We consider cooperative TU-games with unpaid players, which are described by a TUgame and two categories of players, paid and unpaid. Unpaid players participate in the cooperative game but are not rewarded for their participation, for instance for legal reasons. The objective is then to determine how the contributions of unpaid players are redistributed among the paid players. To meet this goal, we introduce and characterize axiomatically three values that are inspired by the Shapley value but differ in the way they redistribute the contributions of unpaid players. These values are unified as instances of a more general two-step allocation procedure.
Béal et al. (Mon,) studied this question.