Linking the Kar and folk solutions through a problem separation property

Minimum cost spanning tree problems try to connect agents e¢ ciently to a source when agents are located at di¤erent points in space and the cost of using an edge is …xed. We examine solution concepts to divide the common cost of connection among users, and revisit the dispute between two of the most familiar solutions: the Kar and folk solutions, both based on the familiar Shapley value. We characterize the family of solutions corresponding to the a¢ ne combination of the Kar and folk solutions. The weights being put on the Kar and folk solutions are related to how we share cost in a simple two-agent problem. This family is characterized using a new property called Weak Problem Separation that allows, under conditions, to divide the problem in two: the connection of an agent to the source and the connection of agents to each other. New characterizations of the Kar and folk solutions are then o¤ered. In addition, a new rule is proposed and characterized.