An exact bound on epsilon for nonemptiness of epsilon cores of games

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 epsilon greater than or equal to ϵ0(π, n0) all games in the collection with at least n0 players have nonempty epsilon-cores. Examples are provided in which the bound on epsilon 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 epsilon is small.