Constrained equal losses

Constrained equal losses (CEL) is a division rule for solving bankruptcy problems. According to this rule, each claimant should lose an equal amount from his or her claim, except that no claimant should receive a negative amount. In the context of taxation, it is known as poll tax.^[1]

Formal definition[edit]

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 CEL rule says that each claimant i should receive $\max(0,c_{i}-r)$ , where r is a constant chosen such that $\sum _{i=1}^{n}\max(0,c_{i}-r)=E$ . The rule can also be described algorithmically as follows:

Initially, all agents are active, and each agent gets his full claim.
While the total allocation is larger than the estate:
- Remove one unit equally from all active agents.
- Each agent whose total allocation drops to zero becomes inactive.

Examples[edit]

Examples with two claimants:

$CEL(60,90;100)=(35,65)$ ; here $r=25$ .
$CEL(50,100;100)=(25,75)$ ; here $r=25$ too.
$CEL(40,80;100)=(30,70)$ ; here $r=10$ .

Examples with three claimants:

$CEL(50,100,150;100)=(0,25,75)$ ; here $r=75$ .
$CEL(50,100,150;200)=(16.667,66.666,116.667)$ ; here $r=33.333$ .
$CEL(50,100,150;300)=(50,100,150)$ ; here $r=0$ .

Usage[edit]

In the Jewish law, if several bidders participate in an auction and then revoke their bids simultaneously, they have to compensate the seller for the loss. The loss is divided among the bidders according to the CEL rule.^[2]^[3]

Characterizations[edit]

The CEL rule has several characterizations. It is the only rule satisfying the following sets of axioms:

Equal treatment of equals, minimal rights first, and composition down;^[4]
Conditional null compensation, and composition up;^[5]
Conditional null compensation, and the dual of claims-monotonicity.^[6]

Dual rule[edit]

The constrained equal awards (CEA) rule is the dual of the CEL rule, that is: for each problem $(c,E)$ , we have $CEA(c,E)=c-CEL(c,\sum c-E)$ .

References[edit]

^ 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.
^ Aumann, Robert J; Maschler, Michael (1985-08-01). "Game theoretic analysis of a bankruptcy problem from the Talmud". Journal of Economic Theory. 36 (2): 195–213. doi:10.1016/0022-0531(85)90102-4. ISSN 0022-0531.
^ Maimonides, Laws of Appraisals and Devoted Property 8:4. "If the first bidder says: "I will [redeem] it for ten selaim," the second: "...for twenty," and a third "...for 24," and the second and third bidders retract at the same time, we enable the first to redeem it for 10, and we expropriate 7 from the property of both the second and the third. Thus, the Temple treasury collects 24. Similarly, if all three of them retract and the consecrated article is [ultimately] sold for 3, we expropriate 7 selaim from the property of all of them."
^ Herrero, Carmen (2003), Sertel, Murat R.; Koray, Semih (eds.), "Equal Awards vs. Equal Losses: Duality in Bankruptcy", Advances in Economic Design, Studies in Economic Design, Berlin, Heidelberg: Springer, pp. 413–426, doi:10.1007/978-3-662-05611-0_22, ISBN 978-3-662-05611-0, retrieved 2021-09-29
^ Herrero, Carmen; Villar, Antonio (2002-12-01). "Sustainability in bankruptcy problems". Top. 10 (2): 261–273. doi:10.1007/BF02579019. ISSN 1863-8279. S2CID 120694615.
^ C-H Yeh, 2001, "Sustainability, claims monotonicity, and the constrained equal award rule", Mimeo.

[: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.

[:2-2] Aumann, Robert J; Maschler, Michael (1985-08-01). "Game theoretic analysis of a bankruptcy problem from the Talmud". Journal of Economic Theory. 36 (2): 195–213. doi:10.1016/0022-0531(85)90102-4. ISSN 0022-0531.

[3] Maimonides, Laws of Appraisals and Devoted Property 8:4. "If the first bidder says: "I will [redeem] it for ten selaim," the second: "...for twenty," and a third "...for 24," and the second and third bidders retract at the same time, we enable the first to redeem it for 10, and we expropriate 7 from the property of both the second and the third. Thus, the Temple treasury collects 24. Similarly, if all three of them retract and the consecrated article is [ultimately] sold for 3, we expropriate 7 selaim from the property of all of them."

[4] Herrero, Carmen (2003), Sertel, Murat R.; Koray, Semih (eds.), "Equal Awards vs. Equal Losses: Duality in Bankruptcy", Advances in Economic Design, Studies in Economic Design, Berlin, Heidelberg: Springer, pp. 413–426, doi:10.1007/978-3-662-05611-0_22, ISBN 978-3-662-05611-0, retrieved 2021-09-29

[5] Herrero, Carmen; Villar, Antonio (2002-12-01). "Sustainability in bankruptcy problems". Top. 10 (2): 261–273. doi:10.1007/BF02579019. ISSN 1863-8279. S2CID 120694615.

[6] C-H Yeh, 2001, "Sustainability, claims monotonicity, and the constrained equal award rule", Mimeo.

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