A value for communication situations with players having different bargaining abilities

The aim of this paper is to extend the Myerson value (Myerson in Math Oper Res 2:225–229, 1977) to situations in which players in a TU-game, in addition to having cooperation possibilities restricted by a graph, also have different bargaining abilities. Then, we will associate to each player in a communication situation a weight in the interval [0, 1] that measures his bargaining ability. A unitary weight corresponds to a fully cooperative player whereas a null weight corresponds to a player that is not willing to cooperate in any way. Intermediate values modulate the bargaining ability. We modify the original TU-game to a new game which is, in turn, a modification of the Myerson’s graph-restricted game. We will assume that the reduction in the will to cooperate implies that players can not obtain the total dividend of the connected coalitions which must be discounted by an appropriate factor. Then, we propose as a solution for these situations the Shapley value (Shapley, in: Kuhn, Tucker (eds) Annals of mathematics studies, Princeton University Press, Princeton, 1953) of the modified game. This solution extends the Myerson value (and also the Shapley value). Moreover it satisfies monotonicity in the weights. Different characterizations of this rule can be obtained. They are based on properties as bargaining component efficiency, fairness, balanced contributions and balanced bargaining ability contributions, and thus they are parallel to those more prominent existing in the literature for the Myerson value.