On cycle-free accessible union stable network structures

We investigate set theoretic properties that characterize the collection of connected coalitions in cycle-free undirected graphs among the accessible union stable network structures. It turns out that the only additional requirement is that every non-unitary feasible coalition can be written in a unique way as a union of non-unitary supports. As a consequence, a fairness axiom for solutions for cooperative games on cycle-free accessible union stable network structures can be defined that, together with the well-known component efficiency, characterizes the Shapley value on the class of cycle-free accessible union stable network structures. Since this fairness axiom combines ideas behind the traditional fairness axiom and balanced contributions, we refer to it as balanced fairness.