The two-step average tree value for graph and hypergraph games

We introduce the two-step average tree value for transferable utility games with restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games. Instead of averaging players' marginal contributions corresponding to all admissible rooted spanning trees of the underlying (hyper)graph, which determines the average tree solution or value, we consider a two-step averaging procedure, in which first, for each player the average of players' marginal contributions corresponding to all admissible rooted spanning trees that have this player as the root is calculated, and second, the average over all players of all the payoffs obtained in the first step is computed. In general these two approaches lead to different solution concepts. Contrary to the average tree value, the new solution satisfies component fairness and the total cooperation equal treatment property on the entire class of hypergraph games. Moreover, the two-step average tree value is axiomatized on the class of semi-cycle-free hypergraph games, which is more general than the class of cycle-free hypergraph games by allowing the underlying hypergraphs to contain certain cycles. The two-step average tree value is also core stable on the subclass of superadditive semi-cycle-free hypergraph games.