On the complexity of core, kernel, and bargaining set

Coalitional games model scenarios where players can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. A fundamental issue of coalitional games is to single out the most desirable outcomes in terms of worth distributions, usually called solution concepts. Since decisions taken by realistic players cannot involve unbounded resources, recent computer science literature advocated the importance of assessing the complexity of computing with solution concepts. In this context, the paper provides a complete picture of the complexity issues arising with three prominent solution concepts for coalitional games with transferable utility, namely, the core, the kernel, and the bargaining set, whenever the game worth-function is represented in some reasonably compact form. The starting points of the investigation are the settings of graph games and of marginal contribution nets, where the worth of any coalition can be computed in polynomial time in the size of the game encoding and for which various open questions were stated in the literature. The paper answers these questions and, in addition, provides new insights on succinctly specified games, by characterizing the computational complexity of the core, the kernel, and the bargaining set in relevant generalizations and specializations of the two settings. Concerning the generalizations, the paper shows that dealing with arbitrary polynomial-time computable worth functions-no matter of the specific game encoding being considered-does not provide any additional source of complexity compared to graph games and marginal contribution nets. Instead, only for the core, a slight increase in complexity is exhibited for classes of games whose worth functions encode NP-hard optimization problems, as in the case of certain combinatorial games. As for specializations, the paper illustrates various tractability results on classes of bounded treewidth graph games and marginal contribution networks.