The Inefficiency of Equilibria in a Network Creation Game with Packet Forwarding

We study a novel variation of network creation games in which the players (vertices) form a graph by building undirected edges to each other with the goal of reducing their costs of using the network. The model we introduce assumes that a minimal set of nodes with high reachability from others are handed the responsibility of routing the traffic alongside the network. For this purpose, we suggest that a minimum dominating set (MDS) of the graph would be a reasonable choice as the intermediate nodes, thus the players in one such set would incur an extra cost for forwarding. We study the Nash equilibrium in this model assuming an extra cost of β is evenly shared among all the nodes in a MDS. We prove upper bounds on the price of anarchy, the worst-case ratio of the social cost of Nash equilibria of the network to that of socially optimum solution, for different values of β. Specifically, we show this inefficiency is modest for β = n since the price of anarchy is O(n). We also prove a tight upper bound of Θ(n) for β = n, and also give some upper bounds when β takes a value between n and n.