The extended Shapley value for generalized cooperative games under precedence constraints

We introduce a new class of games where cooperation among players is restricted by precedence constraints vis á vis the worth of a coalition depends on the order in which the players enter into the coalition. The idea combines two existing classes of cooperative games, namely the cooperative games under precedence constraints due to Faigle and Kern (Int J Game Theory 21(3):249–266, 1992 ) and the games in generalized characteristic function due to Nowak and Radzik (Games Econ Behav 6(1):150–161, 1994 ). A Shapley value for this special class of games is proposed, we call it the extended Shapley value to distinguish it for the existing one. Two axiomatic characterizations of the extended Shapley value are given: one uses Efficiency, Null player, and Linearity; the other uses Efficiency, Marginality, and Null game. Some interesting properties of the extended Shapley value are studied. Furthermore, two extensions of the extended Shapley value, called the extended probabilistic value and the extended order value, are proposed and characterized. Our study shows that the results in cooperative games under precedence constraints cannot have a trivial extension to the generalized constrained games and conversely.