A limitation of transferable utility games is their inability to account for the varying participation levels that players may exhibit in cooperative activities. Multi-choice games address this issue by allowing players to participate at distinct levels. In this extended framework, several extensions of the Shapley value have been introduced. However, the relationships between these extensions remain unclear, and there is currently no solution concept that effectively reconciles them. In this paper, we aim to clarify the connections between two specific extensions of the Shapley value. We show that these extensions share comparable axiomatic characterizations. Furthermore, we propose a family of solutions that provides a consensus between the two extensions. To establish two distinct characterizations for this family, we introduce new axioms.
