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 ' (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by '. Usually, , that is, the estate is insufficient to satisfy all the claims.

The proportional rule says that each claimant i should receive , where r is a constant chosen such that . In other words, each agent gets .

Examples

Examples with two claimants:

. That is: if the estate is worth 100 and the claims are 60 and 90, then , so the first claimant gets 40 and the second claimant gets 60.
, and similarly .

Examples with three claimants:

.
.
.

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 , where . The results are the same for the two-claimant problems above, but for the three-claimant problems we get:

, since all claims are truncated to 100;
, since the claims vector is truncated to (100,200,200).
, 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, . Note that implies .

Then, it revises the claim of agent i to , and the estate to . Note that that .

Finally, it activates the truncated-claims proportional rule, that is, it returns , where .

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

. The minimal rights are . The remaining claims are and the remaining estate is ; it is divided equally among the claimants.
. The minimal rights are . The remaining claims are and the remaining estate is .
. The minimal rights are . The remaining claims are and the remaining estate is .

With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are and thus the outcome is equal to TPROP, for example, .

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.

References