  Throughput and energy efficiency in 3-way relay channels are studied in this
paper. Unlike previous contributions, we consider a circular message exchange.
First, an outer bound and achievable sum rate expressions for different
relaying protocols are derived for 3-way relay channels. The sum capacity is
characterized for certain SNR regimes. Next, leveraging the derived achievable
sum rate expressions, cooperative and competitive maximization of the energy
efficiency are considered. For the cooperative case, both low-complexity and
globally optimal algorithms for joint power allocation at the users and at the
relay are designed so as to maximize the system global energy efficiency. For
the competitive case, a game theoretic approach is taken, and it is shown that
the best response dynamics is guaranteed to converge to a Nash equilibrium. A
power consumption model for mmWave board-to-board communications is developed,
and numerical results are provided to corroborate and provide insight on the
theoretical findings.
