Cooperative game theory has proposed different notions of powerful players. For example, big-boss games (Muto et al. 1988) and clan games (Potters et al. 1989) are particular cases of veto games (Bahel 2016). The present paper extends these veto games by assuming that there is a given subset of powerful (or essential) players, but only a few (as opposed to all) essential players are required for a coalition to have a positive group value. The resulting games, which are called r-essential games, encompass convex games (Shapley 1971) and veto games. We show that r-essential games have a nonempty core. We give a recursive description of the core. Moreover, it is shown that the core and the bargaining set are equivalent for r-essential games. An application to networks is provided.
