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A Talmudic Approach to Bankruptcy Problems

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Game Theory in Management Accounting

Part of the book series: Contributions to Management Science ((MANAGEMENT SC.))

Abstract

Bankruptcy problems arise when agents hold claims against a certain (perfectly divisible) good, and the available amount is not enough to satisfy them all. A great source of inspiration to solve these problems emanates from the Talmud. We survey classical and recent contributions to the literature that constitute this Talmudic approach to bankruptcy problems.

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Notes

  1. 1.

    For each \(N \in \mathcal{ N}\), each MN, and each \(z \in \mathbb{R}^{n}\), let z M ≡ (z i ) iM .

  2. 2.

    These two axioms were studied first by Curiel et al. (1987).

  3. 3.

    The property was introduced by Moreno-Ternero and Villar (2004) under the name of Securement.

  4. 4.

    The constrained equal-awards rule, A, selects, for each \((N,c,E) \in \mathcal{ D}\), the vector (min{c i , λ}) iN , where λ > 0 is chosen so that iN min{c i , λ} = E. The constrained equal-losses rule, L, selects, for each \((N,c,E) \in \mathcal{ D}\), the vector (max{0, c i λ}) iN , where λ > 0 is chosen so that iN max{0, c i λ} = E.

  5. 5.

    The name was coined by Thomson (2003). To ease its presentation, we assume N = {1, 2}, but dismiss it from the definition.

  6. 6.

    See also Moreno-Ternero (2006).

  7. 7.

    See also van den Brink and Moreno-Ternero (2016).

  8. 8.

    This is a rather pessimistic assessment of what a coalition can achieve. Other more optimistic proposals have been considered in the literature (e.g., Driessen 1998).

  9. 9.

    Curiel et al. (1987) showed that the necessary and sufficient condition for a bankruptcy rule to be associated with a coalitional game is precisely Claims Truncation Invariance.

  10. 10.

    The pre-nucleolus (e.g., Schmeidler, 1969) is the set of payoff vectors at which the vector of dissatisfactions is minimized in the lexicographic (maximin) order among all efficient payoff vectors.

  11. 11.

    Quant and Borm (2011) proposed a different generalization of Aumann and Maschler’s procedure.

  12. 12.

    To ease its presentation, we assume N = {1, 2}, but dismiss it from the definition.

  13. 13.

    More recently, Huijink et al. (2015) have identified the rules in such a family as claim-and-right rules, which give a specific interpretation to the concept of baselines formalized earlier by Hougaard et al. (20122013a,b). See also Timoner and Izquierdo (2016) for a related notion.

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Acknowledgements

Financial support from the Spanish Ministry of Economy and Competitiveness (ECO2014-57413-P) is gratefully acknowledged.

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Correspondence to Juan D. Moreno-Ternero .

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Moreno-Ternero, J.D. (2018). A Talmudic Approach to Bankruptcy Problems. In: Mueller, D., Trost, R. (eds) Game Theory in Management Accounting. Contributions to Management Science. Springer, Cham. https://doi.org/10.1007/978-3-319-61603-2_12

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