Efficiently computing the Shapley value of connectivity games in low-treewidth graphs

Abstract

The Shapley value is the solution concept in cooperative game theory that is most used in both theoretical and practical settings. Unfortunately, in general, computing the Shapley value is computationally intractable. This paper focuses on computing the Shapley value of (weighted) connectivity games. For these connectivity games, which are defined on an underlying (weighted) graph, computing the Shapley value is #textsf {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 quickly compute exact Shapley values for these networks, whereas in prior work these values could only be approximated using a heuristic method. Finally, it is demonstrated that our algorithm can also efficiently compute the Shapley value time for several larger (artificial) benchmark graphs from the PACE 2018 challenge.