The Shapley Value and Games with Hierarchies

Abstract

In this paper we focus on restrictions arising from the players belonging to some hierarchical structure that is represented by a digraph. Two of these models are the games with a permission structure and games under precedence constraints. In both cases, the hierarchy can be represented by a directed graph which restricts the possibilities of coalition formation. These two approaches led to two different type of solutions in the literature. The precedence power solutions for games under precedence constraints, are axiomatized with an axiom that applies a network power measure to the precedence constraint. We will show that something similar can be done for games with a permission structure, and obtain a class of permission power solutions. This class contains the (conjunctive) permission value. With this we have two classes of solutions for games with a hierarchy, one based on permission structures and another based on precedence constraints, that are characterized by similar axioms. Moreover, the solutions are linked with network power measures.

Fast Hierarchical Games for Image Explanations

“…Therefore, the exploration of relevant areas of an image can be done in a hierarchical manner. There exists extensive literature on hierarchies of games and their properties [39], [40]. Our contribution is to deploy these ideas for the purpose of image explanations.…”

Networks, Communication and Hierarchy: Applications to Cooperative Games

“…In this setting Algaba, van denBrink and Dietz (2017) define a class of new values in which the removal of certain 'irrelevant' players does not effect the payoffs of the remaining players. A comparison between the two models is given inAlgaba and van den Brink (2019).…”