Games with a Partial Permission Structure and Their Applications

This thesis contains an introduction and six chapters. The introduction helps readers to position this thesis in game theory and graph theory, and Chapter 2 presents preliminaries, including cooperative games, digraphs, and games with a permission structure. Chapters 3, 4 and 5 provide new permission approaches, and characterize some Shapley value type solutions for cooperative games under these permission approaches. More specific, Chapter 3 combines disjunctive permission and local permission features to construct the local disjunctive permission approach. This approach requires that a player needs to get permission from at least one of its direct authorizers to cooperate, which can be used to model the approval right. Chapter 4 generalizes the local disjunctive and the local conjunctive permission approaches. This generalized approach requires that a player needs to get permission from a certain number of its direct authorizers to cooperate, which is suitable to describe certain voting situations. Chapter 5 provides a generalization of the local and the global disjunctive permission approaches, which requires that a non-top player needs permission from a sequence of authorizers that include a top player, or a sequence of at least $\xi$ authorizers to cooperate. This level generalization is commonly used in management. Besides, in those chapters, we provide axiomatizations for Shapley value type solutions for cooperative games under these permission approaches. Chapters 6 and 7 focus on applications of the content studied in the previous chapters. Chapter 6 provides a template for constructing measures based on permission values. In this chapter, we first propose the disjunctive measure to estimate dominance in a digraph by applying the local disjunctive permission value to additive games with a permission structure. Next, we generalize the disjunctive measure based on weak fairness, and extend this measure to weighted digraphs (including weights on nodes and weights on arcs). Finally, we axiomatize all these measures, and apply them to some classical networks in the literature, illustrating how they can be used to identify the key nodes in digraphs. Chapter 7 models highway toll allocation problems and provides a new platform for the application of games with a permission structure. In Chapter 7, we provide three toll allocation methods based on different toll charging rules. Besides, we characterize these methods and investigate the relationships between these methods and (disjunctive and conjunctive) permission values