text: Internet and graphs are very much related. The graphical structure of internet has been studied extensively to provide efficient solutions to routing and other problems. But most of these studies assume a central authority which controls and manages the internet. In the recent years game theoretic models have been proposed which do not require a central authority and the users are assumed to be routing their flows selfishly. The existence of Nash Equilibria, congestion and the amount of inefficiency caused by this selfish routing is a major concern in this field. A type of paradox in the selfish routing networks, Braess' Paradox, first discovered by Braess, is a major contributor to inefficiency. Several pricing mechanisms have also been provided which give a game theoretical model between users(consumers) and ISPs ( Internet Service Providers or sellers) for the internet. We propose a novel pricing mechanism, based on real world Internet network architecture, which reduces the severity of Braess' Paradox in selfish routing game theoretic networks. It's a pricing mechanism between combinatorial users and ISPs. We prove that Nash equilibria exists in this network and provide bounds on inefficiency . We use graphical properties of internet to prove our result. Several interesting extensions and future work have also been discussed.
  @cite_10: The success of the Internet is remarkable in light of the decentralized manner in which it is designed and operated. Unlike small scale networks, the Internet is built and controlled by a large number of disperate service providers who are not interested in any global optimization. Instead, providers simply seek to maximize their own profit by charging users for access to their service. Users themselves also behave selfishly, optimizing over price and quality of service. Game theory provides a natural framework for the study of such a situation. However, recent work in this area tends to focus on either the service providers or the network users, but not both. This paper introduces a new model for exploring the interaction of these two elements, in which network managers compete for users via prices and the quality of service provided. We study the extent to which competition between service providers hurts the overall social utility of the system.
summary: Hayrapetyan et. al @cite_10 analyze among similar lines as us but for their cost function they assume a constant term for the per unit flow price charged by ISPs but in the real world scenario generally the per unit charge decreases if the flow required by the user increases i.e. ISPs provide concession for more flow. Our model captures this. They have not done any analysis of the effect of Braess' Paradox in their network, too. We show that in our model the severity of Braess' Paradox is reduced.