A Game-Theoretic Analysis of the Adversarial Boyd-Kuramoto Model

The "Boyd" model, also known as the "OODA loop", represents the cyclic decision processes of individuals and organisations in a variety of adversarial situations. Combined with the Kuramoto model, which provides a mathematical foundation for describing the behaviour of a set of coupled or networked oscillators, the Boyd-Kuramoto model captures strategic (cyclic) decision making in competitive environments. This paper presents a novel game-theoretic approach to the Boyd-Kuramoto dynamical model in complex and networked systems. A two-player, Red versus Blue, strategic (non-cooperative) game is defined to describe the competitive interactions and individual decision cycles of Red and Blue agent populations. We study the model analytically in the regime of near phase synchrony where linearisation approximations are possible. We find that we can solve for the Nash equilibrium of the game in closed form, and that it only depends on the parameters defining the fixed point of the dynamical system. A detailed numerical analysis of the finite version of the game investigates the behaviour of the underlying networked Kuramoto oscillators and yields a unique, dominant Nash equilibrium solution. The obtained Nash equilibrium is further studied analytically in a region where the underlying Boyd-Kuramoto dynamics are stable. The result suggests that only the fixed point of the dynamical system plays a role, consist with the analytical solution. Finally, the impact of other variations of the Boyd-Kuramoto parameters on the game outcomes are studied numerically, confirming the observations from fixed point approaches. It is observed that many parameters of the Kuramoto model affect the NE solution of the current game formulation much less than initially stipulated, arguably due to the time-scale separation between the underlying Kuramoto model and the static game formulation.