Finding and Recognizing Popular Coalition Structures

An important aspect of multi-agent systems concerns the formation of coalitions that are stable or optimal in some well-defined way. The notion of popularity has recently received a lot of attention in this context. A partition is popular if there is no other partition in which more agents are better off than worse off. In this paper, we study popularity, strong popularity, and mixed popularity (which is particularly attractive because existence is guaranteed by the Minimax Theorem) in a variety of coalition formation settings. Extending previous work on marriage games, we show that mixed popular partitions in roommate games can be found efficiently via linear programming and a separation oracle. This approach is quite universal, leading to efficient algorithms for verifying whether a given partition is popular and for finding strongly popular partitions (resolving an open problem). By contrast, we prove that both problems become computationally intractable when moving from coalitions of size 2 to coalitions of size 3, even when preferences are strict and globally ranked. Moreover, we show that finding popular, strongly popular, and mixed popular partitions in symmetric additively separable hedonic games and symmetric fractional hedonic games is NP-hard. Together, these results indicate strong boundaries to the tractability of popularity in both ordinal and cardinal models of hedonic games.