William (2003-07-01). "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.
- ^ a b Curiel, I. J.; Maschler, M.; Tijs, S. H. (1987-09-01). "Bankruptcy games". Zeitschrift für Operations Research. 31 (5): A143–A159. doi:10.1007/BF02109593. ISSN 1432-5217. S2CID 206811949.
- ^ a b c Aumann, Robert J; Maschler, Michael (1985-08-01). "Game theoretic analysis of a bankruptcy problem from the Talmud". Journal of Economic Theory. 36 (2): 195–213. doi:10.1016/0022-0531(85)90102-4. ISSN 0022-0531.
- ^ Piniles, Zvi Menahem (1863). Darkah Shel Torah (Hebrew). Wien: Forester.
- ^ Chun, Youngsub; Schummer, James; Thomson, William (1998). "Constrained Egalitarianism: A New Solution for Claims Problems".
{{cite journal}}
: Cite journal requires|journal=
(help) - ^ a b O'Neill, Barry (1982-06-01). "A problem of rights arbitration from the Talmud". Mathematical Social Sciences. 2 (4): 345–371. doi:10.1016/0165-4896(82)90029-4. hdl:10419/220805. ISSN 0165-4896.
- ^ a b c Dagan, Nir; Volij, Oscar (1993-11-01). "The bankruptcy problem: a cooperative bargaining approach". Mathematical Social Sciences. 26 (3): 287–297. doi:10.1016/0165-4896(93)90024-D. ISSN 0165-4896.
- ^ Dutta, Bhaskar; Ray, Debraj (1989). "A Concept of Egalitarianism Under Participation Constraints". Econometrica. 57 (3): 615–635. doi:10.2307/1911055. ISSN 0012-9682. JSTOR 1911055.
- ^ Driessen, Theo (1995). "An alternative game theoretic analysis of a bankruptcy problem from the Talmud: the case of the greedy bankruptcy game".
{{cite journal}}
: Cite journal requires|journal=
(help)