We introduce the framework of parameterized collections of games with and without sidepayments and provide three nonemptiness of approximate core theorems for games in parameterized collections. The parameters bound (a) the number of approximate types of players and the size of the approximation and (b) the size of nearly e®ective groups of players and their distance from exact e®ectiveness. The theorems are based on a new notion of partition-balanced pro¯les and approximately partition-balanced pro¯les. The results are 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.