| Summary: | We generalize the class of apex game by combining a winning coalition of symmetric minor players with a collection of apex sets which can form winning coalitions only together with a fixed quota of minor players. By applying power indices to these games and their subgames we generate players' preferences over coalitions which we use to define a coalition formation game. We focus on strongly monotonic power indices and investigate under which conditions on the initial general apex game there are core stable coalitions in the resulting coalition formation game. Besides several general results, we develop condition for the Shapley-Shubik index, the Banzhaf index, and the normalized Banzhaf index in particular. It turns out that many statements can be easily verified for arbitrary collections of apex sets. Nevertheless, we give some relations between the collection of apex sets and the set of core stable coalitions.
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