📈 economics

Dynamic Wholesale Pricing under Censored-Demand Learning

This paper analyzes a finite-horizon dynamic wholesale pricing game between a manufacturer and a retailer facing censored demand, characterizing Markov perfect equilibria and demonstrating that for Weibull and exponential demand distributions, the strategic learning problem can be reduced to efficient, computable recursions.

Original authors: Michalis Deligiannis, Marco Scarsini, Xavier Venel

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michalis Deligiannis, Marco Scarsini, Xavier Venel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a lemonade stand (the Retailer) and you buy your lemons from a big farm (the Manufacturer). You don't know exactly how many people will want lemonade tomorrow. Sometimes it's a hot day, sometimes it's rainy.

In the real world, you only know how many cups you sold. If you run out of lemonade and a customer leaves empty-handed, you never know how many people actually wanted a drink. You just see "sold out." This is called censored demand—you have a partial picture of reality.

This paper is about how the farmer and the lemonade stand owner figure out the best prices and how much to make, even though they are both guessing about the future based on incomplete information.

Here is the story of their partnership, broken down simply:

1. The Setup: A Game of Guessing and Pricing

Every day, the game goes like this:

The Farmer sets a price for a crate of lemons (the Wholesale Price).
The Lemonade Stand sees that price and decides how many crates to order.
The Customers show up. They buy as much as they can.
The Result:
- If they sell everything, the stand owner thinks, "Wow, demand was huge!" (But they don't know how huge).
- If they have leftovers, they know exactly how much demand there was.
The Learning: Both the farmer and the stand owner watch the sales. They update their "guess" about how popular lemonade really is. They share this new information with each other.

2. The Problem: The "Black Box" of Lost Sales

The tricky part is that when they run out of stock, they lose data. It's like trying to learn the weather by only looking at the sky when it's sunny. If it rains, you don't see the clouds; you just see that you got wet.

Because they are both learning at the same time, their decisions are a dance.

If the farmer raises the price, the stand owner orders less.
If the stand owner orders less, they are more likely to run out of stock.
If they run out of stock, the farmer learns less about the true demand, making future guesses harder.

The paper asks: Is there a perfect strategy for both of them to follow every day to maximize their profits, knowing they are learning as they go?

3. The Solution: Finding the "Sweet Spot" (Equilibrium)

The authors found a way to solve this complex puzzle. They call it a Markov Perfect Equilibrium. Think of this as a "rulebook" that tells both the farmer and the stand owner exactly what to do based only on what they know right now (their current belief about demand), without needing to remember every single detail from the past.

They discovered two main scenarios:

Scenario A: The "Weibull" Lemonade (The Complex Case)

Imagine the demand for lemonade follows a specific, slightly complicated curve (like a bell curve that's been squished).

The Magic Trick: The authors realized that even though the math looks scary, they could "scale" the problem. Imagine you have a giant map of the world. Instead of trying to calculate the distance to every single city, you realize that if you know the distance to one city and the shape of the map, you can figure out the rest.
The Result: They proved that the farmer's price doesn't actually depend on how much demand there is (the scale), but only on how sure they are about the shape of the demand (the uncertainty). This simplifies the math massively, turning a giant, impossible calculation into a simple, step-by-step recipe.

Scenario B: The "Exponential" Lemonade (The Simple Case)

Imagine demand is very straightforward (like a straight line dropping off).

The Result: In this case, there is only one perfect strategy. It's unique. The authors showed that you can calculate the perfect price and order quantity by working backward from the last day of the season to the first. It's like solving a maze by starting at the exit and walking backward to the entrance.

4. Why This Matters in the Real World

This isn't just about lemons. Think about:

Walmart and its suppliers: Walmart shares real-time sales data with manufacturers. If a product sells out, the manufacturer sees "0 inventory" but doesn't know if 10 people wanted it or 1,000.
Fashion brands: A new shirt sells out in an hour. Did 50 people want it, or 5,000? The brand and the supplier have to guess the next order based on that "sell-out" signal.

The Big Takeaway:
The paper teaches us that in a partnership where both sides are learning from the same imperfect data:

Shared Learning is Powerful: Even with missing data, if both parties update their beliefs together, they can find a stable, profitable rhythm.
Simplicity Wins: Even in a complex world, the best pricing strategy often depends on just a few key numbers (how sure we are about demand), not on every single detail of the past.
The Farmer's Secret: The farmer (manufacturer) can set a price that ignores the size of the market and focuses only on the uncertainty. The retailer (stand owner) then adjusts their order size based on that price and the market size.

In short, the paper provides a mathematical "GPS" for businesses to navigate the fog of uncertain demand, ensuring they don't order too little (and lose sales) or too much (and waste money), even when they can't see the whole picture.

1. Problem Formulation

The paper investigates a finite-horizon dynamic Stackelberg game between a manufacturer (leader) and a retailer (follower) operating under conditions of censored demand.

Setting: The two parties engage in a wholesale-price contract over $T$ periods. In each period, the manufacturer sets a wholesale price $w_t$ , and the retailer chooses an order quantity $q_t$ .
Censoring: The true demand $D_t$ is unobserved. Only sales $S_t = \min(q_t, D_t)$ are observed. If a stockout occurs ( $D_t > q_t$ ), the unmet demand is lost and unobserved. This creates a "censored" dataset where the true demand distribution cannot be directly inferred without accounting for the order quantity.
Learning: Both parties share the same sales data (Public Information). They update a common Bayesian posterior belief about the unknown demand parameter $\Theta$ .
Demand Structure: The demand follows the Newsvendor distribution family (Braden and Freimer, 1991), defined by the survival function $1 - e^{-\theta \ell(y)}$ . The prior distribution for $\Theta$ is a conjugate Gamma distribution with hyperparameters $(a, b)$ (shape and rate).
Objective: To characterize the Markov Perfect Equilibrium (MPE) where strategies depend only on the current public belief state (the hyperparameters) and time, rather than the entire history of play.

