A model of embedded coalition formation and the Shapley values for games with externalities

This article proposes a stochastic model of embedded coalition formation for cooperative games with externalities. The coalition formation process is sequential and depends on a linear order over the agent set. From this process, Shapley-like values are defined and several equivalent expressions are provided. We introduce a commutativity property for the stochastic processes that guarantees that the Shapley-like values coincide with the Shapley value of an average game (Macho-Stadler et al., 2007) consistent with the underlying stochastic process of embedded coalition formation. Unlike the approach of Macho-Stadler et al. (2007), the probabilities from which the average game is computed are not necessarily symmetric, i.e., they do not necessarily depend on the sizes of the elements of the partition. Next, we provide several axiomatic characterizations for these Shapley-like values that are as close as possible to the best-known axiomatic characterizations for the Shapley value for TU-games. We consider axiomatic characterizations in the two main domains of fixed population and variable population.