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

TL;DR

This paper explores the complexity of computing the Shapley value in bankruptcy games, introduces recursive algorithms and an approximation scheme, and compares it with the Shapley-Shubik index, enhancing computational methods in cooperative game theory.

Contribution

It presents novel recursive algorithms and an FPRAS for calculating the Shapley value in bankruptcy games, addressing computational challenges and establishing new links with voting game indices.

Findings

Computing the Shapley value in bankruptcy games is NP-complete.

Proposed recursive algorithms improve computational efficiency.

Introduced an FPRAS for large-scale approximation.

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.