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Overlapping Coalition Formation (OCF) games, introduced by Chalkiadakis, Elkind, Markakis, Polukarov and Jennings in 2010, are cooperative games where players can simultaneously participate in several coalitions. Capturing the notion of stability in OCF games is a difficult task:deviating players may continue to contribute resources to joint projects with non-deviators, and the crucial question is what payoffs the deviators expect to receive from such projects. Chalkiadakis et al. introduce three stability concepts for OCF games---the conservative core, the refined core, and the optimistic core---that are based on different answers to this question. In this paper, we propose a unified framework for the study of stability in the OCF setting, which encompasses the stability concepts considered by Chalkiadakis et al. as well as a wide variety of alternative stability concepts. Our approach is based on the notion of arbitration functions, which determine the payoff obtained by the deviators, given their deviation and the current allocation of resources. We provide a characterization of stable outcomes under arbitration. We then conduct an in-depth study of four types of arbitration functions, which correspond to four notions of the core; these include the three notions of the core considered by Chalkiadakis et al. Our results complement those of Chalkiadakis et al. and answer questions left open by their work. In particular, we show that OCF games with the conservative arbitration function are essentially equivalent to non-OCF games, by relating the conservative core of an OCF game to the core of a non-overlapping cooperative game, and use this result to obtain a strictly weaker sufficient condition for conservative core non-emptiness than the one given by Chalkiadakis et al.