Core stability in hedonic games among friends and enemies: Impact of Neutrals

We investigate hedonic games under enemies aversion and friends appreciation, where every agent considers other agents as either a friend or an enemy. We extend these simple preferences by allowing each agent to also consider other agents to be neutral. Neutrals have no impact on her preference, as in a graphical hedonic game. Surprisingly, we discover that neutral agents do not simplify matters, but cause complexity. We prove that the core can be empty under enemies aversion and the strict core can be empty under friends appreciation. Furthermore, we show that under both preferences, deciding whether the strict core is nonempty, is NP^NP-complete. This complexity extends to the core under enemies aversion. We also show that under friends appreciation, we can always find a core stable coalition structure in polynomial time.