Settling the Complexity of Popularity in Additively Separable and Fractional Hedonic Games

We study coalition formation in the framework of hedonic games. There, a set of agents needs to be partitioned into disjoint coalitions, where agents have a preference order over coalitions. A partition is called popular if it does not lose a majority vote among the agents against any other partition. Unfortunately, hedonic games need not admit popular partitions and prior work suggests significant computational hardness. We confirm this impression by proving that deciding about the existence of popular partitions in additively separable and fractional hedonic games is ${\mathrm{\Sigma }}_2^p$-complete. This settles the complexity of these problems and is the first work that proves completeness of popularity for the second level of the polynomial hierarchy.