Price of Anarchy of Network Routing Games with Incomplete Information

We consider a class of networks where n agents need to send their traffic from a given source to a given destination over m identical, non-intersecting, and parallel links. For such networks, our interest is in computing the worst case loss in social welfare when a distributed routing scheme is used instead of a centralized one. For this, we use a noncooperative game model with price of anarchy as the index of comparison. Previous work in this area makes the complete information assumption, that is, every agent knows deterministically the amount of traffic injected by every other agent. Our work relaxes this by assuming that the amount of traffic each agent wishes to send is known to the agent itself but not to the rest of the agents; each agent has a belief about the traffic loads of all other agents, expressed in terms of a probability distribution. In this paper, we first set up a model for such network situations; the model is a noncooperative Bayesian game with incomplete information. We study the resulting games using the solution concept of Bayesian Nash equilibrium and a representation called the type agent representation. We derive an upper bound on price of anarchy for these games, assuming the total expected delay experienced by all the agents as the social cost. It turns out that these bounds are independent of the belief probability distributions of the agents. This fact, in particular, implies that the same bounds must hold for the complete information case, which is vindicated by the existing results in the literature for complete information routing games.