Efficient graph topologies in network routing games

A topology is efficient for network games if, for any game over it, every Nash equilibrium is socially optimal. It is well known that many topologies are not efficient for network games. We characterize efficient topologies in network games with a finite set of players, each wishing to transmit an atomic unit of unsplittable flow. We distinguish between two classes of atomic network routing games. In network congestion games a player's cost is the sum of the costs of the edges it traverses, while in bottleneck routing games, it is its maximum edge cost. In both classes, the social cost is the maximum cost among the players' costs. We show that for symmetric network congestion games the efficient topologies are Extension Parallel networks, while for symmetric bottleneck routing games the efficient topologies are Series Parallel networks. In the asymmetric case the efficient topologies include only trees with parallel edges.