Sufficient conditions for nonempty core of minimum cost forest games

This paper studies problems of establishing a minimum cost network and of determining a fair cost allocation among customers. Each supplier offers a different type of service to the customers, and each customer wishes to be connected with the suppliers which he needs. The characteristic function game is reduced from minimum costs for constructing subnetworks. By introducing an equivalence relation on the set of customers, we provide sufficient conditions to have a nonempty core, which solves the above problems. It is shown that the game has a nonempty core as long as the optimal grand network becomes a forest which is composed of the collection of the minimum spanning trees on the above equivalence classes. It is further shown that, whenever the game consists of at most two equivalence classes, the core is nonempty.