Cooperative games with restricted formation of coalitions

In the study of cooperative games, restricted cooperation between players is typically modeled by a set system of feasible coalitions of the players. In this paper, we go one step further and allow for a distinction among players within a feasible coalition, between those who are able to form the coalition and those who are not. This defines a contracting map, a choice function. We introduce the notion of quasi-building system and require that such a choice function gives rise to a quasi-building system. Many known set systems and structures studied in the literature are covered by quasi-building systems. For transferable utility games having a quasi-building system as cooperation structure we take as a solution the average of the marginal vectors that correspond to the set of rooted trees that are compatible with the quasi-building system. Properties of this solution, called the AMV-value, are studied. Relations with other solutions in the literature are also studied. To establish that the AMV-value is an element of the core, we introduce appropriate convexity-type conditions for the game with respect to the underlying quasi-building system. In case of universal cooperation, the AMV-value coincides with the Shapley value.