In addition to previously introduced concept of overlay routing strategy such as the selfish routing, we introduce a new strategy called "overlay optimal routing". Under this routing policy, the overlay seeks to minimize its weighted average delay by splitting its traffic onto multiple paths.
We establish that: (1) The overlay optimal routing can achieve better delay compared to selfish routing, and (II) there exists a Nash equilibrium when multiple overlays adopt this strategy. Although an equilibrium point exists for overlay optimal routing and possibly for selfish routing, we show that the interaction of multiple overlay routing may not be Pareto optimal and that some fairness anomalies of resouce allocation may occur. This is worthy of attention since overlay may not know the existence of other overlays and they will continue to operate at this sub-obtimal point. We explore two pricing schemes to resolve the above issues. We show that by incorporating a proper pricing scheme, the overlay routing game can be led to the desired equilibrium and avoid the problems mentioned above. Extensive fluid-based simulations are performed to support the theoretical claims.