Incentive Stackelberg games for mean-field control with large-scale followers

This paper addresses an incentive Stackelberg game for a class of mean-field stochastic systems with large-scale followers. We formulate a system with one leader and N followers to design team-optimal strategies and incentive mechanisms that minimize the cost functionals. Two conditions for the closed-loop system to be asymptotically mean-square stable under a team-optimal strategy and a follower’s Nash strategy are investigated. This analysis yields two higher-order cross-coupled nonlinear matrix equations (CCNMEs). Directly solving the CCNMEs becomes computationally infeasible for large populations; therefore, partitioned strategy structures and incentive designs within \(\mathbb {R}^{n \times n}\) ${\mathbb{R}}^{n\times n}$ are derived, where n is the dimensionality of the subsystem. Furthermore, by letting the followers’ population size tend to infinity, we develop an approximation method that is independent of \(\varepsilon =1/(N+1)\) $\epsilon =1/(N+1)$ to obtain the reduced-order equations, based on which, population-independent approximate strategies and incentive mechanisms are derived. Explicit linear asymptotic expansions are established for all solution matrices. Moreover, the leader’s and followers’ cost functionals are shown to converge to constant values at rates \(O(\varepsilon )\) $O(\epsilon )$ and \(O(\varepsilon ^2)\) $O({\epsilon }^2)$ , respectively. Finally, a numerical example is presented to validate the effectiveness of the proposed approach and the theoretical results.