We consider a novel formulation of the dynamic pricing and demand learning problem, where the evolution of demand in response to posted prices is governed by a stochastic variant of the popular Bass model with parameters $\alpha, \beta$ that are linked to the so-called "innovation" and "imitation" effects. Unlike the more commonly used i.i.d. and contextual demand models, in this model the posted price not only affects the demand and the revenue in the current round but also the future evolution of demand, and hence the fraction of potential market size $m$ that can be ultimately captured. In this paper, we consider the more challenging incomplete information problem where dynamic pricing is applied in conjunction with learning the unknown parameters, with the objective of optimizing the cumulative revenues over a given selling horizon of length $T$. Equivalently, the goal is to minimize the regret which measures the revenue loss of the algorithm relative to the optimal expected revenue achievable under the stochastic Bass model with market size $m$ and time horizon $T$. Our main contribution is the development of an algorithm that satisfies a high probability regret guarantee of order $\tilde O(m^{2/3})$; where the market size $m$ is known a priori. Moreover, we show that no algorithm can incur smaller order of loss by deriving a matching lower bound. Unlike most regret analysis results, in the present problem the market size $m$ is the fundamental driver of the complexity; our lower bound in fact, indicates that for any fixed $\alpha, \beta$, most non-trivial instances of the problem have constant $T$ and large $m$. We believe that this insight sets the problem of dynamic pricing under the Bass model apart from the typical i.i.d. setting and multi-armed bandit based models for dynamic pricing, which typically focus only on the asymptotics with respect to time horizon $T$.
翻译:我们考虑的是动态定价和需求学习问题的新提法,在这个模型中,对上市价格的需求变化由流行的巴斯斯模式的随机变式调节,其参数为$\alpha,\beta$,与所谓的“创新”和“缩进”效应相联系。与更常用的i.d.和背景需求模型不同,在模型中,已上市价格不仅影响当前一轮的需求和收入,而且影响未来需求的变化,从而影响潜在市场规模的一小部分,最终可以捕捉到美元。在本文中,我们考虑的是更具有挑战性的不完整的信息问题,即动态定价与未知的参数相结合,目标是在销售长度美元的情况下优化累积收入。 与更常用的i.d.d.以及相关的需求模型中,已上市价格损失的多少相比,只能衡量算出最高预期收入的多少,而市场规模为美元和时平下的美元。我们的主要贡献是开发一个更低的算法,即低的基值为美元,而最有可能的市值与前期价格分析结果的多少。