Weak differential marginality and the Shapley value

The principle of differential marginality for cooperative games states that the differential of two players' payoffs does not change when the differential of these players' marginal contributions to coalitions containing neither of them does not change. Together with two standard properties, efficiency and the null player property, differential marginality characterizes the Shapley value. For games that contain more than two players, we show that this characterization can be improved by using a substantially weaker property than differential marginality. Weak differential marginality requires two players' payoffs to change in the same direction when these players' marginal contributions to coalitions containing neither of them change by the same amount.