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.
For each \(N \in \mathcal{ N}\), each M ⊆ N, and each \(z \in \mathbb{R}^{n}\), let z M ≡ (z i ) i ∈ M .
- 2.
These two axioms were studied first by Curiel et al. (1987).
- 3.
The property was introduced by Moreno-Ternero and Villar (2004) under the name of Securement.
- 4.
The constrained equal-awards rule, A, selects, for each \((N,c,E) \in \mathcal{ D}\), the vector (min{c i , λ}) i ∈ N , where λ > 0 is chosen so that ∑ i ∈ N 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}) i ∈ N , where λ > 0 is chosen so that ∑ i ∈ N max{0, c i −λ} = E.
- 5.
The name was coined by Thomson (2003). To ease its presentation, we assume N = {1, 2}, but dismiss it from the definition.
- 6.
See also Moreno-Ternero (2006).
- 7.
See also van den Brink and Moreno-Ternero (2016).
- 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.
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.
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.
Quant and Borm (2011) proposed a different generalization of Aumann and Maschler’s procedure.
- 12.
To ease its presentation, we assume N = {1, 2}, but dismiss it from the definition.
- 13.
References
Aumann RJ (2002) Game theory in the talmud. In: Research bulletin series on jewish law and economics. Bar-Ilan University, Ramat Gan
Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the talmud. JET 36(2):195–213
Chun Y, Schummer J, Thomson W (2001) Constrained egalitarianism: a new solution for claims problems. Seoul J Econ 14(3):269–297
Curiel I, Maschler M, Tijs SH, (1987) Bankruptcy Games. Z Oper Res 31(5):A143–A159
Dagan N (1996) New characterizations of old bankruptcy rules. SCW 13(1):51–59
Dagan N, Volij O (1993) The bankruptcy problem: a cooperative bargaining approach. Math Soc Sci 26(3):287–29
Driessen TSH (1998) The greedy bankruptcy game: an alternative game theoretic analysis of a bankruptcy problem. In: Petrosjan LA, Mazalov VV (eds) Game theory and applications, vol IV. Nova Science Publishers, New York, pp 45–61
Hokari T, Thomson W (2003) Claims problems and weighted generalizations of the Talmud rule. ET 21(2):241–261
Hougaard JL, Moreno-Ternero JD, Østerdal LP (2012) A unifying framework for the problem of adjudicating conflicting claims. JME 48(2):107–114
Hougaard JL, Moreno-Ternero JD, Østerdal LP (2013a) Rationing in the presence of baselines. SCW 40(4):1047–1066
Hougaard JL, Moreno-Ternero JD, Østerdal LP (2013b) Rationing with baselines: the composition extension operator. Ann Oper Res 211(1):179–191
Huijink S, Borm P, Kleppe J, Reijnierse J (2015) Bankruptcy and the per capita nucleolus: the claim-and-right rules family. Math Soc Sci 77:15–31
Moreno-Ternero JD (2006) Composition, securement, and concede-and-divide. Span Econ Rev 8(3):227–237
Moreno-Ternero JD (2007) Bankruptcy rules and coalitional manipulation. IGTR 9(1):105–118
Moreno-Ternero JD (2011a) Voting over piece-wise linear tax methods. JME 47(1):29–36
Moreno-Ternero JD (2011b) A coalitional procedure leading to a family of bankruptcy rules. ORL 39(1):1–3
Moreno-Ternero JD, Villar A (2004) The Talmud rule and the securement of agents’ awards. Math Soc Sci 47(2):245–257
Moreno-Ternero JD, Villar A (2006a) The TAL-family of rules for bankruptcy problems. SCW 27(2):231–249
Moreno-Ternero JD, Villar A (2006b) On the relative equitability of a family of taxation rules. JPubET 8(2):283–291
Moreno-Ternero JD, Villar A (2006c) New characterizations of a classical bankruptcy rule. RED 10(2):73–84
O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2(4):345–371
Perles M, Maschler M (1981) The super-additive solution for the Nash bargaining game. IJGT 10(3):163–193
Quant M, Borm P (2011) Random conjugates of bankruptcy rules. SCW 36(2):249–266
Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appli Math 17(6):1163–1170. doi: 10.1137/0117107
Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45(3):249–297
Thomson W (2008) Two families of rules for the adjudication of conflicting claims. SCW 31(4):667–692
Thomson W (2012) On the Axiomatics of resource allocation: interpreting the consistency principle. Econ Philos 28(3):385–421
Thomson W (2014) How to divide when there isn’t enough: from the Talmud to game theory. Book Manuscript, University of Rochester
Thomson W (2015) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update. Math Soc Sci 74(3):41–59
Thomson W, Yeh C-H (2008) Operators for the adjudication of conflicting claims. JET 143(1):177–198
Timoner P, Izquierdo JM (2016) Rationing problems with ex-ante conditions. Math Soc Sci 79(1):46–52
van den Brink R, Funaki Y, van der Laan G (2013) Characterization of the reverse Talmud bankruptcy rule by exemption and exclusion properties. EJOR 228(2):413–417
van den Brink R, Moreno-Ternero J (2016) The reverse TAL-family or rules for bankruptcy problems. Ann Oper Res. Forthcoming. doi: 10.1007/s10479-017-2455-x
Acknowledgements
Financial support from the Spanish Ministry of Economy and Competitiveness (ECO2014-57413-P) is gratefully acknowledged.
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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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