Colonel William A. Phillips

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, $\sum _{i=1}^{n}c_{i}>E$ , that is, the estate is insufficient to satisfy all the claims.

The proportional rule says that each claimant i should receive $r\cdot c_{i}$ , where r is a constant chosen such that $\sum _{i=1}^{n}r\cdot c_{i}=E$ . In other words, each agent gets ${\frac {c_{i}}{\sum _{j=1}^{n}c_{j}}}\cdot E$ .

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;^[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 $PROP(c_{1}',\ldots ,c_{n}',E)$ , 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, $m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})$ . Note that $\sum _{i=1}^{n}c_{i}\geq E$ 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, $APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)$ .

References

^ 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.
^ 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.
^ Moulin, Hervé (1985). "Egalitarianism and Utilitarianism in Quasi-Linear Bargaining". Econometrica. 53 (1): 49–67. doi:10.2307/1911723. ISSN 0012-9682. JSTOR 1911723.
^ 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.
^ ^a ^b ^c 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.
^ 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.
^ 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.
^ 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.

[:1-1] 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.

[:0-5] 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.

[:5-6] 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] 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.

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