Network uncertainty in selfish routing

We study the problem of selfish routing in the presence of incomplete network information. Our model consists of a number of users who wish to route their traffic on a network of m parallel links with the objective of minimizing their latency. However, in doing so, they face the challenge of lack of precise information on the capacity of the network links. This uncertainty is modelled via a set of probability distributions over all the possibilities, one for each user. The resulting model is an amalgamation of the KP-model of (E. Koutsoupias and C. H. Papadimitriou, 1999) and the congestion games with user-specific functions of (I. Milchtaich, 1996). We embark on a study of Nash equilibria and the price of anarchy in this new model. In particular, we propose polynomial-time algorithms for computing some special cases of pure Nash equilibria and we show that negative results of (I. Milchtaich, 1996), for the non-existence of pure Nash equilibria in the case of three users, do not apply to our model. Consequently, we propose an interesting open problem in this area, that of the existence of pure Nash equilibria in the general case of our model. Furthermore, we consider appropriate notions for the social cost and the price of anarchy and obtain upper bounds for the latter. With respect to fully mixed Nash equilibria, we propose a method to compute them and show that when they exist they are unique. Finally we prove that the fully mixed Nash equilibrium maximizes the social welfare