Incentive Stackelberg games for empirical mean-field decentralised control in large populations

This paper investigates the existence conditions for incentive design in Stackelberg games and their applications to mean-field large-population team-optimal control. We introduce the concept of incentivability, which characterises when a leader can design incentives that induce followers to adopt team-optimal strategies, serving as a structural criterion analogous to controllability and observability. As an application, we analyse a mean-field large-population team-optimal control problem in the asymptotic regime where the number of followers N becomes large. By introducing new parameters, we establish conditions under which feasible strategies exist as the number of decision-makers tends to infinity. To overcome the fundamental physical and computational bottlenecks inherent in conventional centralised designs when N is sufficiently large, we develop an N-independent decentralised incentive strategy. Furthermore, as a novel contribution, we propose two reduced-order strategy sets based on feedback from the population mean and the sample mean, respectively. Finally, a numerical example of a large-population stochastic system demonstrates that the proposed notion of incentivability and the decentralised approximation maintain high accuracy and effectiveness in large-population settings.