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. ^[1]

Formal definition

There is a certain amount of money to divide, denoted by ${\displaystyle E}$ (=Estate or Endowment). There are nclaimants. 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, ${\displaystyle APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)}$.

Characterization

Curiel, Maschler and Tijs ^[8] 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. ^[8]

See also

• Proportional division
• Proportional representation

References

1. William, Thomson (2003-07-01). "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey" . Mathematical Social Sciences. 45 (3): 249–297. doi:10.1016/S0165-4896(02)00070-7. ISSN   0165-4896.
2. Young, H. P (1988-04-01). "Distributive justice in taxation" . Journal of Economic Theory. 44 (2): 321–335. doi:10.1016/0022-0531(88)90007-5. ISSN   0022-0531.
3. Moulin, Hervé (1985). "Egalitarianism and Utilitarianism in Quasi-Linear Bargaining" . Econometrica. 53 (1): 49–67. doi:10.2307/1911723. ISSN   0012-9682. JSTOR   1911723.
4. Moulin, Hervé (1985-06-01). "The separability axiom and equal-sharing methods" . Journal of Economic Theory. 36 (1): 120–148. doi:10.1016/0022-0531(85)90082-1. ISSN   0022-0531.
5. Chun, Youngsub (1988-06-01). "The proportional solution for rights problems" . Mathematical Social Sciences. 15 (3): 231–246. doi:10.1016/0165-4896(88)90009-1. ISSN   0165-4896.
6. O'Neill, Barry (1982-06-01). "A problem of rights arbitration from the Talmud" . Mathematical Social Sciences. 2 (4): 345–371. doi:10.1016/0165-4896(82)90029-4. hdl: 10419/220805 . ISSN   0165-4896.
7. de Frutos, M. Angeles (1999-09-01). "Coalitional manipulations in a bankruptcy problem". Review of Economic Design. 4 (3): 255–272. doi:10.1007/s100580050037. hdl: 10016/4282 . ISSN   1434-4750. S2CID   195240195.
8. Curiel, I. J.; Maschler, M.; Tijs, S. H. (1987-09-01). "Bankruptcy games" . Zeitschrift für Operations Research. 31 (5): A143 –A159. doi:10.1007/BF02109593. ISSN   1432-5217. S2CID   206811949.