A bankruptcy 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
- The proportional rule divides the estate proportionally to each agent's claim. Formally, each claimant i receives <math>r \cdot c_i</math>, where r is a constant chosen such that <math>\sum_{i=1}^n r\cdot c_i = E</math>. We denote the outcome of the proportional rule by <math>PROP(c_1,\ldots,c_n ; E)</math>.
- 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 <math>PROP(c_1',\ldots,c_n',E)</math>, where <math>c'_i := \min(c_i, E)</math>. first gives, to each agent i, his minimal right, which is the amount not claimed by the other agents. Formally, <math>m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that <math>\sum_{i=1}^n c_i \geq E</math> implies <math>m_i \leq c_i</math>. Then, it revises the claim of agent i to <math>c'_i := c_i - m_i</math>, and the estate to <math>E' := E - \sum_i m_i</math>. Note that <math>E' \geq 0</math>. Finally, it activates the truncated-claims proportional rule, that is, it returns <math>TPROP(c_1,\ldots,c_n,E') = PROP(c_1,\ldots,c_n,E')</math>, where <math>c_i := \min(c'_i, E')</math>. 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 <math>\min(c_i, r)</math>, where r is a constant chosen such that <math>\sum_{i=1}^n \min(c_i,r) = E</math>. We denote the outcome of this rule by <math>CEA(c_1,\ldots,c_n ; E)</math>. In the context of taxation, it is known as leveling tax. 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 <math>2 E < \sum_{i=1}^n c_i </math> then <math>CG(c_1,\ldots,c_n; E) = CEA(c_1/2,\ldots,c_n/2; E)</math>; Otherwise, <math>CG(c_1,\ldots,c_n; E) = c/2 + CEL(c_1/2,\ldots,c_n/2; E-\sum_j (c_j/2))</math>.
- The following rule is attributed If the sum of claims is larger than 2E, then it applies the CEA rule on half the claims, that is, it returns <math>CEA(c_1/2,\ldots,c_n/2; E)</math> ; Otherwise, it gives each agent half its claim and then applies CEA on the remainder, that is, it returns <math>(c_1/2,\ldots,c_n/2) + CEA(c_1/2,\ldots,c_n/2; E-\sum_{j=1}^n c_j/2)</math> .
- 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 <math>\min(c_i/2, r)</math>. Otherwise, it gives each agent i <math>\max(c_i/2, \min(c_i, r))</math>, 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:
<math>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))</math>.
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:
- The Nash bargaining solution corresponds to the constrained equal awards rule;
- The lexicographic-egalitarian bargaining solution also corresponds to the constrained equal awards rule;
- The Tau-value solution corresponds to the adjusted proportional rule. is its maximal right - the amount that this coalition can ensure itself if all other claimants drop their claims: <math>v(S) := \min\left(E, \sum_{j\in S}c_j\right)</math>.
Properties of division rules
In most settings, division rules are often required to satisfy the following basic properties: 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