We consider collections of games with and without side payments described by certain natural parameters. Given the parameters π describing a collection of games and a lower bound n₀ on the number of players, we obtain a bound ε₀(π, n₀) so that, for any ε ≥ ε₀(π, n₀), all games in the collection with at least n₀ players have nonempty ε-cores. Examples are provided in which the bound on ε is met. For parameter values ensuring that there are many close substitutes for most players and that relatively small groups of players can realize nearly all gains to collective activities, for games with many players the bound on ε is small.