Design Tradeoffs in Concave Cost-Sharing Games

This note focuses on the design of cost-sharing rules to optimize the efficiency of the resulting equilibria in cost-sharing games with concave cost functions. Our analysis focuses on two well-studied measures of efficiency, termed the price of anarchy and price of stability, which provide worst-case guarantees on the performance of the (worst or best) equilibria. Our first result characterizes the cost-sharing design that optimizes the price of anarchy, followed by the price of stability. This optimal cost-sharing rule is precisely the Shapley value cost-sharing rule. Our second result characterizes the cost-sharing design that optimizes the price of stability, followed by the price of anarchy. This optimal cost-sharing rule is precisely the marginal contribution cost-sharing rule. This analysis highlights a fundamental tradeoff between the price of anarchy and price of stability in the considered class of games. That is, given the optimality of both the Shapley value and marginal cost distribution rules in each of their respective domains, it is impossible to improve either the price of anarchy or price of stability without degrading its counterpart.