  The $1-N$ generalized Stackelberg game (single-leader multi-follower game) is
intricately intertwined with the interaction between a leader and followers
(hierarchical interaction) and the interaction among followers (simultaneous
interaction). However, obtaining the optimal strategy of the leader is
generally challenging due to the complex interactions among the leader and
followers. Here, we propose a general methodology to find a generalized
Stackelberg equilibrium of a $1-N$ generalized Stackelberg game. Specifically,
we first provide the conditions where a generalized Stackelberg equilibrium
always exists using the variational equilibrium concept. Next, to find an
equilibrium in polynomial time, we transformed the $1-N$ generalized
Stackelberg game into a $1-1$ Stackelberg game whose Stackelberg equilibrium is
identical to that of the original. Finally, we propose an effective computation
procedure based on the projected implicit gradient descent algorithm to find a
Stackelberg equilibrium of the transformed $1-1$ Stackelberg game. We validate
the proposed approaches using the two problems of deriving operating strategies
for EV charging stations: (1) the first problem is optimizing the one-time
charging price for EV users, in which a platform operator determines the price
of electricity and EV users determine the optimal amount of charging for their
satisfaction; and (2) the second problem is to determine the spatially varying
charging price to optimally balance the demand and supply over every charging
station.
