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In this paper, we study the interaction between overlay routing and traffic engineering (TE) in a single autonomous system (AS). We formulate this interaction as a two-player non-cooperative non-zero sum game, where the overlay tries to minimize the delay of its traffic and the TE's objective is to minimize network cost. We study a Nash routing game with best-reply dynamics, in which the overlay and TE have equal status, and take turns to compute their optimal strategies based on the response of the other player in the previous round. We prove the existence, uniqueness and global stability of Nash equilibrium point (NEP) for a simple network. For general networks, we show that the selfish behavior of an overlay can cause huge cost increases and oscillations to the whole network. Even worse, we have identified cases, both analytically and experimentally, where the overlay's cost increases as the Nash routing game proceeds even though the overlay plays optimally based on TE's routing at each round. Experiments are performed to verify our analysis.