Bass模型下的动态定价与学习
Dynamic Pricing and Learning under the Bass Model
摘要：
我们考虑一种新的制定的动态定价和需求学习问题，其中的需求响应于公布的价格的演变是由流行的巴斯模型的随机变量与参数$α, β$，被链接到所谓的“创新”和“模仿”的效果。与更常用的i.i.d.和上下文需求模型，在该模型中，公布的价格不仅影响当前回合中的需求和收入，而且影响需求的未来演变，并且因此影响最终可以捕获的潜在市场规模$m$的部分。在本文中，我们考虑更具挑战性的不完全信息问题，动态定价与学习的未知参数相结合，在给定的销售时间长度$T$的目标是优化的累计收入。等价地，目标是最小化后悔，该后悔测量算法相对于在具有市场规模$m$和时间范围$T$的随机Bass模型下可实现的最优预期收入的收入损失。我们的主要贡献是开发的算法，满足高概率的遗憾保证顺序$\tilde O(m^{2/3})$，其中市场规模$m$是已知的先验。此外，我们表明，没有算法可以产生较小的损失，通过推导出匹配的下限。与大多数遗憾的分析结果，在本问题的市场规模$m$的复杂性的根本驱动力，我们的下限，事实上，表明，对于任何固定的$α, β$，大多数非平凡的问题的情况下，有恒定的$T$和大$m$。我们相信，这一见解设置的问题，动态定价下的巴斯模型除了典型的i.i.d.。设置和多臂强盗为基础的模型，动态定价，通常只集中在渐近相对于时间范围$T$。
英文摘要：
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 $α, β$ 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 $α, β$, 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 $T$.
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