This paper provides a systematic study of the robust Stackelberg equilibrium (RSE), which naturally generalizes the widely adopted solution concept of the strong Stackelberg equilibrium (SSE). The RSE accounts for any possible up-to-$\delta$ suboptimal follower responses in Stackelberg games and is adopted to improve the robustness of the leader's strategy. While a few variants of robust Stackelberg equilibrium have been considered in previous literature, the RSE solution concept we consider is importantly different -- in some sense, it relaxes previously studied robust Stackelberg strategies and is applicable to much broader sources of uncertainties. We provide a thorough investigation of several fundamental properties of RSE, including its utility guarantees, algorithmics, and learnability. We first show that the RSE we defined always exists and thus is well-defined. Then we characterize how the leader's utility in RSE changes with the robustness level considered. On the algorithmic side, we show that, in sharp contrast to the tractability of computing an SSE, it is NP-hard to obtain a fully polynomial approximation scheme (FPTAS) for any constant robustness level. Nevertheless, we develop a quasi-polynomial approximation scheme (QPTAS) for RSE. Finally, we examine the learnability of the RSE in a natural learning scenario, where both players' utilities are not known in advance, and provide almost tight sample complexity results on learning the RSE. As a corollary of this result, we also obtain an algorithm for learning SSE, which strictly improves a key result of Bai et al. in terms of both utility guarantee and computational efficiency.

翻译：本文系统研究了鲁棒斯塔克尔伯格均衡（RSE），该概念自然推广了广泛采用的强斯塔克尔伯格均衡（SSE）解概念。RSE考虑了斯塔克尔伯格博弈中追随者可能出现的至多δ次优响应，用于提升领导者策略的鲁棒性。尽管先前文献已探讨过几种鲁棒斯塔克尔伯格均衡变体，但我们所考虑的RSE解概念具有重要区别——从某种意义上说，它松弛了先前研究的鲁棒斯塔克尔伯格策略，适用于更广泛的未知性来源。我们深入研究了RSE的几个基本性质，包括其效用保障、算法与可学习性。首先证明我们定义的RSE总是存在，因此是良定义的。进而刻画了领导者效用随鲁棒性水平变化的关系。在算法方面，研究表明：与SSE计算的易处理性形成鲜明对比，对任意常数鲁棒性水平，获得完全多项式时间近似方案（FPTAS）是NP难的。尽管如此，我们为RSE开发了拟多项式时间近似方案（QPTAS）。最后，在双方效用事先未知的自然学习场景中检验了RSE的可学习性，并给出了学习RSE的几乎紧样本复杂度结果。作为该结果的推论，我们还获得了学习SSE的算法，在效用保障和计算效率两方面严格改进了Bai等人的关键结果。