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.^[1]

Formal definition

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

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

Examples

Examples with two claimants:

• ${\displaystyle PROP(60,90;100)=(40,60)}$. That is: if the estate is worth 100 and the claims are 60 and 90, then ${\displaystyle r=2/3}$, so the first claimant gets 40 and the second claimant gets 60.
• ${\displaystyle PROP(50,100;100)=(33.333,66.667)}$, and similarly ${\displaystyle PROP(40,80;100)=(33.333,66.667)}$.

Examples with three claimants:

• ${\displaystyle PROP(100,200,300;100)=(16.667,33.333,50)}$.
• ${\displaystyle PROP(100,200,300;200)=(33.333,66.667,100)}$.
• ${\displaystyle PROP(100,200,300;300)=(50,100,150)}$.

Characterizations

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

• Self-duality and composition-up;^[2]
• Self-duality and composition-down;
• No advantageous transfer;^[3][4][5]
• Resource linearity;^[5]
• No advantageous merging and no advantageous splitting.^[5][6][7]

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 ${\displaystyle PROP(c_{1}',\ldots ,c_{n}',E)}$, where ${\displaystyle c'_{i}:=\min(c_{i},E)}$. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:

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

Adjusted-proportional rule

The adjusted proportional rule^[8] first gives, to each agent i, their 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 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. Examples:

• ${\displaystyle APROP(60,90;100)=(35,65)}$. The minimal rights are ${\displaystyle (m_{1},m_{2})=(10,40)}$. The remaining claims are ${\displaystyle (c_{1}',c_{2}')=(50,50)}$ and the remaining estate is ${\displaystyle E'=50}$; it is divided equally among the claimants.
• ${\displaystyle APROP(50,100;100)=(25,75)}$. The minimal rights are ${\displaystyle (m_{1},m_{2})=(0,50)}$. The remaining claims are ${\displaystyle (c_{1}',c_{2}')=(50,50)}$ and the remaining estate is ${\displaystyle E'=50}$.
• ${\displaystyle APROP(40,80;100)=(30,70)}$. The minimal rights are ${\displaystyle (m_{1},m_{2})=(20,60)}$. The remaining claims are ${\displaystyle (c_{1}',c_{2}')=(20,20)}$ and the remaining estate is ${\displaystyle E'=20}$.

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