[Paper Review] Dynamic Wholesale Pricing under Censored-Demand Learning

Michalis Deligiannis, Marco Scarsini+1 more|arXiv (Cornell University)|Mar 13, 2026

Supply Chain and Inventory Management0 citations

TL;DR

The paper analyzes a finite-horizon dynamic wholesale-price contract where both manufacturer and retailer learn from censored sales data, establishing Markov perfect equilibria under Weibull and exponential demand with dimensionality reduction.

ABSTRACT

We study a finite-horizon dynamic wholesale-price contract between a manufacturer and a retailer, both of whom observe only sales, rather than the true demand. When the retailer stocks out, unmet demand is unobserved, so both parties update a common posterior over the demand distribution from sales data. Each period, the manufacturer sets the wholesale price, the retailer chooses an order quantity, and the public belief state is updated. We characterize Markov perfect equilibria as functions of this public belief. Our main results are as follows: for Weibull demand, we extend the well-known scaling approach to this strategic learning setting, prove the existence of an equilibrium, and reduce computation to a standardized one-parameter recursion; for exponential demand, we show that the equilibrium is unique and computable via a simple backward recursion.

Motivation & Objective

• Model a dynamic wholesale-price contract where demand is unknown and only sales are observed.
• Enable joint Bayesian learning from censored demand via a common posterior.
• Prove existence of a Markov perfect equilibrium (MPE) and develop tractable computation for Weibull demand.
• Characterize the equilibrium under exponential demand and provide a backward-recursive solution.

Proposed method

• Adopt a dynamic Stackelberg game where the manufacturer sets wholesale prices and the retailer orders quantities.
• Use a newsvendor-family demand with a conjugate prior to preserve Bayesian conjugacy under censoring.
• Employ Markov perfect equilibrium with state as the public belief (a,b) and backward induction equations.
• Apply dimensionality reduction (Weibull case) to reduce the dynamic program to a one-parameter recursion.
• Derive explicit equilibrium conditions for exponential demand showing unique, computable backward recursion.

Experimental results

Research questions

• RQ1 How can a manufacturer and retailer optimally interact over a finite horizon when true demand is unknown and only censored sales are observed?
• RQ2 Does a Markov perfect equilibrium exist in this censored-demand learning setting, and under which demand distributions is it unique or computable?
• RQ3 Can dimensionality reduction render the Weibull-demand-based dynamic program tractable and scalable?
• RQ4 What are the equilibrium characterizations and computation methods for the exponential-demand case?

Key findings

• For Weibull demand with a gamma prior, an MPE exists and can be computed via a reduced one-parameter recursion.
• In the Weibull setting, the manufacturer’s wholesale price at equilibrium depends only on the initial belief and the count of uncensored observations.
• For exponential demand, the MPE is unique and computable by a simple backward recursion without period-by-period optimization.
• Under exponential demand, the equilibrium order quantity is given by a closed-form-like recursion related to the value function differences.
• The manufacturer’s pricing is invariant to the demand scale parameter in the standardized formulation, while retailer profits scale with it through larger orders.