Marginality Approach to Shapley Value in Games with Externalities

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities. In particular, when externalities occur, a direct translation of Shapley's axioms does not imply a unique value. In this paper we study the marginality approach to this problem, based on the idea of an alpha-parametrized definition of the marginal contribution, where alpha is a vector of weights associated with an agent joining/leaving a coalition. We prove that all values that satisfy the direct translation of Shapley's axioms can be obtained using the marginality approach. Moreover, we show that every such value can be uniquely derived using marginality approach by choosing appropriate weights alpha. Next, we analyze how properties of a value translate to the requirements on the definition of the marginal contribution (i.e. weights alpha). Building upon this analysis, we show that under certain conditions, two other axiomatizations of the Shapley value (i.e., Young's marginality axiomatization and Myerson's axiomatization based on the concept of balanced contributions), translated to games with externalities using the proper definition of the alpha-parametrized marginal contribution, are equivalent to Shapley's axiomatization.