Power Measures and Solutions for Games under Precedence Constraints

In the literature, there exist several models where a cooperative TU-game is enriched with a hierarchical structure on the player set that is represented by a digraph. We consider games under precedence constraints introduced by Faigle and Kern (1992) who also introduce a generalization of the Shapley value for such games. They characterized this solution by efficiency, linearity, the null player property and hierarchical strength which states that in unanimity games the payoffs are allocated among the players in the unanimity coalition proportional to their hierarchical strength in the corresponding coalition. The hierarchical strength of a player in a coalition in an acyclic digraph is the number of admissible permutations (those in which successors in the digraph enter before predecessors) where this player is the last of that coalition to enter. We introduce and axiomatize a new solution for games under precedence constraints, called hierarchical solution. Unlike the precedence Shapley value, this new solution satisfies the desirable axiom of irrelevant player independence meaning that payoffs assigned to relevant players are not affected by the presence of irrelevant players. This hierarchical solution is defined in a similar spirit as the precedence Shapley value but is a precedence power solution, and thus allocates the dividend of a coalition proportionally to a power measure for acyclic digraphs, specifically proportionally to the hierarchical measure. We give an axiomatization of this measure and extend it to regular set systems. Finally, we consider the normalized hierarchical measure on the subclasses of forests and sink.