Proportional rule (bankruptcy)
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 <math>E</math> (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by <math>c_i</math>. Usually, <math>\sum_{i=1}^n c_i > E</math>, that is, the estate is insufficient to satisfy all the claims.
The proportional rule says that each claimant i should receive <math>r \cdot c_i</math>, where r is a constant chosen such that <math>\sum_{i=1}^n r\cdot c_i = E</math>. In other words, each agent gets <math>\frac{c_i}{\sum_{j=1}^n c_j}\cdot E</math>.
Examples Examples with two claimants: * <math>PROP(60,90; 100) = (40,60)</math>. That is: if the estate is worth 100 and the claims are 60 and 90, then <math>r = 2/3</math>, so the first claimant gets 40 and the second claimant gets 60. * <math>PROP(50,100; 100) = (33.333,66.667)</math>, and similarly <math>PROP(40,80; 100) = (33.333,66.667)</math>.
Examples with three claimants: <math>PROP(100,200,300; 100) = (16.667, 33.333, 50)</math>. <math>PROP(100,200,300; 200) = (33.333, 66.667, 100)</math>. * <math>PROP(100,200,300; 300) = (50, 100, 150)</math>.
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;
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 <math>PROP(c_1',\ldots,c_n',E)</math>, where <math>c'_i := \min(c_i, E)</math>. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:
- <math>TPROP(100,200,300; 100) = (33.333, 33.333, 33.333)</math>, since all claims are truncated to 100;
- <math>TPROP(100,200,300; 200) = (40, 80, 80)</math>, since the claims vector is truncated to (100,200,200).
- <math>TPROP(100,200,300; 300) = (50, 100, 150)</math>, 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, <math>m_i := \max(0, E-\sum_{j\neq i} c_j)</math>. Note that <math>\sum_{i=1}^n c_i \geq E</math> implies <math>m_i \leq c_i</math>.
Then, it revises the claim of agent i to <math>c'_i := c_i - m_i</math>, and the estate to <math>E' := E - \sum_i m_i</math>. Note that that <math>E' \geq 0</math>.
Finally, it activates the truncated-claims proportional rule, that is, it returns <math>TPROP(c_1,\ldots,c_n,E') = PROP(c_1,\ldots,c_n,E')</math>, where <math>c*_i := \min(c'_i, E')</math>.
With two claimants, the revised claims are always equal, so the remainder is divided equally. Examples:
- <math>APROP(60,90; 100) = (35,65)</math>. The minimal rights are <math>(m_1,m_2) = (10,40)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>; it is divided equally among the claimants.
- <math>APROP(50,100; 100) = (25,75)</math>. The minimal rights are <math>(m_1,m_2) = (0,50)</math>. The remaining claims are <math>(c_1',c_2') = (50,50)</math> and the remaining estate is <math>E'=50</math>.
- <math>APROP(40,80; 100) = (30,70)</math>. The minimal rights are <math>(m_1,m_2) = (20,60)</math>. The remaining claims are <math>(c_1',c_2') = (20,20)</math> and the remaining estate is <math>E'=20</math>.
With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are <math>(0,0,0)</math> and thus the outcome is equal to TPROP, for example, <math>APROP(100,200,300; 200) = TPROP(100,200,300; 200) = (20, 40, 40)</math>.
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.