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 in the first step 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 in the second step the average over all players of all the payoff is obtained in the first step is computed. In general these two approaches lead to different solution concepts. When each component in the underlying communication structure is cycle-free, a linear cactus with cycles, or the complete graph, the two-step average tree value coincides with the average tree value. A comparative analysis of both solution concepts is done and an axiomatization of the the two-step average tree value on the subclass of TU games with semi-cycle-free hypergraph communication structure, which is more general than that given by a cycle-free hypergraph, is obtained