Finding and verifying the nucleolus of cooperative games

• 成果类型: Article
• 署名作者: Benedek, Marton; Fliege, Jorg; Tri-Dung Nguyen
• 署名单位: HUN-REN; HUN-REN Centre for Economic & Regional Studies; Institute of Economics - HAS; Hungarian Academy of Sciences; Corvinus University Budapest; Budapest University of Technology & Economics; University of Southampton; University of Southampton; University of Southampton
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01527-9
• 发表日期: 2021
• 页码: 135-170
• 关键词: characterization sets

摘要

The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties-it always exists and lies in the core (if the core is nonempty), and it is unique. The nucleolus is considered as the most `stable' solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n <= 15). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the `balancedness' of at most (n - 1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the newalgorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.