Necessary players and values

In this paper we introduce the Gamma value, a new value for cooperative games with transferable utility. We also provide an axiomatic characterization of the Gamma 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 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 value, the equal division value and the Shapley value in two different scenarios. In addition, we propose an extension of the Gamma value for cooperative games with transferable utility and with a coalition structure. Finally, we provide axiomatic characterizations of the coalitional Gamma value and of the Owen and Banzhaf-Owen values using alternative properties concerning necessary players.