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The $1-N$ generalized Stackelberg game (single-leader multi-follower game) isintricately intertwined with the interaction between a leader and followers(hierarchical interaction) and the interaction among followers (simultaneousinteraction). However, obtaining the optimal strategy of the leader isgenerally challenging due to the complex interactions among the leader andfollowers. Here, we propose a general methodology to find a generalizedStackelberg equilibrium of a $1-N$ generalized Stackelberg game. Specifically,we first provide the conditions where a generalized Stackelberg equilibriumalways exists using the variational equilibrium concept. Next, to find anequilibrium in polynomial time, we transformed the $1-N$ generalizedStackelberg game into a $1-1$ Stackelberg game whose Stackelberg equilibrium isidentical to that of the original. Finally, we propose an effective computationprocedure based on the projected implicit gradient descent algorithm to find aStackelberg equilibrium of the transformed $1-1$ Stackelberg game. We validatethe proposed approaches using the two problems of deriving operating strategiesfor EV charging stations: (1) the first problem is optimizing the one-timecharging price for EV users, in which a platform operator determines the priceof electricity and EV users determine the optimal amount of charging for theirsatisfaction; and (2) the second problem is to determine the spatially varyingcharging price to optimally balance the demand and supply over every chargingstation.

Computing Algorithm for an Equilibrium of the Generalized Stackelberg Game