On Computing the Shapley Value in Bankruptcy Games -llustrated by Rectified Linear Function Game-
Shunta Yamazaki, Tomomi Matsui
TL;DR
The paper studies the computational problem of the Shapley value in bankruptcy games, establishing NP-completeness for the decision version and revealing a close link to Shapley–Shubik indices via a dual-game formulation. It provides a dynamic-programming framework with $O(n^2E)$ per-player complexity, leveraging the connection to weighted voting games, and introduces two recursive algorithms (Recursive Completion and Dual-Game) with memoisation to enable efficient exact computation. Additionally, it offers a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) based on Monte Carlo sampling to approximate Shapley values for large instances, with rigorous probabilistic guarantees. Together, these results advance both the theoretical understanding and practical computation of Shapley values in bankruptcy scenarios, enabling scalable analysis of large claim sets and estates.
Abstract
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.