  Hedonic games provide a natural model of coalition formation among
self-interested agents. The associated problem of finding stable outcomes in
such games has been extensively studied. In this paper, we identify simple
conditions on expressivity of hedonic games that are sufficient for the problem
of checking whether a given game admits a stable outcome to be computationally
hard. Somewhat surprisingly, these conditions are very mild and intuitive. Our
results apply to a wide range of stability concepts (core stability, individual
stability, Nash stability, etc.) and to many known formalisms for hedonic games
(additively separable games, games with W-preferences, fractional hedonic
games, etc.), and unify and extend known results for these formalisms. They
also have broader applicability: for several classes of hedonic games whose
computational complexity has not been explored in prior work, we show that our
framework immediately implies a number of hardness results for them.
