Sociotropy-Inspired Potential Game for Cooperative MIMO Beamforming

This paper introduces a sociotropy-inspired game-theoretic framework for distributed multiple-input multiple-output (MIMO) beamforming systems, where each antenna element is modeled as a strategic agent that adapts its beamforming parameters by balancing individual transmission performance with coordinated interaction among neighboring antennas. The resulting distributed beamforming problem is formulated as an exact potential game, enabling a unified analysis of cooperative antenna behavior under per-antenna power constraints. A complete mathematical formulation is developed, including the derivation of the utility and potential functions, the associated KKT stationarity conditions, and distributed projected gradient dynamics for beamforming adaptation. In addition, a graph-based multi-agent coordination mechanism is introduced to incorporate structured information exchange among antennas through similarity-driven message passing. Numerical simulations compare the proposed sociotropic strategy against both a selfish non-cooperative baseline and a graph-regularized multi-agent learning approach. Results demonstrate that sociotropic coordination improves interference management, convergence stability, and robustness under dynamic channel conditions, while maintaining lower computational complexity than learning-based coordination methods. Finally, the proposed distributed sociotropic beamforming framework is evaluated against classical maximum ratio transmission (MRT), zero-forcing (ZF), and regularised zero-forcing (RZF) beamforming schemes under identical time-varying channel dynamics. Results demonstrate that while conventional baselines exhibit performance saturation under channel fluctuations, the proposed method achieves continuous adaptation and improved sum-rate evolution over time.