Efficiently Computing the Shapley Value of Connectivity Games in Low-Treewidth Graphs

TL;DR

This paper introduces an efficient algorithm for calculating the Shapley value in connectivity games on graphs with low treewidth, enabling exact solutions in practical, real-world networks where previous methods were limited.

Contribution

The paper presents a novel algorithm that computes the Shapley value efficiently on graphs with bounded treewidth, improving over prior intractable approaches for connectivity games.

Findings

Efficient exact computation of Shapley values on low-treewidth graphs

Successful application to real-world covert networks

Ability to handle larger benchmark graphs from PACE 2018 challenge

Abstract

The Shapley value is the solution concept in cooperative game theory that is most used in both theoretical as practical settings. Unfortunately, computing the Shapley value is computationally intractable in general. This paper focuses on computing the Shapley value of (weighted) connectivity games. For these connectivity games, that are defined on an underlying (weighted) graph, computing the Shapley value is #P-hard, and thus (likely) intractable even for graphs with a moderate number of vertices. We present an algorithm that can efficiently compute the Shapley value if the underlying graph has bounded treewidth. Next, we apply our algorithm to several real-world (covert) networks. We show that our algorithm can compute exact Shapley values for these networks quickly, whereas in prior work these values could only be approximated using a heuristic method. Finally, it is shown that our algorithm can also compute the Shapley value time efficiently for several larger (artificial) benchmark graphs from the PACE 2018 challenge.