On the Shapley value of liability games

• Published in: European journal of operational research Vol. 300; no. 1; pp. 378 - 386
• Main Authors: Csóka, Péter, Illés, Ferenc, Solymosi, Tamás
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2022
• Subjects: Constant-sum game; Game theory; Insolvency; Liability game; Shapley value; Shapley value; Constant-sum game; Game theory; Liability game; Insolvency
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2021.10.012

•We propose a basis for the linear vector space of constant-sum games.•The Shapley value in liability games satisfies order preservation.•Liability monotonicity, asset monotonicity, and super-modularity also holds.•Calculating the Shapley value is non-deterministic polynomial-time hard. 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.