# 17-016/II (2017-01-30)

Author(s)
René van den Brink, VU University Amsterdam, The Netherlands
Keywords:
Cooperative TU-game, hierarchy, permission structure, antimatroid, local permission structure, applications.
JEL codes:
C71

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions.

In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize exisiting solutions for TU-games. In this survey we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures.

A central model of games with hierarchies are the games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.