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 Template:Anchor

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 $PROP({c_1}^{\prime },\dots ,{c_n}^{\prime },E)$, where $c{\text{'}}_i:=min(c_i,E)$. 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 Template:Anchor

The adjusted proportional rule first gives, to each agent i, their minimal right, which is the amount not claimed by the other agents. Formally, $m_i:=max(0,E-\sum _{j\ne i}{c}_j)$. Note that $\sum _{i=1}^n{c}_i\ge E$ implies $m_i\le c_i$.

Then, it revises the claim of agent i to $c{\text{'}}_i:=c_i-m_i$, and the estate to $E^{\prime }:=E-\sum _i{m}_i$. Note that that $E^{\prime }\ge 0$.

Finally, it activates the truncated-claims proportional rule, that is, it returns $TPROP(c_1,\dots ,c_n,E^{\prime })=PROP({c_1}^{″},\dots ,{c_n}^{″},E^{\prime })$, where $c^{\prime }{\text{'}}_i:=min(c{\text{'}}_i,E^{\prime })$.

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 $({c_1}^{\prime },{c_2}^{\prime })=(50,50)$ and the remaining estate is $E^{\prime }=50$; 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 $({c_1}^{\prime },{c_2}^{\prime })=(50,50)$ and the remaining estate is $E^{\prime }=50$.
• $APROP(40,80;100)=(30,70)$. The minimal rights are $(m_1,m_2)=(20,60)$. The remaining claims are $({c_1}^{\prime },{c_2}^{\prime })=(20,20)$ and the remaining estate is $E^{\prime }=20$.

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

Characterization

Curiel, Maschler and Tijs prove that the AP-rule returns the tau-value of the coalitional game associated with the bankruptcy problem.

The AP-rule is self-dual. In addition, it is the only rule satisfying the following properties:

• Minimal rights (-separability): the outcome remains the same if we first handle each claimant his minimal right and then apply the same rule to the remainder.
• Equal treatment of equals (-symmetry): claimants with identical claim get identical award.
• Additivity of claims: if one of the claims is partitioned into sub-claims (e.g. one of the claimants dies and leaves his claim to his heirs), the allocation to the other claimants does not change.
• Independence of irrelevant claims (also called "game-theoretic"): the outcome does not change if we truncate each claim larger than E to E.

In contrast, the truncated-proportional rule violates minimal-rights, and the proportional rule violates also Independence-of-irrelevant-claims.

See also

• Proportional division
• Proportional representation