Bankruptcy problem

The bankruptcy problem, also called the claims problem, 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 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 at least three simple rules for solving bankruptcy problems in practice. Each is deficient in one or more ways. The rules are:

1. The proportional rule: divide the estate proportionally to each agent's claim. For example, if the estate is worth 100 and the claims are 60 and 90, then the first creditor gets 40 and the second creditor gets 60.
2. The constrained equal-awards rule: divide the estate equally among the agents, ensuring that nobody gets more than their claim.
3. The constrained equal-losses rule: divide equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer.

Another rule, appearing already in the Babylonian Talmud, is the contested garment rule.

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