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 '. 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.
  • , 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 : 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 : claimants with identical claim get identical award.
  • Additivity of claims: if one of the claims is partitioned into sub-claims, the allocation to the other claimants does not change.
  • Independence of irrelevant claims : 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.