On Computing the Shapley Value in Bankruptcy Games -llustrated by Rectified Linear Function Game-

2025-11-27

AI Summary

This paper investigates the computational complexity of the Shapley value in bankruptcy games, establishing for the first time that its exact computation is NP-complete. To address this intractability, we propose three algorithmic approaches: (1) a novel recursive algorithm leveraging the dual game structure, enhanced with memoization to significantly improve exact computation efficiency; (2) an optimized dynamic programming method; and (3) the first fully polynomial-time randomized approximation scheme (FPRAS) tailored to bankruptcy games, based on Monte Carlo sampling for efficient approximation. This work pioneers the integration of dual-game modeling and Monte Carlo techniques into bankruptcy game analysis, unifying exact and approximate algorithms within a single theoretical framework. It achieves a rigorous balance between computational tractability and theoretical soundness, yielding a scalable computational toolkit for practical bankruptcy allocation analysis.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Computing the Shapley value in bankruptcy games is NP-complete.
• Developing dynamic programming and recursive algorithms for efficient Shapley value calculation.
• Providing an FPRAS approximation method for large-scale bankruptcy game instances.

Innovation

• Dynamic programming technique for Shapley value calculation
• Recursive algorithms with dual game formulation
• FPRAS using Monte Carlo sampling for approximation