# The Three Musketeers. Old Solutions to Bankruptcy Problems

## Author Info

• Carmen Herrero

(University of Alicante and IVIE)

## Abstract

This paper concentrates on a comparative analysis of three basic rules to solve bankruptcy problems from an axiomatic viewpoint. These rules are: (i) The proportional rule, that divides the estate proportionally to agents' claims. (ii) The constrained equal-awards rule, that divides equally the estate among the agents under the condition that nobody gets more than her claim. (iii) The constrained equal-losses rule, that divides equally the diference between the aggregate claims and the budget, provided no agent ends up with a negative award. The comparative analysis of the three aforementioned rules aims at clarifying the class of real life problems for which each of these solutions is better. With this purpose in mind and following the recommendations given above, we concentrate on those characterizations that permit an easy comparison of these three rules. In particular, we focus on a family of results that characterize each of these rules by three independent axioms, two of which are common to all of them. As the Three Musketeers were four so are our three rules. The Talmud rule here will play the role of D'Artagnan. This is an appealing allocation rule that amounts to solve bankruptcy problems by combining the constrained equal awards rule and the constrained equal losses rule. We start by formally introducing the family of bankruptcy problems and the three basic rules. Then we present several appealing properties for bankruptcy rules. We offer a joint characterization of the three bankruptcy rules in terms of some of those properties, as well as independent characterizations. Previous properties help us to analyze also the contested garment rule, and its consistent extension, the Talmud rule (the rule playing here the role of D'Artagnan). Variants and extensions of the aforementioned rules are also analyzed. The paper ends by providing noncooperative support of the constrained equal-awards rule and of the constrained equal-losses rule.

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## Bibliographic Info

Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0609.

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Handle: RePEc:ecm:wc2000:0609

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## References

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1. Young, H. P., 1988. "Distributive justice in taxation," Journal of Economic Theory, Elsevier, vol. 44(2), pages 321-335, April.
2. Dagan, Nir & Serrano, Roberto & Volij, Oscar, 1997. "A Noncooperative View of Consistent Bankruptcy Rules," Games and Economic Behavior, Elsevier, vol. 18(1), pages 55-72, January.
3. Antonio Villar Notario & Carmen Herrero Blanco & Michael Maschler, 1996. "Individual rights and collective responsibility: The rights-egalitarian solution," Working Papers. Serie AD 1996-04, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
4. Carmen Herrero & Antonio Villar, 2002. "Sustainability in bankruptcy problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 10(2), pages 261-273, December.
5. Carmen Herrero, 1997. "Endogenous reference points and the adjusted proportional solution for bargaining problems with claims," Social Choice and Welfare, Springer, vol. 15(1), pages 113-119.
6. Nir Dagan & Oscar Volij, 1993. "The Bankruptcy Problem: a Cooperative Bargaining Approach," Economic theory and game theory 001, Nir Dagan.
7. Antonio Villar Notario & Carmen Herrero Blanco, 1996. "Agenda independence in allocation problems with single-peaked preferences," Working Papers. Serie AD 1996-14, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
8. Chun, Youngsub, 1989. "A noncooperative justification for egalitarian surplus sharing," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 245-261, June.
9. Damme, E.E.C. van, 1986. "The Nash bargaining solution is optimal," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154426, Tilburg University.
10. M. Angeles de Frutos, 1999. "Coalitional manipulations in a bankruptcy problem," Review of Economic Design, Springer, vol. 4(3), pages 255-272.
11. Nir Dagan, 1996. "New characterizations of old bankruptcy rules," Social Choice and Welfare, Springer, vol. 13(1), pages 51-59, January.
12. Eric van Damme, 1984. "The Nash Bargaining Solution is Optimal," Discussion Papers 597, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
13. Chun, Youngsub, 1988. "The equal-loss principle for bargaining problems," Economics Letters, Elsevier, vol. 26(2), pages 103-106.
14. Herrero, Carmen & Marco, Maria Carmen, 1993. "Rational equal-loss solutions for bargaining problems," Mathematical Social Sciences, Elsevier, vol. 26(3), pages 273-286, November.
15. Carmen Herrero Blanco, 1998. "- Minimal Rights In Claims Problems," Working Papers. Serie AD 1998-20, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
16. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
17. O'Neill, Barry, 1982. "A problem of rights arbitration from the Talmud," Mathematical Social Sciences, Elsevier, vol. 2(4), pages 345-371, June.
18. Chun, Youngsub, 1988. "The proportional solution for rights problems," Mathematical Social Sciences, Elsevier, vol. 15(3), pages 231-246, June.
19. William Thomson, 2001. "On the axiomatic method and its recent applications to game theory and resource allocation," Social Choice and Welfare, Springer, vol. 18(2), pages 327-386.
20. Damme, Eric van, 1986. "The Nash bargaining solution is optimal," Journal of Economic Theory, Elsevier, vol. 38(1), pages 78-100, February.
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