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A Tale of Two Cities: Pessimism and Opportunism in Offline Dynamic Pricing

This paper proposes a nonparametric partial identification framework for offline dynamic pricing in environments with incomplete historical data coverage, introducing distinct pessimistic and opportunistic decision rules that provide finite-sample regret bounds and outperform standard offline reinforcement learning baselines while mapping a firm's risk posture to its optimal pricing strategy.

Original authors: Zeyu Bian, Zhengling Qi, Lan Wang

Published 2026-05-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zeyu Bian, Zhengling Qi, Lan Wang

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 the captain of an airline trying to sell tickets for a flight. Your goal is to set the perfect price: high enough to make money, but low enough that people actually buy the tickets.

Usually, captains learn by experimenting: "Let's try $200 today, $300 tomorrow, and see what happens." But in the real world, companies often can't experiment. They can't afford to lose money on bad prices, or they simply don't have time to test every possible price point. Instead, they have to look at a history book of past sales to figure out what to do next. This is called "offline dynamic pricing."

The Problem: The "Missing Pages" in the History Book

The authors of this paper point out a major flaw in how companies usually use these history books.

Imagine your history book only has entries for prices of $100, $200, and $400. It has no data for $300.

The Old Way: Most computer algorithms say, "We have no data for $300, so we can't use it. Let's just pick the best price from the ones we do have ($100, $200, or $400)."
The Danger: What if the perfect price was actually $300? By ignoring it because it's "missing" from the book, the company leaves money on the table.

The paper calls this a "no-coverage" problem. The data is incomplete, and the best price might be entirely missing from the records.

The Solution: Using "Common Sense" (Monotonicity)

The authors propose a new way to read the history book. They rely on a simple rule of economics called monotonicity: All else being equal, if you raise the price, fewer people will buy.

Think of it like a staircase. If you know how many people bought tickets at $200 and how many bought at $400, you can make a very educated guess about the $300 price, even if no one ever bought at exactly $300.

The demand at $300 must be lower than at $200.
The demand at $300 must be higher than at $400.

So, instead of guessing a single number for $300, the authors create a safe zone (an interval). They say, "We don't know the exact number, but we know it's somewhere between the $200 result and the $400 result."

The Two Strategies: The Turtle and the Fox

Once they have these "safe zones" instead of single numbers, the authors create two different decision-making styles for the company.

1. The Pessimistic Policy (The Turtle) 🐢

Mindset: "Worst-case scenario."
How it works: The Turtle looks at the "safe zone" for every price and asks, "What is the lowest amount of money I could possibly make if I pick this price?" It then picks the price that guarantees the highest of those worst-case amounts.
Who is this for? A company that is terrified of losing money. If you are a small business that can't afford a bad day, you want the Turtle. It protects you from the downside. It's safe, stable, and boring.

2. The Opportunistic Policy (The Fox) 🦊

Mindset: "What am I missing out on?"
How it works: The Fox looks at the "safe zones" and asks, "If I pick this price, how much worse could I do compared to the best possible alternative?" It tries to minimize the "regret" of making the wrong choice.
The Difference: If a price has a low "worst-case" but a huge "best-case" (a huge potential upside), the Fox might still pick it. The Turtle would reject it immediately because of the low floor. The Fox is willing to take a calculated risk to catch a big win.
Who is this for? A big, established company with deep pockets that wants to grow. If you have the resources to absorb a small loss for a chance at a massive gain, you want the Fox.

The Results: What Happened in the Experiments?

The authors tested these ideas using computer simulations and real data from airline tickets (specifically flights from New York to Charlotte).

The Old Methods: When the data was missing (like the $300 price), standard computer programs failed. They stuck to the prices they knew, even if those prices were suboptimal.
The New Methods:
- The Opportunistic (Fox) strategy was the winner in most cases. It was brave enough to pick the "missing" prices that turned out to be the most profitable. In many tests, it performed almost as well as if it had perfect data.
- The Pessimistic (Turtle) strategy was better than the old standard methods but sometimes too cautious. However, it was excellent at protecting against disaster.
- Crucially, the old "pessimistic" methods used by other researchers never picked a missing price. The new "Refined Pessimistic" method could, but only in very specific situations (like if the missing price was the highest possible one).

The Takeaway

This paper gives companies a practical map for when their data is incomplete:

Don't ignore missing prices. Use the relationship between high and low prices to estimate what the missing ones might be.
Choose your personality.
- If you need safety and hate losing money, use the Pessimistic rule.
- If you want growth and are willing to risk a little to get a lot, use the Opportunistic rule.

The paper proves mathematically that these two rules are the best possible ways to handle this specific problem, giving companies a clear choice based on how much risk they are willing to take.

Technical Summary: A Tale of Two Cities: Pessimism and Opportunism in Offline Dynamic Pricing

Problem Statement
This paper addresses the challenge of offline dynamic pricing in environments where historical data provides incomplete coverage of the price space. In many practical applications, firms cannot afford the costs or risks of extensive experimentation, leading to datasets where certain candidate prices—including potentially the optimal ones—are entirely unobserved. Standard offline reinforcement learning (RL) methods, including those relying on full coverage or partial coverage assumptions (where the optimal policy's actions must be represented in the data), fail in these "no-coverage" settings. When the optimal price is missing, demand and Q-functions are no longer point-identified, rendering standard estimators suboptimal or inapplicable. The authors aim to develop a statistically sound framework for offline dynamic pricing that operates without any data coverage assumptions, leveraging the fundamental economic regularity that demand is non-increasing in price.

