Trouble Comes in Threes: Core stability in Minimum Cost Connection Networks

We consider a generalization of the Minimum Cost Spanning Tree (MCST) model dubbed the Minimum Cost Connection Network (MCCN) model, where network users have connection demands in the form of a pair of target nodes they want connected directly, or indirectly. Given a network which satisfies all connection demands at minimum cost, the problem consists of allocating the total cost of the efficient network among its users. As such, every MCCN problem induces a cooperative cost game where the cost of each each coalition of users is given by the cost of an efficient network satisfying the demand of the users in the coalition. Unlike in the MCST model we show that the core of the induced cost game in the MCCN model can be empty (without introducing Steiner nodes). We therefore consider sufficient conditions for non-empty core. Theorem 1 shows that when the efficient network and the demand graph consist of the same components, the induced cost game has non-empty core. Theorem 2 shows that when the demand graph has at most two components the induced cost game has non-empty core.