Conjectural Variations in Competitive Dynamic Pricing: A Learning Foundation via Experimentation Design and Feedback Structure

Abstract

We study competitive dynamic pricing among multiple sellers, motivated by the rise of large-scale experimentation and algorithmic pricing in retail and online marketplaces. Sellers repeatedly set prices using simple learning rules and observe their own realized demand, while possibly observing only a subset of rivals' prices, even though demand depends on all sellers' prices and is subject to random shocks. Each seller runs local price experiments, such as switchback-style designs, and updates a focal price using a linear demand estimate fitted to its own demand data and the competitor prices it observes. Under certain conditions on demand, the resulting dynamics converge to a Conjectural Variations (CV) equilibrium, a classic static equilibrium notion in which each seller best responds under a conjecture that rivals' prices co-move systematically to changes in its own price. Unlike standard CV models that treat conjectures as behavioral primitives, we show that these conjectures arise endogenously from the interaction between the feedback structure and the correlation structure of experimentation. When a seller does not observe some rivals' prices, correlated experimentation induces an omitted-variable bias in demand estimation. We show that this bias determines the conjectures that govern the long-run equilibrium. Notably, when this learning bias vanishes, for example under full price feedback or independent experimentation of unobserved rivals, the learning dynamics converge to the standard Nash equilibrium. We provide simple sufficient conditions on demand for convergence in standard models and establish a finite-sample guarantee, showing that the mean squared price error decays at a rate of $\widetilde O (T^{-1/2})$.