Coalition formation in general apex games under monotonic power indices

- Apex games are generalized, the apex player is replaced by a collection of apex sets.
- Monotonic power indices are applied to derive a hedonic coalition formation game.
- Core stable coalitions (if any) contain at least half the minor players.
- For simple apex games and the Shapley-Shubik index the core is empty.
- For simple apex games and the Banzhaf index the core is non-empty.

An apex game consists of one apex player and a set of minor players. We identify two key properties of apex games and use them to introduce the class of general apex games. We derive players' preferences over winning coalitions by applying strongly monotonic power indices on such a game and all its subgames and investigate whether there are core stable coalitions in the induced hedonic coalition formation game. Besides several general results, in particular, we develop conditions on the game for the Shapley-Shubik index, the Banzhaf index, and the normalized Banzhaf index.