Optimal Coalition Structure Generation In Partition Function Games

The authors are grateful for financial support received from the UK EP-SRC through the project Market-Based Control of Complex Computational Systems (GR/T10657/01). The authors are also thankful to Jennifer McManus, School of English, University of Liverpool for excellent editorial assistance. In multi-agent systems (MAS), coalition formation is typically studied using characteristic function game (CFG) representations, where the performance of any coalition is independent from co-existing coalitions in the system. However, in a number of environments, there are significant externalities from coalition formation where the effectiveness of one coalition may be affected by the formation of other distinct coalitions. In such cases, coalition formation can be modeled using partition function game (PFG) representations. In PFGs, to accurately generate an optimal division of agents into coalitions (so called CSG problem), one would have to search through the entire search space of coalition structures since, in a general case, one cannot predict the values of the coalitions affected by the externalities a priori. In this paper we consider four distinct PFG settings and prove that in such environments one can bound the values of every coalition. From this insight, which bridges the gap between PFG and CFG environments, we modify the existing state-of-the-art anytime CSG algorithm for the CFG setting and show how this approach can be used to generate the optimal CS in the PFG settings.