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For state-of-the-art network function virtualization (NFV) systems, it remains a key challenge to conduct effective service chain composition for different network services (NSs) with ultra-low request latencies and minimum network congestion. To this end, existing solutions often require full knowledge of the network state, while ignoring the privacy issues and overlooking the non-cooperative behaviors of users. What is more, they may fall short in the face of unexpected failures such as user unavailability and virtual machine breakdown. In this paper, we formulate the problem of service chain composition in NFV systems with failures as a non-cooperative game. By showing that such a game is a weighted potential game and exploiting the unique problem structure, we propose two effective distributed schemes that guide the service chain compositions of different NSs towards the Nash equilibrium (NE) state with both near-optimal latencies and minimum congestion. Besides, we develop two novel learning-aided schemes as comparisons, which are based on deep reinforcement learning (DRL) and Monte Carlo tree search (MCTS) techniques, respectively. Our theoretical analysis and simulation results demonstrate the effectiveness of our proposed schemes, as well as the adaptivity when faced with failures.