Shapley Value for Games with Externalities and Games on Graphs

The Shapley value [46] is one of the most important solution concepts in coalitional game theory. It was originally defined for classical model of a coalitional game, which is relevant to a wide range of economic and social situations. However, while in certain cases the simplicity is the strength of the classical coalitional game model, it often becomes a limitation. To address this problem, a number of extensions have been proposed in the literature. In this thesis, we study two important such extensions – to games with externalities and graph-restricted games. Games with externalities [53] are a richer model of coalitional games in which the value of a coalition depends not only on its members, but also on the arrangement of other players. Unfortunately, four axioms that uniquely determine the Shapley value in classical coalitional games are not enough to imply a unique value in games with externalities. In this thesis, we study a method of strengthening the Null-Player Axiom by using α-parameterized definition of the marginal contribution in games with externalities. We prove that this approach yields a unique value for every α. Moreover, we show that this method is indeed general, in that all the values that satisfy the direct translation of Shapley's axioms to games with externalities can be obtained using this approach. Graph-restricted games [36] model naturally-occurring scenarios where coordination between any two players within a coalition is only possible if there is a communication channel between them. Two fundamental solution concepts that were proposed for such a game are the Shapley value and its particular extension – the Myerson value. In this thesis we develop algorithms to compute both values. Since the computation of either value involves visiting all connected induced subgraphs of the graph underlying the game, we start by developing a dedicated algorithm for this purpose and show that it is the fastest known in the literature. Then, we use it as the cornerstone upon which we build algorithms for the Shapley and Myerson values. ACKNOWLEDGEMENTS I would like to thank Dr. Tomasz Michalak, my mentor, co-author of all my papers, and a friend, for introducing me to research and his guidance through all these years. He has always motivated me to work, and when I was working-to work even harder. I thank for all the skype discussions and showing me " how a good paper should be written ". I would like …