Abstract
We follow the Nash program to provide a new strategic justification of the Talmud rule in bankruptcy problems. The design of our game is based on a focal axiomatization of the rule, which combines consistency with meaningful lower and upper bounds to all creditors. Our game actually considers bilateral negotiations, inspired by those bounds, which are extended to an arbitrary number of creditors, by means of consistency.
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Notes
Aumann and Maschler (1985) mention that a bankruptcy problem can be handled from either the gain perspective or the loss perspective. The gain perspective focuses on dividing the liquidation value, and the loss perspective on dividing the shortfall.
Bounds constitute one of the central ideas in the axiomatics of fair allocation. They can be interpreted as a first attempt of a compromise to solve the problems at stake. In bankruptcy, they protect creditors with smaller or bigger claims from receiving too little. Invariance under claims truncation and minimal rights first also convey a similar implicit protection, but it is a byproduct of the invariance requirements they formalize, by which the rule should ignore certain changes in the data of the problem. These normative aspects motivate the two games, but also allow to differentiate among them.
This feature is also shared by Tsay and Yeh (2019).
As \(y_{p}+y_{l}\le c_{p}\), it follows that \(y_{p}+y_{l}\le q\).
As \(c_{p}<y_{p}+y_{l}\le c_{l}\), it follows that \(c_{p}\le c_{p}+c_{l}-(y_{p}+y_{l})\le c_{l}\) and, therefore, \(q\ge c_{p}\).
As \(y_{p}+y_{l}> c_{l}\), it follows that \(c_l+c_p-(y_{p}+y_{l})< c_{p}\), and, therefore, \(q\ge c_l+c_p-(y_{p}+y_{l})\).
To see this, if \(\pi ^{\sigma }(1)=i\), then obviously \(y^{\sigma '}=y^{\sigma }\). If \(\pi ^{\sigma }(1)=k\ne i\), then by the game rule, either \(y^{\sigma }=y^{\sigma _{i}}\ne y^{\sigma _{k}}\) or \(y^{\sigma }=y^{\sigma _{k}}\). In the former case, as for each \(l\in N\setminus \{k\}\), \(y^{\sigma '_{l}}=y^{\sigma }\ne y^{\sigma _{k}}=y^{\sigma '_{k}}\), then \(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\). In the latter case, if there is \(l\in N\setminus \{k\}\) such that \(y^{\sigma _{l}}\ne y^{\sigma _{k}}\), then \(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\); otherwise, \(y^{\sigma '}=y^{\sigma _{k}}=y^{\sigma }\). Thus, \(y^{\sigma '}=y^{\sigma }\).
To see this, suppose that by following \(\sigma\), \(p\in N\) is the P-creditor. If for each \(k,h\in N\), \(y^{\sigma _{k}}=y^{\sigma _{h}}\), then the original proposal (\(y^{\sigma }\)) coincides with the new proposal (\(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\)). If for each \(k,h\in N\setminus \{p\}\), \(y^{\sigma _{k}}=y^{\sigma _{h}}\) and \(y^{\sigma _{p}}\ne y^{\sigma _{i}}\), then the original proposal (\(y^{\sigma }\)) still coincides with the new proposal (\(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\)). If for each \(k,h\in N\setminus \{i\}\), \(y^{\sigma _{k}}=y^{\sigma _{h}}\) and \(y^{\sigma _{p}}\ne y^{\sigma _{i}}\), then the original proposal (\(y^{\sigma }\)) again coincides with the new proposal (\(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\)). If for some \(k,h\in N\setminus \{i,p\}\), \(y^{\sigma _{k}}\ne y^{\sigma _{h}}\), then the original proposal (\(y^{\sigma }\)) coincides with the new proposal (\(y^{\sigma '}=y^{\sigma '_{i}}=y^{\sigma }\)).
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Acknowledgements
We would like to thank William Thomson, Tien-Wang Tsaur, an Associate Editor of this journal and two anonymous referees for their helpful comments. We also acknowledge the comments made by participants at seminars and conferences where earlier versions of this article have been presented at Budapest, Hokkaido, Seoul and Turku. Moreno-Ternero acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (ECO2017-83069-P). Tsay acknowledges financial support from the grant of Ministry of Science and Technology, Taiwan (MOST-107-2628-H-194-001-MY2). Yeh acknowledges financial support from Career Development Award of Academia Sinica, Taiwan (AS-99-CDA-H01) and the grant of Ministry of Science and Technology, Taiwan (MOST-108-2410-H-001-027-MY3). We are responsible for any remaining deficiencies.
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Moreno-Ternero, J.D., Tsay, MH. & Yeh, CH. A strategic justification of the Talmud rule based on lower and upper bounds. Int J Game Theory 49, 1045–1057 (2020). https://doi.org/10.1007/s00182-020-00727-z
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DOI: https://doi.org/10.1007/s00182-020-00727-z