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Wikipedia / 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 (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by .

, that is, the total claims are (weakly) larger than the estate.

A division rule for all i. That is: each claimant receives at most its claim, and the sum of allocations is exactly the estate E.

Generalizations

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.

  • The proportional rule divides the estate proportionally to each agent's claim. Formally, each claimant i, where r.

  • There is a variant called truncated-claims proportional rule, in which each claim larger than E is truncated to E.

  • The adjusted proportional rule first gives, to each agent i, his minimal right. Then, it revises the claim of agent i. 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, where r. In the context of taxation, it is known as leveling tax.

  • 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, where r. This rule was discussed by Maimonides. In the taxation context, it is known as poll tax.

  • The contested garment rule (also called the Talmud rule.

  • The following rule is attributed to Piniles. If the sum of claims is larger than 2E .

  • 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, 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. Formally:

.

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 gamescoalitional

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. 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.

Properties of division rules

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

  • Feasibility.

  • Efficiency.

  • Non-negativity.

  • Claims-boundedness.

Minimal-rights.

  • Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights.

  • Equal treatment of equals (ETE). 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.

  • Group order preservation: stronger than both group-ETE and order preservation: it requires order-preservation among every two subsets of agents.

Repeated bankruptcy problems

A repeated bankruptcy problem (aka dynamic claims problem) is an extension of the basic model, in which a bankruptcy problem with the same set of players (but different claims and different endowments) occurs repeatedly. For example, in refugee camps, the available food and money is rationed among the refugees every month. The allocation at past occurrences might affect the allocation at the present occurrence. Moreno-Ternero and Vidal-Puga presents several reasonable axioms that should be satisfied by an allocation rule that takes history into account.

See also

References