Minimum cost spanning tree problems with indifferent agents

We consider an extension of minimum cost spanning tree (mcst) problems in which some agents do not need to be connected to the source, but might reduce the cost of others to do so. Even if the cost usually cannot be computed in polynomial time, we extend the characterization of the Kar solution (Kar, 2002) for classic mcst problems. It is obtained by adapting the Equal treatment property: if the cost of the edge between two agents changes, their cost shares are affected in the same manner if they have the same demand. If not, their changes are proportional to each other. We obtain a family of weighted Shapley values. Three interesting solutions in that family are characterized using stability, fairness and manipulation-proofness properties.