Necessary players and values

In this paper we introduce the $$\Gamma $$ Γ value, a new value for cooperative games with transferable utility. We also provide an axiomatic characterization of the $$\Gamma $$ Γ value based on a property concerning the so-called necessary players. A necessary player of a game is one without which the characteristic function is zero. We illustrate the performance of the $$\Gamma $$ Γ value in a particular cost allocation problem that arises when the owners of the apartments in a building plan to install an elevator and share its installation cost; in the resulting example we compare the proposals of the $$\Gamma $$ Γ value, the equal division value and the Shapley value in two different scenarios. In addition, we propose an extension of the $$\Gamma $$ Γ value for cooperative games with transferable utility and with a coalition structure. Finally, we provide axiomatic characterizations of the coalitional $$\Gamma $$ Γ value and of the Owen and Banzhaf-Owen values using alternative properties concerning necessary players.