Approximating the Shapley Value of Minimum Cost Spanning Tree Games: An FPRAS for Saving Games

2026-03-24

AI Summary

This work addresses the computational challenge of efficiently approximating the Shapley value in minimum-cost spanning tree games. By reformulating the original problem as a savings game with favorable structural properties, the authors present the first multiplicative (relative-error) approximation scheme for the Shapley value in this setting. Building on this transformation, they develop a fully polynomial-time randomized approximation scheme (FPRAS) that combines Monte Carlo sampling to produce, with high probability, an approximate solution within any prescribed relative error in fully polynomial time. This approach overcomes the limitations of prior methods, which were restricted to additive approximations, thereby providing both theoretical guarantees and a practical algorithm for large-scale computation of Shapley values in cooperative games.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/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 address the problem of computing the Shapley value in minimum-cost spanning tree (MCST) games. We introduce the saving game as a key framework for approximating the Shapley value. By reformulating MCST games into their saving-game counterparts, we obtain structural properties that enable multiplicative (relative-error) approximation. Building on this reformulation, we develop a Monte Carlo based Fully Polynomial-time Randomized Approximation Scheme (FPRAS) for the Shapley value.

Problem

• Shapley value
• minimum-cost spanning tree games
• saving games
• approximation

Innovation

• Shapley value
• minimum-cost spanning tree games
• saving game
• FPRAS
• Monte Carlo approximation