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Equipping routers in Wireless Mesh Networks with multiple radios tuned to multiple non-overlapping channels increases network connectivity and improves system throughput. On the other hand, selfish behavior of nodes during channel assignment can drastically decrease the network performance in a non-cooperative environment through protocol deviation. In this paper, we have addressed the problem of channel assignment to multiple radios in a multiple-collision domain Wireless Mesh Networks as a non-cooperative game. Analytical results show that Nash Equilibrium, which is not always social optimal due to multiple collision domains, exists under certain conditions. End users non-cooperative bargaining mechanism has been developed to increase the end-to-end achieved data rate of source nodes. Experimental results show that non-cooperative games with bargaining always perform better than Nash Equilibrium.