Proportional rule (bankruptcy) explained

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

E (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by c _{i} . Usually, \sum ^{n} _{i =1} 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 ⋅ c _{i} , where r is a constant chosen such that \sum ^{n} _{i =1} r ⋅ c _{i} = E . In other words, each agent gets (c _{i})/(\sum ^{n} _{j =1} c _{j}) ⋅ 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;^[2]
• Self-duality and composition-down;
• No advantageous transfer;^{[3] [4] [5]}
• Resource linearity;
• No advantageous merging and no advantageous splitting.^{[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

PROP(c _{1',\ldots,c} _{n',E)} , where 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: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

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,

m _{i} : = max ( 0 , E - \sum _{j ≠} c _{j)} . Note that \sum ^{n} _{i =1} c _{i} \geq E implies m _{i} \leq c _{i} .

Then, it revises the claim of agent i to

c' _{i} : = c _{i} - m _{i} , and the estate to E' : = E - \sum _{i} m _{i} . Note that that E' \geq 0 .

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

TPROP(c _{1,\ldots,c} _{n,E')} = PROP(c _{1'',\ldots,c} _{n'',E')} , where c'' _{i} : = min (c' _{i,} E') .

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',c} _{2')} = ( 50 , 50 ) and the remaining estate is E' =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',c} _{2')} = ( 50 , 50 ) and the remaining estate is E' =50 .APROP( 40 , 80 ; 100 ) = ( 30 , 70 ) . The minimal rights are (m _{1,m} _{2)} = ( 20 , 60 ) . The remaining claims are (c _{1',c} _{2')} = ( 20 , 20 ) and the remaining estate is E' =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 ) .

See also

• Proportional division
• Proportional representation

Notes and References

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