An explicit bound on e for non-emptiness of the e-core of an arbitrary game with side payments

We introduce the concept of a parameterized collection of games with side payments and determine a bound on e so that all sufficiently large games in such a collection have non-empty e -cores. A parameterized collection of games is described by (a) the number of approximate player types and the accuracy of this approximation, (b) the size of approximately effective small groups, and (c) the size of the bound on maximal per capita payoffs achievable in coalitions. An explicit relationship between the required size of e the parameters describing the collection and the number of players in the game under consideration is established. Given the parameters, the larger the number of players in a game, the smaller the e that can be chosen.