Shapley Meets Shapley

TL;DR

This paper analyzes the computational complexity of the Shapley value in matching games, providing polynomial-time algorithms for special cases, proving #P-completeness in general, and offering an approximation scheme.

Contribution

It introduces polynomial-time algorithms for specific graph classes, proves #P-completeness for unweighted cases, and develops an FPRAS for approximating the Shapley value.

Findings

Polynomial-time computation for graphs with degree two and small modular decompositions.

#P-completeness of computing the Shapley value in unweighted matching games.

An FPRAS for approximating the Shapley value in general cases.

Abstract

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.