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

Takumi Jimbo, Tomomi Matsui

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.

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

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.
Paper Structure (12 sections, 11 theorems, 24 equations, 2 figures, 3 algorithms)

This paper contains 12 sections, 11 theorems, 24 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Null Players in MCST-Saving Games
  4. Monte Carlo Method
  5. 0-1 Weighted Complete Graph
  6. Non-negatively Weighted Complete Graph
  7. Computational Experiments
  8. Relationship between the Sample Size $M$ and $1/\varepsilon^{2}$
  9. Relationship between the Number of Players $n$ and the Required Sample Size $M$
  10. Summary of Contributions and Conclusions
  11. MCST Game
  12. Concluding Remarks

Key Result

Lemma 2.1

In an MCST-saving game, a player $i \in N$ is a null player if and only if his/her Shapley value is equal to zero, i.e. $\phi_i=0.$

Figures (2)

  • Figure 1: Relationship between the sample size $M$ and $1/\varepsilon^{2}$($n=3$).
  • Figure 2: Relationship between the sample size $M$ and $n$($\varepsilon=0.1$).

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 10 more