In this research, we discuss a problem of calculating the Shapley value in bankruptcy games. We show that the decision problem of computing the Shapley value in bankruptcy games is NP-complete. We also investigate the relationship between the Shapley value of bankruptcy games and the Shapley-Shubik index in weighted voting games. The relation naturally implies a dynamic programming technique for calculating the Shapley value. We also present two recursive algorithms for computing the Shapley value: the first is the recursive completion method originally proposed by O'Neill, and the second is our novel contribution based on the dual game formulation. These recursive approaches offer conceptual clarity and computational efficiency, especially when combined with memoisation technique. Finally, we propose a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) based on Monte Carlo sampling, providing an efficient approximation method for large-scale instances.

翻译：本研究探讨了破产博弈中夏普利值的计算问题。我们证明了计算破产博弈夏普利值的判定问题属于NP完全问题。同时，我们研究了破产博弈夏普利值与加权投票博弈中夏普利-舒比克指数之间的关系，该关系自然地引出了计算夏普利值的动态规划技术。此外，我们提出了两种计算夏普利值的递归算法：第一种是奥尼尔最初提出的递归补全法，第二种是我们基于对偶博弈公式提出的创新方法。这些递归方法结合记忆化技术，在概念清晰性和计算效率方面具有优势。最后，我们提出了一种基于蒙特卡洛采样的完全多项式时间随机近似方案，为大规模实例提供了高效的近似计算方法。