In this paper, all the rate-pairs that are achievable at a Nash equilibrium (NE) in the two-user linear deterministic symmetric decentralized interference channel (LD-S-DIC) with noisy feedback are identified. More specifically, the Nash region (NR) of the LD-S-DIC with noisy feedback is fully characterized. The relevance of these rate-pairs is that once they are achieved by using NE transmit-receive configurations, none of the transmitter-receiver pairs can increase their individual rates by unilaterally changing their configurations. More importantly, it is shown that the NR of the LD-S-DIC with noisy feedback is larger than the NR of the LD-S-DIC without feedback only in certain cases. When interference is stronger than the desired signals, a larger NR is observed only if the signal to noise ratios (SNRs) of the feedback links are higher than the SNRs of the direct links. Conversely, when desired signals are stronger than interference, a larger NR is observed only if the SNRs of the feedback links are higher than both the signal to interference ratios (SIRs) and the interference to noise ratios (INRs) of the direct links. Previous results, namely the NE region of the two-user LD-S-DIC without feedback and with perfect output feedback are obtained as special cases of the results presented in this contribution.
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