Nonreciprocal field theory for decision-making in multi-agent control systems.

Field theories for complex systems traditionally focus on collective behaviours emerging from simple, reciprocal pairwise interaction rules. However, many natural and artificial systems exhibit behaviours driven by microscopic decision-making processes that introduce both nonreciprocity and many-body interactions, challenging these conventional approaches. We develop a theoretical framework to incorporate decision-making into field theories using the shepherding problem from swarm robotics as a paradigmatic example of a multi-agent control system, where agents, the herders, must coordinate to confine another group of agents, the targets, within a prescribed region. By introducing continuous approximations of two key decision-making elements - target selection and trajectory planning - we derive field equations that capture the essential features of this distributed control problem. Our theory reveals that different decision-making strategies emerge at the continuum level, from average attraction to highly selective choices, and from undirected to goal-oriented motion, driving transitions between homogeneous and confined configurations. The resulting nonreciprocal field theory not only describes the shepherding problem but provides a general framework for incorporating decision-making into continuum theories of collective behaviour, with implications for applications ranging from robotic swarms to traffic and crowd management systems.