2. Methodology

The authors employ Dynamic Programming (DP) and Bayesian Learning techniques within a game-theoretic framework.

State Variable: The state of the game is defined by the hyperparameters $(a_t, b_t)$ $(a_{t}, b_{t})$ of the posterior distribution.
- $a_t$ (shape) increases by 1 only when an uncensored demand observation occurs ( $S_t < q_t$ ).
- $b_t$ (rate) increases by $\ell(S_t)$ in every period.
Solution Concept: The authors seek a Markov Perfect Equilibrium (MPE), defined by value functions for the manufacturer ( $V^M$ ) and retailer ( $V^R$ ) that satisfy Bellman equations involving backward induction.
Dimensionality Reduction: A core methodological challenge is that the state space $(a_t, b_t)$ $(a_{t}, b_{t})$ is two-dimensional and grows complex. The authors extend the scaling approach (originally by Lariviere and Porteus, 1999, for single-agent problems) to this strategic setting.
- They construct a Standardized Markov Perfect Solution (SMPS) where the rate parameter $b$ is normalized to 1.
- They prove that the original game's equilibrium can be recovered from the standardized solution via a scaling transformation involving $b^{1/k}$ (where $k$ is the Weibull shape parameter).
Specific Cases:
- Weibull Demand: $\ell(y) = y^k$ . The paper proves the existence of an MPE using the dimensionality reduction technique.
- Exponential Demand: $\ell(y) = y$ (a special case of Weibull with $k=1$ ). The paper proves uniqueness and provides a closed-form recursive characterization.

3. Key Contributions

Strategic Learning with Censored Data: The paper bridges the gap between supply chain contracting and censored-demand learning. Unlike previous literature where learning is often exogenous or single-agent, this model features strategic interaction where the retailer's ordering decision (which causes censoring) directly influences the information available to both parties.
Existence of MPE in Strategic Settings: The authors prove the existence of an MPE for Weibull demand in a two-tier supply chain. This is the first application of the dimensionality-reduction technique to prove equilibrium existence in a strategic (multi-agent) newsvendor context.
Uniqueness and Computability for Exponential Demand: For the exponential case, the authors demonstrate that the MPE is unique. Furthermore, they show that the equilibrium can be computed via a simple backward recursion without solving complex optimization problems at every step, significantly reducing computational complexity.
Structural Insights on Pricing: A counter-intuitive finding is that in the equilibrium, the manufacturer's wholesale price depends only on the shape parameter $a$ (the number of uncensored observations) and time, but is invariant to the rate parameter $b$ (which scales the demand magnitude).

4. Key Results

A. Dimensionality Reduction and Existence (Weibull Demand)

Theorem 4.2: Establishes that if a solution exists for the standardized problem (where $b=1$ ), an MPE exists for the original problem.
Scaling Relation: The equilibrium wholesale price $w^*_t(a, b)$ in the original game is identical to the standardized price $w^*_t(a)$ . The order quantity scales as $q^*_t(a, b) = b^{1/k} \tilde{q}^*_t(a)$ .
Implication: The manufacturer does not need to adjust prices based on the scale of demand ( $b$ ), only on the precision of the belief regarding the demand shape ( $a$ ). However, the retailer's order quantity and both parties' profits scale with demand magnitude.

B. Characterization of MPE (Exponential Demand)

Theorem 5.1: For exponential demand ( $k=1$ ), the MPE is unique.
Recursive Formulas: The equilibrium strategies are given by explicit formulas:
- Retailer's Order: $\tilde{q}^*_t(a, w) = \max\left( \left( \frac{p + \Delta V^R}{w} \right)^{1/a} - 1, 0 \right)$ , where $\Delta V^R$ represents the value of future learning.
- Manufacturer's Price: Determined by a first-order condition that balances immediate margin against the value of information generation.
Computational Efficiency: The solution requires only backward recursion on the scalar state $a$ , avoiding the curse of dimensionality associated with the continuous $b$ parameter.

5. Significance and Implications

Operational Insight: The results suggest that in volatile markets with censored data, manufacturers can simplify their pricing policies. They do not need to constantly re-optimize prices based on the absolute volume of sales (scale), but rather focus on the frequency of stockouts (which updates the shape parameter $a$ ).
Data Sharing: The model validates the value of platforms like Walmart's Luminate, where shared POS data allows both parties to learn from the same censored observations, leading to a common belief state.
Theoretical Advancement: By extending the Lariviere and Porteus (1999) scaling technique to a dynamic Stackelberg game, the paper provides a tractable framework for analyzing dynamic contracts under uncertainty, which was previously considered analytically intractable due to the non-conjugacy of censored data in standard distributions.
Future Directions: The paper opens avenues for studying asymmetric information (where one party sees more data), alternative contract forms (revenue sharing), and infinite-horizon dynamics.

In summary, this paper provides a rigorous game-theoretic foundation for dynamic pricing in supply chains where demand is unknown and data is censored, offering both existence proofs and computationally efficient algorithms for equilibrium strategies.