  Hedonic games provide a general model of coalition formation, in which a set
of agents is partitioned into coalitions, with each agent having preferences
over which other players are in her coalition. We prove that with additively
separable preferences, it is $\Sigma_2^p$-complete to decide whether a core- or
strict-core-stable partition exists, extending a result of Woeginger (2013).
Our result holds even if valuations are symmetric and non-zero only for a
constant number of other agents. We also establish $\Sigma_2^p$-completeness of
deciding non-emptiness of the strict core for hedonic games with dichotomous
preferences. Such results establish that the core is much less tractable than
solution concepts such as individual stability.
