Complexity of constructing solutions in the core based on synergies among coalitions

Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can accomplish them more efficiently. Motivating the agents to abide by a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory: the set of solutions that satisfy it is known as the core. The computational questions around the core have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise, natural, general representation for games in characteristic form that relies on superadditivity. In our representation, individual agents' values are given as well as values for those coalitions that introduce synergies. We show that this representation allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is NP-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition); we do so by showing that if these are given, the problem becomes solvable in time polynomial in the size of the representation in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains NP-complete even when the collaborative possibilities are given. Finally, we show that for convex characteristic functions, a solution in the core can be computed efficiently (in O(nl^2) time, where n is the number of agents and l is the number of synergies), even when the collaborative possibilities are not given in advance.

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