Methodology
The authors propose a nonparametric partial-identification framework that exploits the monotonicity of demand with respect to price to bound the value of unobserved prices.

Partial Identification Framework:
Instead of extrapolating via parametric models, the method constructs interval bounds for the demand distribution function $F_t(d|a)$ at unobserved prices $a$ . These bounds are derived from the observed demand distributions at the nearest lower ( $a^L_t$ ) and upper ( $a^U_t$ ) observed prices. Specifically, for an unobserved price $a$ , the conditional CDF is bounded by $F_t(d|a^L_t) \le F_t(d|a) \le F_t(d|a^U_t)$ . To account for sampling errors, the authors refine these bounds using confidence intervals derived from the Dvoretzky–Kiefer–Wolfowitz inequality, ensuring the true CDF lies within the constructed interval with high probability.
Decision Rules:
Within this framework, the authors formulate two distinct dynamic decision rules tailored to sequential no-coverage environments:
- Pessimistic Policy: This policy maximizes the worst-case revenue. It selects the price that maximizes the lower bound of the Q-function, constructed by minimizing the Q-function over the confidence set of possible demand distributions. This approach offers protection against downside risk when uncertainty about unobserved prices is substantial.
- Opportunistic Policy: This policy minimizes the worst-case regret relative to the best feasible alternative. Unlike the pessimistic approach which focuses on the lower bound of revenue, the opportunistic approach considers the maximum regret an action might incur compared to the best alternative action under the worst-case realization of the unknown demand. This allows firms to pursue potential upside from under-explored prices.
Algorithms:
The authors develop efficient algorithms for both policies using backward induction (dynamic programming). The core computational step involves solving a linear program to find the worst-case Q-value (for the pessimistic policy) or the worst-case regret (for the opportunistic policy) within the constructed confidence sets.

Key Contributions

No-Coverage Framework: The paper presents the first offline dynamic pricing technique that remains valid without any data coverage assumption. It introduces a partial-identification framework for policy learning when optimal prices are missing from offline data, imposing no parametric restrictions on demand, the policy class, or the Q-function class.
Theoretical Guarantees: The authors establish finite-sample regret bounds for both the pessimistic and opportunistic policies. The bounds decompose into two terms: one capturing estimation error for observed prices and another quantifying the additional cost induced by unobserved optimal prices.
- When the optimal price is covered, the regret recovers the standard rate of $O_p(\sqrt{\log N / N})$ .
- When coverage fails, the results quantify the precise cost of learning under partial identification, showing the bound is tight (attainable).
Managerial Mapping: The framework provides a practical mapping from a firm's risk posture to its pricing policy. It distinguishes between firms seeking revenue stability (pessimistic policy) and those willing to bear measured risk for potential gains (opportunistic policy).

Results
The authors evaluate their methods through simulations and a real-data application involving airline ticket pricing.

Simulations: Using Poisson and Negative Binomial demand models with various missing-data mechanisms, the proposed methods are compared against standard offline RL baselines (Greedy, Vanilla Pessimistic, Conservative Q-Learning, and Batch-Constrained Q-learning). The results show that the proposed opportunistic method often outperforms baselines, particularly in scenarios where the behavior policy is suboptimal or data coverage is sparse. The opportunistic method frequently achieves performance comparable to an oracle optimal policy even with missing prices. The refined pessimistic method also outperforms the vanilla pessimistic approach in several cases. Notably, the opportunistic method is the only one capable of selecting unobserved prices that are optimal, whereas pessimistic baselines (including the refined version in most cases) tend to exclude them.
Real Data Application: In an analysis of Expedia airline ticket data (JFK to CLT), where certain price points were under-represented, the opportunistic method achieved the highest empirical total revenue in Monte Carlo simulations. The refined pessimistic method ranked third, performing slightly worse than CQL, while the vanilla pessimistic approach yielded the lowest revenue. The opportunistic approach successfully selected the unobserved price point in specific states, whereas CQL did not.

Significance and Claims
The paper claims to bridge the gap between offline dynamic pricing and partial identification in a sequential, no-coverage environment. It argues that existing offline RL methods are inadequate for pricing problems where the optimal price may be entirely absent from historical data. By leveraging demand monotonicity, the proposed framework allows for policy learning beyond the support of observed data without relying on parametric assumptions. The authors emphasize that their regret bounds are tight and rate-optimal. Managerially, the work offers a nuanced tool for decision-makers, allowing them to choose between a conservative strategy (pessimism) for downside protection and an aggressive strategy (opportunism) for potential gains, depending on their risk tolerance and resource availability. The paper concludes that while no single strategy uniformly outperforms the other, the choice depends on the specific context and the firm's strategic goals regarding risk and uncertainty.