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This paper studies the outage performance of a two-way relay transmission, where two user terminals exchange information through a shared relay that exploits analog network coding to deal with the multi-access signals. Starting from some definitions, an exact outage probability of the conventional three- node scenario is derived over frequency-nonselective Rayleigh fading channels. To gain more insight, an approximate expression is also formulated. Following this analysis, opportunistic relaying is considered for multi-relay networks to make a good use of the available diversity degrees of the channel. Then, three significant relay selection strategies, i.e., the max-min, the max-hms and the min-scp policies, are studied and analyzed in terms of outage probability. Simulation experiments are performed and performance comparisons are conducted. The results show that the proposed policies are efficient and appropriate methods to implement relay selection and can achieve significant performance gains in terms of outage probability across the whole range of signal-to-noise ratio (SNR) as well as the sources' target rates, which validates the studies in this paper.