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The Stackelberg game depicts a leader-follower relationship wherein decisionsare made sequentially, and the Stackelberg equilibrium represents an expectedoptimal solution when the leader can anticipate the rational response of thefollower. Motivated by control of network systems with two levels ofdecision-making hierarchy, such as the management of energy networks and powercoordination at cellular networks, a networked multi-leaders andmulti-followers Stackelberg game is proposed. Due to the constraint of limitedinformation interaction among players, a clustered information structure isassumed that each leader can only communicate with a portion of overallfollowers, namely its subordinated followers, and also only with its localneighboring leaders. In this case, the leaders cannot fully anticipate thecollective rational response of all followers with its local information. Toaddress Stackelberg equilibrium seeking under this partial informationstructure, we propose a distributed seeking algorithm based on implicitgradient estimation and network consensus mechanisms. We rigorously prove theconvergence of the algorithm for both diminishing and constant step sizes understrict and strong monotonicity conditions, respectively. Furthermore, the modeland the algorithm can also incorporate linear equality and inequalityconstraints into the followers' optimization problems, with the approach of theinterior point barrier function. Finally, we present numerical simulations inapplications to corroborate our claims on the proposed framework.

Distributed Stackelberg Equilibrium Seeking for Networked Multi-Leader Multi-Follower Games with A Clustered Information Structure