In the absence of externalities, marginality is equivalent to an independence property that rests on Harsanyi's dividends. These dividends identify the surplus inherent to each coalition. Independence states that a player's payoff stays the same if only dividends of coalitions to which this player does not belong to change. We introduce notions of marginality and independence for games with externalities. We measure a player's contribution in an embedded coalition by the change in the worth of this coalition that results when the player is removed from the game. We provide a characterization result using efficiency, anonymity, and marginality or independence, which generalizes Young's characterization of the Shapley value. An application of our result yields a new characterization of the solution put forth by Macho-Stadler et al. (J Econ Theor, 135, 2007, 339-356) without linearity, as well as for almost all generalizations put forth in the literature. The introduced method also allows us to investigate egalitarian solutions and to reveal how accounting for externalities may result in a deviation from the Shapley value. This is exemplified with a new solution that is designed in a way to not reward external effects, while at the same time it cannot be assumed that any partition is the default partition.
shapley value potential restriction operator partition function form game externalities