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Social Choice and Welfare

Erschienen in:

01.08.2014 | Original Paper

A new approach for bounding awards in bankruptcy problems

verfasst von: José-Manuel Giménez-Gómez, M. Carmen Marco-Gil

Erschienen in: Social Choice and Welfare | Ausgabe 2/2014

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Abstract

The solution for the contested garment problem, proposed in the Babylonic Talmud, suggests that each agent should receive at least some part of the resources whenever the claim exceeds the available amount. In this context, we propose a new method to define lower bounds on awards, an idea that has underlied the theoretical analysis of bankruptcy problems from its beginning (O’Neill, Math Soc Sci 2:345–371, 1982) to present day (Dominguez and Thomson, Econ Theory 28:283–307, 2006). Specifically, starting from the fact that a society establishes its own set of commonly accepted equity principles, our proposal ensures to each agent the smallest amount she gets according to all the admissible rules. We analyze its recursive application for different sets of equity principles.

Vorheriger Artikel Policy convergence in a two-candidate probabilistic voting model

Nächster Artikel Choice in ordered-tree-based decision problems

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For each \(i\in N\), if \(m_{i}(E,c)>0\) and \(\varphi _{i}(E,c)<m_{i}(E,c)\) either \(\sum \nolimits _{i\in N}\varphi _{i}(E,c)<E\), contradicting efficiency, or there is \(j\ne i\) such that \(\varphi _{j}(E,c)>c_{j}\), contradicting claim-boundedness. Otherwise, that is, if \(m_{i}(E,c)=0\), by non-negativity \(\varphi _{i}(E,c)\ge m_{i}(E,c)\).

In this sense, the recursive application of the minimal rights does not satisfy efficiency, since from the second step on, each agent receives nothing.

The Talmud rule (Aumann and Maschler 1985) assigns the awards that the Constrained Equal Awards rule recommends for \((E, c/2)\), when the endowment is less than the half-sum of the claims. Otherwise, each agent receives her half-claim plus the amount provided by the Constrained Equal Losses rule when it is applied to the residual problem (E-C/2, c/2).

The proofs of Remarks 1 and 2 are available from the authors under request.

Self-duality requires invariance regarding the perspective from which the problem is derived, that is, dividing ‘what is available’ or ‘what is missing’. Formally, two properties, \(\mathcal {P}\) and \(\mathcal {P}^{\prime },\) are dual if whenever a rule, \(\varphi ,\) satisfies \(\mathcal {P}\), its dual, \(\varphi ^{d},\) satisfies \(\mathcal {P}^{\prime }\). A property, \(\mathcal {P}\), is self-dual when it coincides with its dual.

Formal definitions of these properties can be found in Thomson (2003).

If \((E,c)\in \fancyscript{B}\backslash \fancyscript{B}_{0}\), there is a permutation \(\pi \) such that \(\pi (c)\) is increasingly ordered and we can compute \(\varphi (E,c)=\pi ^{-1}[\varphi (E,\pi (c))].\) Where a permutation is a bijection applying \(\mathcal {N}\) to itself and, abusing notation, \(\pi (c)\) will denote the claim vector obtained by applying permutation \(\pi \) to its components, i.e. the i-th component of \( \pi (c)\) is \(c_{j}\) whenever \(j=\pi (i)\). Similar considerations apply for \(\pi [\varphi (E,c)]\).

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Titel: A new approach for bounding awards in bankruptcy problems
verfasst von: José-Manuel Giménez-Gómez
M. Carmen Marco-Gil
Publikationsdatum: 01.08.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2014
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0789-0

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