Merge-proofness and cost solidarity in shortest path games

We study cost-sharing rules in network problems where agents seek to ship quantities of some good to their respective locations, and the cost on each arc is linear in the flow crossing it. In this context, Core Selection requires that each subgroup of agents pay a joint cost share that is not higher than its stand-alone cost. We prove that the demander rule, under which each agent pays the cost of her shortest path for each unit she demands, is the unique cost-sharing rule satisfying both Core Selection and Merge Proofness. The Merge Proofness axiom prevents distinct nodes from reducing their joint cost share by merging into a single node. An alternative characterization of the demander rule is obtained by combining Core Selection and Cost Solidarity. The Cost Solidarity axiom says that each agent's cost share should be weakly increasing in the cost matrix.