Resumen de Values for games with authorization structure
In a general way, game theory studies cooperation and conflict models, using mathematical methods. This thesis is about cooperative game theory. A cooperative game consists of a set of players and a characteristic function which determines the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate, regardless of the actions taken by the other players. It is often assumed that the players are free to participate in any coalition, but in some situations there are dependency relationships among the players that restrict their capacity to cooperate within some coalitions. Those relationships must be taken into account if we want to distribute the profits fairly. Myerson studied games in which communication between players is restricted. He considered graphs to model those restraints. Subsequently, different kinds of limitations on cooperation among players have been studied, and various structures have been used for that, like convex geometries, matroids, antimatroids or augmenting systems. A particularly interesting case of limited cooperation arises when we consider veto relationships between players. In this regard, Gilles, Owen and van den Brink modeled situations in which a hierarchical structure imposes some constraints on the behavior of the players in the game. They introduced games with permission structure, that consist of a set of players, a cooperative game and a mapping that assigns to every player a subset of direct subordinates. They defined and characterized values for games with permission structure. Subsequently, Derks and Peters generalized that model by considering the so-called restrictions. They obtain and characterize a value for games with restricted coalitions. Although the model considered by Derks and Peters is more general, the characterization of the value proposed is not as intuitive and straightforward as that given by Gilles, Owen and van den Brink. Our aim in this work is to propose a new model of games with restricted cooperation. This new model will fulfill three requirements. Firstly, it will be even more general than the one given by Derks and Peters. Secondly, it will allow us to define and axiomatize a sharing value in a similar way as in the case of games with permission structure. And, in the third place, it will be applicable to fuzzy permission relationships.
The thesis is organized as follows:
In chapter 1 we present the theoretical preliminaries and provide a historical framework for games with restricted cooperation.
In chapter 2 we introduce the so called authorization structures. We define and characterize a Shapley value and a Banzhaf value for games with authorization structure.
In chapter 3 we aim to model situations in which the dependency relationships among the players are not complete. To that end we introduce fuzzy authorization structures. A Shapley value and a Banzhaf value for games with fuzzy authorization structure are obtained and characterized. We use the Choquet integral to define a new auxiliary game that combines the information from the original game and from the fuzzy dependency relationships.
In chapter 4 the power in authorization structures is studied. Using the Shapley value and the Banzhaf value we measure how favorable the situation of each agent in an authorization structure is. In each case, a fuzzy digraph is assigned to each authorization structure, measuring the dependence or the dominance relationship between any two agents. Moreover, a characterization of those measures is given. In a similar way, the power in fuzzy authorization structures is also discussed.
In chapter 5 a particular type of authorization structure, called interior operator structure, is analyzed. Interior operator structures are characterized in terms of transitivity of the veto relationships, and a Shapley value for games with interior operator structure is studied. Because of the transitivity of the veto relationships, this value turns out to satisfy a property of structural monotonicity. Finally, fuzzy interior operator structures are analyzed.
In chapter 6 we focus on the most simple authorization structures: conjunctive authorization structures. We characterize them by proving that an authorization structure is conjunctive if and only if all the dependency relationships induced are bilateral. A simplified expression of the Shapley power measures defined in chapter 4 is given for these structures. We give a Shapley value and a Banzhaf value for games with conjunctive authorization structure. We also study fuzzy conjunctive authorization structures.
In chapter 7 we deal with NTU games. NTU games with authorization structure are introduced, and a Shapley solution is obtained. We analyze the fuzzy case as well. Lastly, a Harsanyi solution for NTU games with interior operator structure is presented.
