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In this paper, we present a game-theoretic study of the problem of routing in networks with max-min fair congestion control at the link level. The problem is formulated as a noncooperative game, in which each user aims to maximize its own bandwidth by selecting its routing path. We first prove the existence of Nash equilibria. This is important, because at a Nash equilibrium (NE), no user has any incentive to change its routing strategy-leading to a stable state. In addition, we investigate how the selfish behavior of 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, when paths of other users are fixed. 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 simulations show that the algorithm converges to an NE within 10 iterations and also achieves better fairness compared to other algorithms .