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The bankruptcy problem,^[1] also called the claims problem,^[2] is the 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

The amount available to divide is denoted by E (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by c_i.

It is assumed that $\sum _{i=1}^{n}c_{i}\geq E$ , that is, the total claims are (weakly) larger than the estate.

A division rule is a function that maps a problem instance $(c_{1},\ldots ,c_{n},E)$ to a vector $(x_{1},\ldots ,x_{n})$ such that $\sum _{i=1}^{n}x_{i}=E$ and $0\leq x_{i}\leq c_{i}$ for all i. That is: each claimant receives at most its claim, and the sum of allocations is exactly the estate E.

There are generalized variants in which the total claims might be smaller than the estate. In these generalized variants, $\sum _{i=1}^{n}c_{i}\geq E$ is not assumed and $0\leq x_{i}\leq c_{i}$ is not required.

Rules

There are several rules for solving bankruptcy problems in practice.^[1]

Proportional rule

The proportional rule divides the estate proportionally to each agent's claim. Formally, each claimant i receives $r\cdot c_{i}$ , where r is a constant chosen such that $\sum _{i=1}^{n}r\cdot c_{i}=E$ . We denote the outcome of the proportional rule by $PROP(c_{1},\ldots ,c_{n};E)$ .

Examples:

$PROP(60,90;100)=(40,60)$ . That is: if the estate is worth 100 and the claims are 60 and 90, then $r=2/3$ , so the first claimant gets 40 and the second claimant gets 60.
$PROP(50,100;100)=(33.333,66.667)$ .

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 $PROP(c_{1}',\ldots ,c_{n}',E)$ , where $c'_{i}:=\min(c_{i},E)$ .^[2]

Adjusted proportional rule

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, $m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})$ . Note that $\sum _{i=1}^{n}c_{i}\geq E$ implies $m_{i}\leq c_{i}$ .

Then, it revises the claim of agent i to $c'_{i}:=c_{i}-m_{i}$ , and the estate to $E':=E-\sum _{i}m_{i}$ . Note that that $E'\geq 0$ .

Finally, it activates the truncated-claims proportional rule, that is, it returns $PROP(c_{1}'',\ldots ,c_{n}'',E')$ , where $c''_{i}:=\min(c'_{i},E')$ .

Examples:

$PROP(60,90;100)=(35,65)$ . The minimal rights are $(m_{1},m_{2})=(10,40)$ . The remaining claims are $(c_{1}',c_{2}')=(50,50)$ and the remaining estate is $E'=50$ ; it is divided equally among the claimants.
$PROP(60,90;100)=(25,75)$ . The minimal rights are $(m_{1},m_{2})=(0,50)$ . The remaining claims are $(c_{1}',c_{2}')=(50,50)$ and the remaining estate is $E'=50$ ; it is divided equally among the claimants.

Constrained equal awards

The constrained equal-awards rule divides the estate equally among the agents, ensuring that nobody gets more than their claim.

Constrained equal losses

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.

Contested garment rule

Another rule, appearing already in the Babylonian Talmud, is the contested garment rule.^[4]^[5]

References

^ ^a ^b Alcalde, José; Peris, Josep E. (2017-02-17). "Equal Awards vs. Equal Losses in Bankruptcy Problems". SSRN.
^ ^a ^b ^c "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. 2003-07-01. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.
^ 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.
^ Dagan, Nir (1996). "New characterizations of old bankruptcy rules". Social Choice and Welfare.
^ 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. ISSN 0165-4896.

This applied mathematics–related article is a stub. You can help Wikipedia by expanding it.

[:0-1] Alcalde, José; Peris, Josep E. (2017-02-17). "Equal Awards vs. Equal Losses in Bankruptcy Problems". SSRN.

[:1-2] "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. 2003-07-01. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.

[3] 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.

[4] Dagan, Nir (1996). "New characterizations of old bankruptcy rules". Social Choice and Welfare.

[5] 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. ISSN 0165-4896.

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@@ Line 25: / Line 25: @@
 === Proportional rule ===
-The '''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>
+The '''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>.
+Examples:
-For example, 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.
+* <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.
-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 is the proportional rule with claims  <math>c'_i := \min(c_i, E)</math>. <ref name=":1" />
+* <math>PROP(50,100; 100) = (33.333,66.667)</math>.
+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" />
+=== Adjusted proportional rule ===
+The '''adjusted proportional rule'''<ref>{{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>.
+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>.
+Finally, it activates the truncated-claims proportional rule, that is, it returns <math>PROP(c_1'',\ldots,c_n'',E')</math>, where <math>c''_i := \min(c'_i, E')</math>.
+Examples:
+* <math>PROP(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.
+* <math>PROP(60,90; 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>; it is divided equally among the claimants.
 === Constrained equal awards ===
@@ Line 38: / Line 53: @@
 === Contested garment rule ===
-Another rule, appearing already in the [[Babylonian Talmud]], is 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>
+Another rule, appearing already in the [[Babylonian Talmud]], is 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><ref>{{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>
 == See also ==

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Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game Mechanism design
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Theorems	Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Search optimizations	Alpha–beta pruning Aspiration window Principal variation search max^n algorithm Paranoid algorithm Lazy SMP
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