Coalitional games induced by matching problems: Complexity and islands of tractability for the Shapley value

The paper focuses on cooperative games where the worth of any coalition of agents is determined by the optimal value of a matching problem on (possibly weighted) graphs. These games come in different forms that can be grouped in two broad classes, namely of matching and allocation games, and they have a wide spectrum of applications, ranging from two-sided markets where buyers and sellers are encoded as vertices in a graph, to allocation problems where indivisible goods have to be assigned (matched) to agents in a fair way, possibly using monetary compensations. The Shapley value and the related notion of Banzhaf value have often been identified as appropriate solution concepts for many applications of matching/allocation games, but their computation is intractable in general. It is known that these concepts can be computed in polynomial time for matching games on unweighted trees and on graphs having degree at most two. However, it was open whether or not such positive results could be extended to the more general case of graphs having bounded treewidth, and to the case of allocation problems on weighted graphs. The paper provides a positive answer to these questions, by showing that computing the Shapley value and the Banzhaf value is feasible in polynomial time for the following classes of games: matching games over unweighted graphs having bounded treewidth, allocation games over weighted graphs having bounded treewidth, and allocation games over weighted graphs and such that each good is of interest for two agents at most. Without such structural restrictions, computing these solution concepts on allocation games is instead shown to be #P-hard, even in the case of unweighted graphs.