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

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.