Proportional rule (bankruptcy)

The proportional rule is a division rule for solving bankruptcy problems. According to this rule, each claimant should receive an amount proportional to their claim. In the context of taxation, it corresponds to a proportional tax.

Formal definition
There is a certain amount of money to divide, denoted by $$E$$ (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by $$c_i$$. Usually, $$\sum_{i=1}^n c_i > E$$, that is, the estate is insufficient to satisfy all the claims.

The proportional rule says that each claimant i should receive $$r \cdot c_i$$, where r is a constant chosen such that $$\sum_{i=1}^n r\cdot c_i = E$$. In other words, each agent gets $$\frac{c_i}{\sum_{j=1}^n c_j}\cdot E$$.

Examples
Examples with two claimants:
 * $$PROP(60,90; 100) = (40,60)$$. That is: if the estate is worth 100 and the claims are 60 and 90, then $$r = 2/3$$, so the first claimant gets 40 and the second claimant gets 60.
 * $$PROP(50,100; 100) = (33.333,66.667)$$, and similarly $$PROP(40,80; 100) = (33.333,66.667)$$.

Examples with three claimants:
 * $$PROP(100,200,300; 100) = (16.667, 33.333, 50)$$.
 * $$PROP(100,200,300; 200) = (33.333, 66.667, 100)$$.
 * $$PROP(100,200,300; 300) = (50, 100, 150)$$.

Characterizations
The proportional rule has several characterizations. It is the only rule satisfying the following sets of axioms:


 * Self-duality and composition-up;
 * Self-duality and composition-down;
 * No advantageous transfer;
 * Resource linearity;
 * No advantageous merging and no advantageous splitting.

Truncated-proportional rule
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 $$PROP(c_1',\ldots,c_n',E)$$, where $$c'_i := \min(c_i, E)$$. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:


 * $$TPROP(100,200,300; 100) = (33.333, 33.333, 33.333)$$, since all claims are truncated to 100;
 * $$TPROP(100,200,300; 200) = (40, 80, 80)$$, since the claims vector is truncated to (100,200,200).
 * $$TPROP(100,200,300; 300) = (50, 100, 150)$$, since here the claims are not truncated.

Adjusted-proportional rule
The adjusted proportional rule first gives, to each agent i, their minimal right, which is the amount not claimed by the other agents. Formally, $$m_i := \max(0, E-\sum_{j\neq i} c_j)$$. Note that $$\sum_{i=1}^n c_i \geq E$$ implies $$m_i \leq c_i$$.

Then, it revises the claim of agent i to $$c'_i := c_i - m_i$$, and the estate to $$E' := E - \sum_i m_i$$. Note that that $$E' \geq 0$$.

Finally, it activates the truncated-claims proportional rule, that is, it returns $$TPROP(c_1,\ldots,c_n,E') = PROP(c_1,\ldots,c_n,E')$$, where $$c''_i := \min(c'_i, E')$$.

With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:


 * $$APROP(60,90; 100) = (35,65)$$. The minimal rights are $$(m_1,m_2) = (10,40)$$. The remaining claims are $$(c_1',c_2') = (50,50)$$ and the remaining estate is $$E'=50$$; it is divided equally among the claimants.
 * $$APROP(50,100; 100) = (25,75)$$. The minimal rights are $$(m_1,m_2) = (0,50)$$. The remaining claims are $$(c_1',c_2') = (50,50)$$ and the remaining estate is $$E'=50$$.
 * $$APROP(40,80; 100) = (30,70)$$. The minimal rights are $$(m_1,m_2) = (20,60)$$. The remaining claims are $$(c_1',c_2') = (20,20)$$ and the remaining estate is $$E'=20$$.

With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are $$(0,0,0)$$ and thus the outcome is equal to TPROP, for example, $$APROP(100,200,300; 200) = TPROP(100,200,300; 200) = (20, 40, 40)$$.