Designing Fast Converging Cost Sharing Methods for Multicast Transmissions

We study a multicast game in non-cooperative directed networks in which a source sends the same message or service to a set of r receiving users and the cost of the used links is divided among the receivers according to a given cost sharing method. By following the approach recently proposed by Chen et al. (Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 854–863, 2008), we analyze the performances of a family of methods satisfying certain desiderata, namely, weak and strong budget-balance, fairness and separability. We show that any fair method may require an arbitrary number of selfish moves in order to converge to a pure Nash equilibrium, hence we focus on the solutions obtained after one round of selfish moves. We evaluate their quality according to two global social functions: the overall cost of the solution and the maximum shared cost of users. The only method satisfying all the properties is the well-known Shapley value for which we show an approximation ratio of the solutions reached after a one round walk equal to \Theta(r2). We then prove that relaxing the strong budget balance and separability properties (we call feasible any method satisfying weak budget balance and fairness) leads to improved performances since we determine a feasible method achieving an approximation ratio of the solutions reached after a one round walk equal to O(r). This bound is asymptotically optimal since we also show that any method satisfying weak budget balance cannot achieve an approximation ratio of the solutions reached after a one round walk smaller than r. Finally, we prove the NP-hardness of computing the sequence of moves leading to the best possible global performance and extend most of the results to undirected networks.