In this paper, we study the problem of routing in networks with max-min fair congestion control at the link level. The goal of each user is to maximize its own bandwidth by selecting its path. The problem is formulated as a non-cooperative game. We first prove the existence of Nash Equilibria. This is important, because at a Nash Equilibrium (NE), no user has the incentive to change its routing strategy. In addition, we investigate how the selfish behavior of the users may affect the performance of the network as a whole. We next introduce a novel concept of observed available bandwidth on each link. It allows a user to find a path with maximum bandwidth under max-min fair congestion control in polynomial time. We then present a game based algorithm to compute an NE and prove that by following the natural game course the network converges to an NE. Extensive experiments show that the network can converge to an NE in less than 10 iterations and also significantly improves the fairness compared with other algorithms. Our results have the implication for the future routing protocol design.