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From Confounding to Learning: Dynamic Service Fee Pricing on Third-Party Platforms

Rui Ai, David Simchi-Levi, Feng Zhu

TL;DR

This work addresses dynamic service-fee pricing on third-party platforms under confounded demand by leveraging instrumental variables formed from the platform’s own actions. It introduces a doubly robust learning framework that combines ERM with IV regression, augmented by carefully tuned noise injection and a low-switching design to handle strategic buyers and deep-function classes. A key theoretical contribution is a phase-transition result: supply noise can dramatically reduce regret, with a critical threshold at $\sigma_S^2 \approx 1/\sqrt{T}$, and a novel homeomorphic construction that enables efficiency guarantees for learning demand with deep neural networks without star-shapedness. Empirically, the method demonstrates substantial revenue gains on real data from Zomato and Lyft, validating the practical viability of using actions as instruments for demand learning in dynamic pricing settings.

Abstract

We study the pricing behavior of third-party platforms facing strategic agents. Assuming the platform is a revenue maximizer, it observes market features that generally affect demand. Since only the equilibrium price and quantity are observable, this presents a general demand learning problem under confounding. Mathematically, we develop an algorithm with optimal regret of $\Tilde{\cO}(\sqrt{T}\wedgeσ_S^{-2})$. Our results reveal that supply-side noise fundamentally affects the learnability of demand, leading to a phase transition in regret. Technically, we show that non-i.i.d. actions can serve as instrumental variables for learning demand. We also propose a novel homeomorphic construction that allows us to establish estimation bounds without assuming star-shapedness, providing the first efficiency guarantee for learning demand with deep neural networks. Finally, we demonstrate the practical applicability of our approach through simulations and real-world data from Zomato and Lyft.

From Confounding to Learning: Dynamic Service Fee Pricing on Third-Party Platforms

TL;DR

This work addresses dynamic service-fee pricing on third-party platforms under confounded demand by leveraging instrumental variables formed from the platform’s own actions. It introduces a doubly robust learning framework that combines ERM with IV regression, augmented by carefully tuned noise injection and a low-switching design to handle strategic buyers and deep-function classes. A key theoretical contribution is a phase-transition result: supply noise can dramatically reduce regret, with a critical threshold at , and a novel homeomorphic construction that enables efficiency guarantees for learning demand with deep neural networks without star-shapedness. Empirically, the method demonstrates substantial revenue gains on real data from Zomato and Lyft, validating the practical viability of using actions as instruments for demand learning in dynamic pricing settings.

Abstract

We study the pricing behavior of third-party platforms facing strategic agents. Assuming the platform is a revenue maximizer, it observes market features that generally affect demand. Since only the equilibrium price and quantity are observable, this presents a general demand learning problem under confounding. Mathematically, we develop an algorithm with optimal regret of . Our results reveal that supply-side noise fundamentally affects the learnability of demand, leading to a phase transition in regret. Technically, we show that non-i.i.d. actions can serve as instrumental variables for learning demand. We also propose a novel homeomorphic construction that allows us to establish estimation bounds without assuming star-shapedness, providing the first efficiency guarantee for learning demand with deep neural networks. Finally, we demonstrate the practical applicability of our approach through simulations and real-world data from Zomato and Lyft.
Paper Structure (60 sections, 28 theorems, 148 equations, 12 figures, 2 tables, 3 algorithms)

This paper contains 60 sections, 28 theorems, 148 equations, 12 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Instrumental Variables in Machine Learning
  4. Pricing with Strategic Buyers
  5. Demand Learning under Uncertainty
  6. Nonparametric Regression and Covering Number
  7. Model and Methodology
  8. The Model
  9. Sellers
  10. The Platform
  11. Buyers
  12. Strategic Behavior
  13. General Feature Vectors
  14. Our Methodology
  15. Regret Analysis
  16. ...and 45 more sections

Key Result

Theorem 1

Under ass:context, with probability at least $1-\frac{1}{T}$, alg:AaaIV achieves at most $\mathcal{O}\left(dim\log(dim)+dim\log^2 T+ \frac{\eta_S^2\log^3 T}{\sigma_S^4} +\frac{\eta_S^2\log^2T}{\sigma_S^4(1-\gamma)^2} +\frac{\eta_S^2\log T}{\sigma_S^4(1-\gamma)^3}\right)$ regret against any buyer who

Figures (12)

  • Figure 1: Uber ride receipts from the same city: one from February 2024 (left) and two from March 2024 (right)
  • Figure 2: Phase transition with respect to $\sigma_S^2$.
  • Figure 3: $\mathcal{F}$ is only star-shaped around 0.
  • Figure 4: Regret of \ref{['alg:AaaIV']} when $\sigma_S^2 = 1$.
  • Figure 5: Regret of \ref{['alg:AaaIV']} when $\sigma_S^2 = 0$.
  • ...and 7 more figures

Theorems & Definitions (41)

  • Example 1
  • Theorem 1: $\sigma_S>0$
  • Theorem 2: $\sigma_S=0$
  • Theorem 3
  • Theorem 4: Unknown $\sigma_S$
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1: Corollary 1.7. in rigollet2023high
  • Lemma 2: Theorem 1.13 in rigollet2023high
  • ...and 31 more