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On the Shapley value of liability games

Author

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  • Csóka, Péter
  • Illés, Ferenc
  • Solymosi, Tamás

Abstract

In a liability problem, the asset value of an insolvent firm must be distributed among the creditors and the firm itself, when the firm has some freedom in negotiating with the creditors. We model the negotiations using cooperative game theory and analyze the Shapley value to resolve such liability problems. We establish three main monotonicity properties of the Shapley value. First, creditors can only benefit from the increase in their claims or of the asset value. Second, the firm can only benefit from the increase of a claim but can end up with more or with less if the asset value increases, depending on the configuration of small and large liabilities. Third, creditors with larger claims benefit more from the increase of the asset value. Even though liability games are constant-sum games and we show that the Shapley value can be calculated directly from a liability problem, we prove that calculating the Shapley payoff to the firm is NP-hard.

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  • Csóka, Péter & Illés, Ferenc & Solymosi, Tamás, 2022. "On the Shapley value of liability games," European Journal of Operational Research, Elsevier, vol. 300(1), pages 378-386.
    • Handle: RePEc:eee:ejores:v:300:y:2022:i:1:p:378-386
    DOI: 10.1016/j.ejor.2021.10.012
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    1. Bas Dietzenbacher & Peter Borm & Arantza Estévez-Fernández, 2020. "NTU-bankruptcy problems: consistency and the relative adjustment principle," Review of Economic Design, Springer;Society for Economic Design, vol. 24(1), pages 101-122, June.
    2. Gustavo Bergantiños & Silvia Lorenzo-Freire, 2008. "A characterization of optimistic weighted Shapley rules in minimum cost spanning tree problems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 35(3), pages 523-538, June.
    3. Kohlberg, Elon & Neyman, Abraham, 2018. "Games of threats," Games and Economic Behavior, Elsevier, vol. 108(C), pages 139-145.
    4. Nimrod Megiddo, 1978. "Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 189-196, August.
    5. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Arantza Estévez-Fernández & Peter Borm & M. Gloria Fiestras-Janeiro, 2020. "Nontransferable utility bankruptcy games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 154-177, April.
    7. William Thomson, 2013. "Game-Theoretic Analysis Of Bankruptcy And Taxation Problems: Recent Advances," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(03), pages 1-14.
    8. S. C. Littlechild & G. Owen, 1973. "A Simple Expression for the Shapley Value in a Special Case," Management Science, INFORMS, vol. 20(3), pages 370-372, November.
    9. Anna B. Khmelnitskaya, 2003. "Shapley value for constant-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(2), pages 223-227, December.
    10. Thomson, William, 2015. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 41-59.
    11. Guni Orshan & Federico Valenciano & José M. Zarzuelo, 2003. "The Bilateral Consistent Prekernel, the Core, and NTU Bankruptcy Problems," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 268-282, May.
    12. Nowak, Andrzej S & Radzik, Tadeusz, 1994. "A Solidarity Value for n-Person Transferable Utility Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(1), pages 43-48.
    13. Daniel Granot & Jeroen Kuipers & Sunil Chopra, 2002. "Cost Allocation for a Tree Network with Heterogeneous Customers," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 647-661, November.
    14. Bergantinos, Gustavo & Lorenzo-Freire, Silvia, 2008. ""Optimistic" weighted Shapley rules in minimum cost spanning tree problems," European Journal of Operational Research, Elsevier, vol. 185(1), pages 289-298, February.
    15. Bilbao, J.M. & Ordóñez, M., 2009. "Axiomatizations of the Shapley value for games on augmenting systems," European Journal of Operational Research, Elsevier, vol. 196(3), pages 1008-1014, August.
    16. Yang Cai & Ozan Candogan & Constantinos Daskalakis & Christos Papadimitriou, 2016. "Zero-Sum Polymatrix Games: A Generalization of Minmax," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 648-655, May.
    17. Xiaotie Deng & Christos H. Papadimitriou, 1994. "On the Complexity of Cooperative Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 257-266, May.
    18. Balog, Dóra & Bátyi, Tamás László & Csóka, Péter & Pintér, Miklós, 2017. "Properties and comparison of risk capital allocation methods," European Journal of Operational Research, Elsevier, vol. 259(2), pages 614-625.
    19. Pradeep Dubey, 1982. "The Shapley Value as Aircraft Landing Fees--Revisited," Management Science, INFORMS, vol. 28(8), pages 869-874, August.
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    2. Wei Li & Wolfgang Karl Hardle & Stefan Lessmann, 2022. "A Data-driven Case-based Reasoning in Bankruptcy Prediction," Papers 2211.00921, arXiv.org.
    3. Munich, Léa, 2024. "Schedule situations and their cooperative game theoretic representations," European Journal of Operational Research, Elsevier, vol. 316(2), pages 767-778.
    4. Léa Munich, 2023. "Schedule Situations and their Cooperative Game Theoretic Representations," Working Papers 2023-08, CRESE.

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    More about this item

    Keywords

    Game theory; Shapley value; Constant-sum game; Liability game; Insolvency;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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