Abstract: A class of cooperative games with graph communication structure is studied in this paper by considering some important players, namely essential players. Under the assumption that only connected coalitions containing essential players are able to cooperate and obtain their worths, the class of graph games with essential players is proposed as well as an allocation rule. The proposed value follows the spirit of the Myerson value defined by applying the Shapley value on a modified game. Three properties, feasible component efficiency, the inessential component property, and fairness, are provided to fully characterize this value, where feasible component efficiency and fairness follows the same ideas of component efficiency and fairness for classical graph games, and the inessential component property says that the total payoffs of the players in a non-feasible component is zero. Moreover, some computational aspects of the proposed value and comparisons with disjunctive permission value for games with permission structure are also studied, respectively.

Key words: Cooperative game, Graph structure, Essential player, Myerson value, Computation, Permission structure