  We study the adaptive modulation (AM) problem in a network-coded two-way
relay channel (NC-TWRC), where each of the two users controls its own bit rate
in the $m$-ary quadrature amplitude modulation ($m$-QAM) to minimize the
transmission error rate and enhance the spectral efficiency. We show that there
exists a strategic complementarity, one user tends to transmit while the other
decides to do so in order to enhance the overall spectral efficiency, which is
beyond the scope of the conventional single-agent AM scheduling method. We
propose a two-player game model parameterized by the signal-to-noise ratios
(SNRs) of two user-to-user channels and prove that it is a supermodular game
where there always exist the extremal pure strategy Nash equilibria (PSNEs),
the largest and smallest PSNEs. We show by simulation results that the extremal
PSNEs incur a similar bit error rate (BER) as the conventional single-agent AM
scheme, but significantly improve the spectral efficiency in the NC-TWRC
system. The study also reveals the Pareto order of the extremal PSNEs: The
largest and smallest PSNEs are Pareto worst and best PSNEs, respectively.
Finally, we derive the sufficient conditions for the extremal PSNEs to be
symmetric and monotonic in channel SNRs. We also discuss how to utilize the
symmetry and monotonicity to relieve the complexity in the PSNE learning
process.
