Group Strategyproofness and No Subsidy via LP-Duality

We make two contributions to cooperative game theory, both of which rely heavily on linear programming duality theory. First, we introduce a fairness criterion on service providers, which we call No Subsidy. This condition ensures that a subset of the users is not overcharged in order to subsidize another set of users. We completely characterize cost functions that allow cost sharing mechanisms satisfying this condition. This class of cost functions contains the submodular functions, and can serve the purpose of formally capturing our intuitive notion of functions that exhibit economies of scale. Besides the well-studied class of non-decreasing, submodular cost functions, the only other class known to admit a budget balanced and group strategyproof cost sharing mechanism is the class of supermodular functions. In our second result, we give a budget balanced and group strategyproof cost sharing method for a cost function that is not in either of these classes. Our cost function arises naturally in the context of correcting an ineeciency of the scheme of multicast routing. Our cost sharing method utilizes the dual solution produced by the primal-dual algorithm for the minimum spanning tree problem, based on the bidirected cut relaxation for this problem.