The price of anarchy of serial cost sharing and other methods

Users share an increasing marginal cost technology. A method charges non negative cost shares covering costs. We look at the worst surplus (relative to the efficient surplus) in a Nash equilibrium of the demand game, where the minimum is taken over all convex preferences quasilinear in cost shares. We compare two budget-balanced methods, average cost pricing and serial cost sharing, and two methods collecting a budget surplus, marginal cost pricing and incremental cost sharing. In the latter case we count the budget surplus as a loss. For any convex cost function, the average cost and serial methods guarantee a (relative) surplus no less than 1 n , where n is the number of users. Neither marginal cost pricing, nor incremental cost sharing guarantees any positive gain. With quadratic costs, the surplus guaranteed by serial cost sharing is O( 1 logn ), and O( 1 n ) for the three other methods. This generalizes if the marginal cost is convex or concave, and its elasticity is bounded.