Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with level structure of cooperation

We present parallel characterizations of two different values in the framework of restricted cooperation games. The restrictions are introduced as a finite sequence of partitions defined on the player set, each of them being coarser than the previous one, hence forming a structure of different levels of a priori unions. On the one hand, we consider a value first introduced in Ref. [18], which extends the Shapley value to games with different levels of a priori unions. On the other hand, we introduce another solution for the same type of games, which extends the Banzhaf value in the same manner. We characterize these two values using logically comparable properties.

► A level structure is a sequence of partitions that limits agents' cooperation.
► The Banzhaf value is generalized to games with level structure of cooperation.
► The new value is characterized by means of six independent properties.
► A new characterization of the Shapley levels value is provided.
► Both characterizations use logically comparable axioms.