A Shapley value for games with expanded coalitions

Abstract Derks and Peters (1993) introduce games with restricted coalitions defined by means of restrictions, which are monotonic projections of the set of players that assign to each coalition a subcoalition. They introduce a Shapley value considering these projections. In our work, we also consider a monotonic projection of the set of players, but instead of taking subcoalitions of coalitions, we take coalitions that contain them. That is, coalitions are expanded. In this way, we model situations in which some players are necessary to make a coalition effective. We give a Shapley value in this framework and obtain two axiomatic characterizations that determine endogenously the projection that expands the coalitions. In the first characterization, we employ a generalization of a monotonicity axiom used by (Young, (1985)) to characterize the Shapley value. In the second one, a dummy player axiom is required, together with additivity. Additionally, we provide an axiomatic characterization of the value introduced by Derks and Peters (1993), employing a suitably adapted dummy player axiom.