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.

翻译：$1-N$ 广义Stackelberg博弈（单领导者多跟随者博弈）同时交织着领导者与跟随者之间的层级交互以及跟随者之间的同时交互。然而，由于领导者与跟随者间复杂的交互作用，获取领导者的最优策略通常具有挑战性。本文提出了一种通用方法，用于求解$1-N$广义Stackelberg博弈的广义Stackelberg均衡。具体而言，我们首先利用变分均衡概念给出了广义Stackelberg均衡始终存在的条件。接着，为在多项式时间内求得均衡，我们将$1-N$广义Stackelberg博弈转化为一个$1-1$ Stackelberg博弈，且该博弈的Stackelberg均衡与原博弈完全一致。最后，我们提出了一种基于投影隐式梯度下降算法的有效计算流程，用于求解转化后$1-1$ Stackelberg博弈的Stackelberg均衡。通过电动汽车充电站运营策略推导中的两个问题验证了所提方法：（1）第一个问题优化电动汽车用户的单次充电定价，其中平台运营商决定电价，电动汽车用户根据自身满意度确定最优充电量；（2）第二个问题通过空间差异化定价，在每个充电站实现供需最优平衡。