Cores of partitioning games

A generalization of assignment games, called partitioning games, is introduced. Given a finite set N of players, there is an a priori given set π of coalitions of N and only coalitions in π play an essential role. Necessary and sufficient conditions for the nonemptiness of the cores of all games with essential coalitions π are developed. These conditions appear extremely restrictive. However when N is 'large', there are relatively few 'types' of players, and members of π are 'small' and defined in terms of numbers of players of each type contained in subsets, then approximate cores are nonempty.