Balanced Nontransferable Utility Games in Graph Structure

By a game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a nonempty set of payo®s for each coalition of players. It is well-known that balancedness is a su±cient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. But any information on the internal organization of the coalition is neglected in this set-up. We generalize the concept of coalitional games to games in graph structure. In such a graph game we consider the set of all graphs on any coalition of players. Every graph on a coalition represents an internal organization within the coalition. To every graph on any coalition a possibly empty set of payo® vectors is assigned. A payo® vector lies in the core of the game if no coalition can make all of its members better o® by organizing themselves in some graph. By exploiting the additional structure of graph games, we can introduce the so-called balanced-core of a graph game, a re ̄nement of the core. To do de ̄ne the balanced-core, we de ̄ne for every graph on any coalition a power vector that re°ects the relative positions of the players within the graph. We also introduce the concepts of a balanced collection of graphs and a balanced graph game. A payo® vector is in the balanced-core if it lies in the core and belongs to the intersection of the payo® sets of a balanced collection of graphs. We prove that any balanced graph game has a nonempty balanced-core. Formulating the classical Cournot duopoly problem as a graph game we show that the unique element of the balanced core corresponds to the Cournot-Nash equilibrium of the well-known noncooperative quantity duopoly game. This places the paper in the Nash research program, looking for a unifying approach to cooperative and noncooperative behavior in which each theory helps to justify and clarify the other.