We consider dynamic pricing strategies in a streamed longitudinal data set-up where the objective is to maximize, over time, the cumulative profit across a large number of customer segments. We consider a dynamic probit model with the consumers' preferences as well as price sensitivity varying over time. Building on the well-known finding that consumers sharing similar characteristics act in similar ways, we consider a global shrinkage structure, which assumes that the consumers' preferences across the different segments can be well approximated by a spatial autoregressive (SAR) model. In such a streamed longitudinal set-up, we measure the performance of a dynamic pricing policy via regret, which is the expected revenue loss compared to a clairvoyant that knows the sequence of model parameters in advance. We propose a pricing policy based on penalized stochastic gradient descent (PSGD) and explicitly characterize its regret as functions of time, the temporal variability in the model parameters as well as the strength of the auto-correlation network structure spanning the varied customer segments. Our regret analysis results not only demonstrate asymptotic optimality of the proposed policy but also show that for policy planning it is essential to incorporate available structural information as policies based on unshrunken models are highly sub-optimal in the aforementioned set-up.

翻译：我们考虑一个流式纵向数据设置下的动态定价策略，目标是在时间维度上最大化大量客户细分市场的累积利润。我们采用一个动态Probit模型，其中消费者的偏好以及价格敏感性随时间变化。基于消费者具有相似特征从而表现出相似行为这一已知结论，我们考虑一种全局收缩结构，该结构假设不同细分市场中消费者的偏好可通过空间自回归模型得到良好近似。在此类流式纵向设置中，我们通过遗憾（即与预先知晓模型参数序列的“全知者”相比的预期收益损失）来衡量动态定价策略的效果。我们提出一种基于惩罚随机梯度下降的定价策略，并明确刻画其遗憾作为时间、模型参数的时间变异性以及连接不同客户细分市场的自相关网络结构强度的函数。我们的遗憾分析结果不仅证明了所提策略的渐近最优性，还表明在策略规划中纳入可用结构信息至关重要，因为基于非收缩模型的策略在上述设置中高度次优。