Bankruptcy problem

A bankruptcy problem, also called a claims problem, 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 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.

Proportional

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 with two claimants:

• $PROP(60,90;100)=(40,60)$. 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.
• $PROP(50,100;100)=(33.333,66.667)$, and similarly $PROP(40,80;100)=(33.333,66.667)$.

Examples with three claimants:

• $PROP(100,200,300;100)=(16.667,33.333,50)$.
• $PROP(100,200,300;200)=(33.333,66.667,100)$.
• $PROP(100,200,300;300)=(50,100,150)$.

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 $c'_{i}:=\min(c_{i},E)$. The results are the same for the two-claimant problems, but for the three-claimant problems we get:

• $TPROP(100,200,300;100)=(33.333,33.333,33.333)$, since all claims are truncated to 100;
• $TPROP(100,200,300;200)=(40,80,80)$, since the claims vector is truncated to (100,200,200).
• $TPROP(100,200,300;300)=(50,100,150)$, since here the claims are not truncated.

Adjusted proportional

The adjusted proportional rule 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 $TPROP(c_{1},\ldots ,c_{n},E')=PROP(c_{1}'',\ldots ,c_{n}'',E')$, where $c''_{i}:=\min(c'_{i},E')$.

With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:

• $APROP(60,90;100)=(35,65)$. 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.
• $APROP(50,100;100)=(25,75)$. 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 $E'=50$.
• $APROP(40,80;100)=(30,70)$. The minimal rights are $(m_{1},m_{2})=(20,60)$. The remaining claims are $(c_{1}',c_{2}')=(20,20)$ and the remaining estate is $E'=20$.

With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are $(0,0,0)$ and thus the outcome is equal to TPROP, for example, $APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)$.

Constrained equal awards

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 $\min(c_{i},r)$, where r is a constant chosen such that $\sum _{i=1}^{n}\min(c_{i},r)=E$. We denote the outcome of this rule by $CEA(c_{1},\ldots ,c_{n};E)$. In the context of taxation, it is known as leveling tax. Examples with two claimants:

• $CEA(60,90;100)=CEA(50,100;100)=(50,50)$; here $r=50$.
• $CEA(40,80;100)=(40,60)$; here $r=60$.

Examples with three claimants:

• $CEA(50,100,150;100)=(33.333,33.333,33.333)$; here $r=33.333$.
• $CEA(50,100,150;200)=(50,75,75)$; here $r=75$.
• $CEA(50,100,150;300)=(50,100,150)$; here $r=150$.

Piniles' rule

The following rule is attributed to Piniles.

• If the sum of claims is larger than 2E, then it runs the CEA rule on half the claims, that is, it returns $CEA(c_{1}/2,\ldots ,c_{n}/2;E)$ .
• Otherwise, it gives each agent half its claim and then runs CEA on the remainder, that is, it returns $(c_{1}/2,\ldots ,c_{n}/2)+CEA(c_{1}/2,\ldots ,c_{n}/2;E-\sum _{j=1}^{n}c_{j}/2)$ .

Examples with two claimants:

• $PINI(60,90;100)=(42.5,57.5)$. Initially the claimants get (30,45). The remaining claims are (30,45) and the remaining estate is 25, so it is divided equally.
• $PINI(50,100;100)=(37.5,62.5)$. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally.
• $PINI(50,100;100)=(37.5,62.5)$. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally.

Examples with three claimants:

• $PINI(100,200,300;100)=(33.333,33.333,33.333)$. Here the sum of claims is more than twice the estate, so the outcome is $CEA(50,100,150;100)=(33.333,33.333,33.333)$.
• $PINI(100,200,300;200)=(20,40,40)$. Again the sum of claims is more than twice the estate, so the outcome is $CEA(50,100,150;200)=(50,75,75)$.
• $PINI(100,200,300;300)=(50,100,150)$. Again the sum of claims is more than twice the estate, so the outcome is $CEA(50,100,150;300)=(50,100,150)$.

Constrained egalitarianism

The constrained egalitarian rule 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 $\min(c_{i}/2,r)$.
• Otherwise, it gives each agent i $\max(c_{i}/2,\min(c_{i},r))$,

In both cases, r is a constant chosen such that the sum of allocations equals E.

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. Formally, each claimant i receives $\max(0,c_{i}-r)$, where r is chosen such that $\sum _{i=1}^{n}\max(0,c_{i}-r)=E$. This rule was discussed by Maimonides. In the taxation context, it is known as poll tax. Examples with two claimants:

• $CEL(60,90;100)=(35,65)$; here $r=25$.
• $CEL(50,100;100)=(25,75)$; here $r=25$ too.
• $CEL(40,80;100)=(30,70)$; here $r=10$.

Examples with three claimants:

• $CEL(50,100,150;100)=(0,25,75)$; here $r=75$.
• $CEL(50,100,150;200)=(16.667,66.666,116.667)$; here $r=33.333$.
• $CEL(50,100,150;300)=(50,100,150)$; here $r=0$.

Talmud rule

This rule generalizes examples from the Babylonian Talmud.

• If the sum of claims is larger than 2E, then it returns $CEA(c_{1}/2,\ldots ,c_{n}/2;E)$;
• Otherwise, it returns $c/2+CEL(c_{1}/2,\ldots ,c_{n}/2;E-\sum _{j}(c_{j}/2))$.

The rule can also be explained constructively. Suppose E increases from 0 to the half-sum of the claims: the first units are divided equally, until each claimant receives $\min _{i}(c_{i}/2)$. Then, the claimant with the smallest $c_{i}$ is put on hold, and the next units are divided equally among the remaining claimants until each of them up to the next-smallest $c_{i}$. Then, the claimant with the second-smallest $c_{i}$ is put on hold too. This goes on until either the estate is fully divided, or each claimant gets $c_{i}/2$. 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.

Examples with two claimants:

• $TAL(50,100;100)=(25,75)$; here CEL is used.

Examples with three claimants:

• $TAL(100,200,300;100)=(33.333,33.333,33.333)$; here CEA is used.
• $TAL(100,200,300;200)=(50,75,75)$; here CEA is used.
• $TAL(100,200,300;300)=(50,100,150)$; here either CEA or CEL can be used (the result is the same).

The two-claimant version of the Talmud rule is called the contested garment rule.

Random arrival rule

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. Formally:

$RA(c_{1},\ldots ,c_{n};E)={\frac {1}{n!}}\sum _{\pi \in {\text{permutations}}}\min(c_{i},\max(0,E-\sum _{\pi (j)<\pi (i)}c_{j}))$.

See also

• Entitlement (fair division)
• Proportional cake-cutting with different entitlements

References

• Additive rules in bankruptcy problems and other related problems
• The Bankruptcy Problem: a Cooperative Bargaining Approach