Bankruptcy problem

The amount available to divide is denoted by ${\displaystyle E}$ (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by ${\displaystyle c_{i}}$.

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

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

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

• The proportional rule divides the estate proportionally to each agent's claim. Formally, each claimant i receives ${\displaystyle r\cdot c_{i}}$, where r is a constant chosen such that ${\displaystyle \sum _{i=1}^{n}r\cdot c_{i}=E}$. We denote the outcome of the proportional rule by ${\displaystyle PROP(c_{1},\ldots ,c_{n};E)}$.
• There is a variant called truncated-claims proportional rule, in which each claim larger than E is truncated to E, and then the proportional rule is activated. That is, it equals ${\displaystyle PROP(c_{1}',\ldots ,c_{n}',E)}$, where ${\displaystyle c'_{i}:=\min(c_{i},E)}$.[2]
• 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, ${\displaystyle m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})}$. Note that ${\displaystyle \sum _{i=1}^{n}c_{i}\geq E}$ implies ${\displaystyle m_{i}\leq c_{i}}$. Then, it revises the claim of agent i to ${\displaystyle c'_{i}:=c_{i}-m_{i}}$, and the estate to ${\displaystyle E':=E-\sum _{i}m_{i}}$. Note that ${\displaystyle E'\geq 0}$. Finally, it activates the truncated-claims proportional rule, that is, it returns ${\displaystyle TPROP(c_{1},\ldots ,c_{n},E')=PROP(c_{1}'',\ldots ,c_{n}'',E')}$, where ${\displaystyle c''_{i}:=\min(c'_{i},E')}$. 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 receives ${\displaystyle \min(c_{i},r)}$, where r is a constant chosen such that ${\displaystyle \sum _{i=1}^{n}\min(c_{i},r)=E}$. We denote the outcome of this rule by ${\displaystyle CEA(c_{1},\ldots ,c_{n};E)}$. In the context of taxation, it is known as leveling tax.[2]
• 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 ${\displaystyle \max(0,c_{i}-r)}$, where r is chosen such that ${\displaystyle \sum _{i=1}^{n}\max(0,c_{i}-r)=E}$. This rule was discussed by Maimonides.[4] In the taxation context, it is known as poll tax.
• The contested garment rule (also called the Talmud rule) uses the CEA rule on half the claims if the estate is smaller than half the total claim; otherwise, it gives each claimant half their claims, and applies the CEL rule. Formally, if ${\displaystyle 2E<\sum _{i=1}^{n}c_{i}}$ then ${\displaystyle CG(c_{1},\ldots ,c_{n};E)=CEA(c_{1}/2,\ldots ,c_{n}/2;E)}$; Otherwise, ${\displaystyle CG(c_{1},\ldots ,c_{n};E)=c/2+CEL(c_{1}/2,\ldots ,c_{n}/2;E-\sum _{j}(c_{j}/2))}$.
• The following rule is attributed[2] to Piniles.[5] If the sum of claims is larger than 2E, then it applies the CEA rule on half the claims, that is, it returns ${\displaystyle CEA(c_{1}/2,\ldots ,c_{n}/2;E)}$ ; Otherwise, it gives each agent half its claim and then applies CEA on the remainder, that is, it returns ${\displaystyle (c_{1}/2,\ldots ,c_{n}/2)+CEA(c_{1}/2,\ldots ,c_{n}/2;E-\sum _{j=1}^{n}c_{j}/2)}$ .
• The constrained egalitarian rule[6] 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 ${\displaystyle \min(c_{i}/2,r)}$. Otherwise, it gives each agent i ${\displaystyle \max(c_{i}/2,\min(c_{i},r))}$, 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.[7] Formally:

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

In most settings, division rules are often required to satisfy the following basic properties:[2]

• Feasibility: the sum of allocations is at most the total estate, ${\displaystyle \sum _{i=1}^{n}x_{i}\leq E}$.
• Efficiency: stronger than feasibility: the sum of allocations equals the total estate, ${\displaystyle \sum _{i=1}^{n}x_{i}=E}$.
• Non-negativity: each claimant should get a non-negative amount, ${\displaystyle \forall i:x_{i}\geq 0}$.
• Claims-boundedness: each claimant should get at most his claim, ${\displaystyle \forall i:x_{i}\leq c_{i}}$.
• Minimal-rights: stronger than non-negativity: each claimant should get at least his minimal right, which is what's left if all other agents get their full claims: ${\displaystyle \forall i:x_{i}\geq m_{i},{\text{ where }}m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})}$.
  • Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights.
• Equal treatment of equals (ETE): two claimants with identical claims should get identical allocations: ${\displaystyle c_{i}=c_{j}\implies x_{i}=x_{j}}$. 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: stronger than ETE: agents with weakly-higher claims should get weakly-more and should lose weakly-more: ${\displaystyle c_{i}\geq c_{j}\implies (x_{i}\geq x_{j}{\text{ and }}c_{i}-x_{i}\geq c_{j}-x_{j})}$.
• Group order preservation: stronger than both group-ETE and order preservation: it requires order-preservation among every two subsets of agents.

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

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