We introduce the framework of parameterized collections of games and provide three nonemptiness of approximate core theorems for arbitrary games with and without sidepayments. The parameters bound (a) the number of approximate types of players and the size of the approximation and (b) the size of nearly effective groups of players and their distance from exact effectiveness. The theorems are based on a new notion of partition-balanced profiles and approximately partition-balanced profiles. The results are then applied to a new model of an economy with clubs. In contrast to the extant literature, our approach allows both widespread externalities and uniform results
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