Related characterizations of the Shapley value and the weighted Shapley values via relaxations of differential marginality

Casajus (2018) [7] provides a characterization of the class of positively weighted Shapley values for finite games from an infinite universe of players via three properties: efficiency, the null player out property, and superweak differential marginality. The latter requires two players’ payoffs to change in the same direction whenever only their joint productivity changes, that is, their individual productivities stay the same. Strengthening this property into (weak) differential marginality yields a characterization of the Shapley value. We suggest a relaxation of superweak differential marginality into two subproperties: (i) hyperweak differential marginality and (ii) superweak differential marginality for infinite subdomains. The former (i) only rules out changes in the opposite direction. The latter (ii) requires changes in the same direction for players within certain infinite subuniverses. Together with efficiency and the null player out property, these properties characterize the class of weighted Shapley values.