Social choice theory and cooperative (coalitional) game theory have become important foundations for the design and analysis of multiagent systems. In this paper, we use cooperative game theory tools in order to explore the coalition formation process in the coalitional manipulation problem. Unlike earlier work on a cooperative-game-theoretic approach to the manipulation problem , we consider a model where utilities are not transferable. We investigate the issue of stability in coalitional manipulation voting games; we define two notions of the core in these domains, the α-core and the β-core. For each type of core, we investigate how hard it is to determine whether a given candidate is in the core. We prove that for both types of core, this determination is at least as hard as the coalitional manipulation problem. On the other hand, we show that for some voting rules, the α- and the β-core problems are no harder than the coalitional manipulation problem. We also show that some prominent voting rules, when applied to the truthful preferences of voters, may produce an outcome not in the core, even when the core is not empty.