Symmetry, mutual dependence, and the weighted Shapley values

We pinpoint the position of the (symmetric) Shapley value within the class of positively weighted Shapley value to their treatment of symmetric versus mutually dependent players. While symmetric players are equally productive, mutually dependent players are only jointly (hence, equally) productive. In particular, we provide a characterization of the whole class of positively weighted Shapley values that uses two standard properties, efficiency and the null player out property, and a new property called superweak differential marginality. Superweak differential marginality is a relaxation of weak differential marginality (Casajus and Yokote, 2017). It requires two players' payoff for two games to change in the same direction whenever only their joint productivity changes, i.e., their individual productivities stay the same. In contrast, weak differential marginality already requires this when their individual productivities change by the same amount. The Shapley value is the unique positively weighted Shapley value that satisfies weak differential marginality.