We study the dynamic pricing problem where the demand function is nonparametric and H\"older smooth, and we focus on adaptivity to the unknown H\"older smoothness parameter $\beta$ of the demand function. Traditionally the optimal dynamic pricing algorithm heavily relies on the knowledge of $\beta$ to achieve a minimax optimal regret of $\widetilde{O}(T^{\frac{\beta+1}{2\beta+1}})$. However, we highlight the challenge of adaptivity in this dynamic pricing problem by proving that no pricing policy can adaptively achieve this minimax optimal regret without knowledge of $\beta$. Motivated by the impossibility result, we propose a self-similarity condition to enable adaptivity. Importantly, we show that the self-similarity condition does not compromise the problem's inherent complexity since it preserves the regret lower bound $\Omega(T^{\frac{\beta+1}{2\beta+1}})$. Furthermore, we develop a smoothness-adaptive dynamic pricing algorithm and theoretically prove that the algorithm achieves this minimax optimal regret bound without the prior knowledge $\beta$.

翻译：我们研究需求函数为非参数且具有赫尔德光滑性的动态定价问题，重点关注对未知赫尔德光滑参数$\beta$的自适应性。传统上，最优动态定价算法严重依赖对$\beta$的已知信息，以实现$\widetilde{O}(T^{\frac{\beta+1}{2\beta+1}})$的极小化最优遗憾。然而，我们通过证明无定价策略能在未知$\beta$的情况下自适应达到该极小化最优遗憾，凸显了动态定价问题中自适应性的挑战。基于这一不可能性结论，我们提出自相似条件以实现自适应性。重要的是，我们证明自相似条件不会降低问题的内在复杂度，因为它保留了$\Omega(T^{\frac{\beta+1}{2\beta+1}})$的遗憾下界。此外，我们设计了一种光滑自适应动态定价算法，并从理论上证明该算法无需先验知识$\beta$即可达到该极小化最优遗憾界。