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, 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

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

There are various rules for solving bankruptcy problems in practice. .

Bankruptcy rules and cooperative games

Bargaining games

It is possible to associate each bankruptcy problem with a cooperative bargaining problem, and use a bargaining rule to solve the bankruptcy problem. Then:

Coalitional games

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, 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: An alternative way to associate a claims problem with a cooperative game is its maximal right - the amount that this coalition can ensure itself if all other claimants drop their claims:.

Properties of division rules

In most settings, division rules are often required to satisfy the following basic properties: