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

Zeyu Bian, Zhengling Qi, Cong Shi, Lan Wang

Abstract

This paper studies offline dynamic pricing without data coverage assumption, thereby allowing for any price including the optimal one not being observed in the offline data. Previous approaches that rely on the various coverage assumptions such as that the optimal prices are observable, would lead to suboptimal decisions and consequently, reduced profits. We address this challenge by framing the problem to a partial identification framework. Specifically, we establish a partial identification bound for the demand parameter whose associated price is unobserved by leveraging the inherent monotonicity property in the pricing problem. We further incorporate pessimistic and opportunistic strategies within the proposed partial identification framework to derive the estimated policy. Theoretically, we establish rate-optimal finite-sample regret guarantees for both strategies. Empirically, we demonstrate the superior performance of the newly proposed methods via a synthetic environment. This research provides practitioners with valuable insights into offline pricing strategies in the challenging no-coverage setting, ultimately fostering sustainable growth and profitability of the company.

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

Abstract

This paper studies offline dynamic pricing without data coverage assumption, thereby allowing for any price including the optimal one not being observed in the offline data. Previous approaches that rely on the various coverage assumptions such as that the optimal prices are observable, would lead to suboptimal decisions and consequently, reduced profits. We address this challenge by framing the problem to a partial identification framework. Specifically, we establish a partial identification bound for the demand parameter whose associated price is unobserved by leveraging the inherent monotonicity property in the pricing problem. We further incorporate pessimistic and opportunistic strategies within the proposed partial identification framework to derive the estimated policy. Theoretically, we establish rate-optimal finite-sample regret guarantees for both strategies. Empirically, we demonstrate the superior performance of the newly proposed methods via a synthetic environment. This research provides practitioners with valuable insights into offline pricing strategies in the challenging no-coverage setting, ultimately fostering sustainable growth and profitability of the company.

Paper Structure

This paper contains 25 sections, 8 theorems, 71 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Pessimism versus Opportunism
  3. Contributions
  4. Related Works
  5. Models and Assumptions
  6. Background
  7. Baseline Methods: Greedy and Pessimistic Approaches
  8. Limitations of Baseline Methods
  9. A Partial Identification Framework
  10. Pessimistic and Opportunistic Strategies
  11. A Refined Pessimistic Strategy
  12. Opportunistic Strategy
  13. Comparison of Vanilla Pessimistic, Refined Pessimistic, and Opportunistic Methods
  14. Numerical Studies
  15. Discussion
  16. ...and 10 more sections

Key Result

Proposition 1

Under Assumption assumption:poisson, Equation eq: bellman becomes for any $0 \leq t \leq T$ and $x, a$.

Figures (3)

  • Figure 1: An illustrative example demonstrating the partial identification task, where only intervals of the revenue under the two prices can be obtained: $Q_a \in [Q_a^L,Q_a^U]$, for $a=0 \hbox{and}1$.
  • Figure 2: Value functions for the optimal policy and three types of suboptimal policies: Type I excludes price 10 from the optimal policy, while Type II and Type III exclude prices 9 and 8, respectively.
  • Figure 3: Comparison of empirical value functions among vanilla pessimistic, refined pessimistic, and the opportunistic methods across five scenarios. We also include the oracle optimal policy for comparison.

Theorems & Definitions (25)

  • Definition 1: Identifiability
  • Example 1
  • Example 2
  • Remark 1: Markov Property
  • Proposition 1
  • Remark 2
  • Lemma 1: Decomposition Lemma
  • Remark 3
  • Theorem 1
  • Remark 4
  • ...and 15 more