# 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, $APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)$.