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Switchback Price Experiments with Forward-Looking Demand

Yifan Wu, Ramesh Johari, Vasilis Syrgkanis, Gabriel Y. Weintraub

Abstract

We consider a retailer running a switchback experiment for the price of a single product, with infinite supply. In each period, the seller chooses a price $p$ from a set of predefined prices that consist of a reference price and a few discounted price levels. The goal is to estimate the demand gradient at the reference price point, with the goal of adjusting the reference price to improve revenue after the experiment. In our model, in each period, a unit mass of buyers arrives on the market, with values distributed based on a time-varying process. Crucially, buyers are forward looking with a discounted utility and will choose to not purchase now if they expect to face a discounted price in the near future. We show that forward-looking demand introduces bias in naive estimators of the demand gradient, due to intertemporal interference. Furthermore, we prove that there is no estimator that uses data from price experiments with only two price points that can recover the correct demand gradient, even in the limit of an infinitely long experiment with an infinitesimal price discount. Moreover, we characterize the form of the bias of naive estimators. Finally, we show that with a simple three price level experiment, the seller can remove the bias due to strategic forward-looking behavior and construct an estimator for the demand gradient that asymptotically recovers the truth.

Switchback Price Experiments with Forward-Looking Demand

Abstract

We consider a retailer running a switchback experiment for the price of a single product, with infinite supply. In each period, the seller chooses a price from a set of predefined prices that consist of a reference price and a few discounted price levels. The goal is to estimate the demand gradient at the reference price point, with the goal of adjusting the reference price to improve revenue after the experiment. In our model, in each period, a unit mass of buyers arrives on the market, with values distributed based on a time-varying process. Crucially, buyers are forward looking with a discounted utility and will choose to not purchase now if they expect to face a discounted price in the near future. We show that forward-looking demand introduces bias in naive estimators of the demand gradient, due to intertemporal interference. Furthermore, we prove that there is no estimator that uses data from price experiments with only two price points that can recover the correct demand gradient, even in the limit of an infinitely long experiment with an infinitesimal price discount. Moreover, we characterize the form of the bias of naive estimators. Finally, we show that with a simple three price level experiment, the seller can remove the bias due to strategic forward-looking behavior and construct an estimator for the demand gradient that asymptotically recovers the truth.

Paper Structure

This paper contains 40 sections, 13 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Forward-Looking Demand and Intertemporal Substitution
  4. Interference in Pricing Experiments
  5. No-regret reserve price optimization with strategic buyers
  6. Dynamic pricing and demand learning
  7. Model and Preliminaries
  8. Market Model
  9. Forward-Looking Buyer Behavior
  10. Estimands
  11. Demand Gradient
  12. Revenue Gradient
  13. Switchback Experiment Design
  14. Switchback Experiment
  15. Local Price Experiments
  16. ...and 25 more sections

Key Result

Lemma 3.1

Consider a switchback experiment with parameters $(S, \boldsymbol{q}, \delta)$. At period $t$, for a buyer with value $v$ and patience level $\gamma$, according to the buyer's private belief $\phi_{t+\tau|t}$ about post-experimental prices conditioning on the experiment ends at period $t+\tau$, we w Under Assumption ass:rational, on each day $t$, a buyer that has not purchased yet, accepts the cur

Figures (2)

  • Figure 1: Price tracker for a paper towel product on Amazon. The history around October 2021 shows switching prices. Source: www.camelcamelcamel.com.
  • Figure 2: Visual representation of main debiasing argument. For simplicity of the figure, $\Delta_i$ quantities on the figure represent the absolute value of the $\Delta_i$ quantities in the text. The solid line depicts that true demand curve, which for simplicity is linear and the small dots on the line are the true static demands for these price points. The thick solid line represents the final estimate of the demand gradient that we uncover. The larger dots represent observed demands which are distorted by the forward looking behavior, albeit at different intensities. The two dotted lines represent the biased estimates of the demand gradient that each of $\Delta_1$ or $\Delta_2$ would uncover. We see that the thick solid line is parallel to the true demand curve, thereby uncovering the correct gradient.

Theorems & Definitions (26)

  • Remark 2.6: Static Demand vs. Demand under Constant Experimentation
  • Definition 2.7: Asymptotically Unbiased Estimator
  • Lemma 3.1: Buyer Waiting-Bias Behavior
  • Corollary 3.2
  • Theorem 4.1: Non-Existence of Asymptotically Unbiased Estimators for Two-Price Experiments
  • Theorem 4.2: Bias of Two-Price Experiments
  • Theorem 4.3: Bias of Two-Price Experiments with Arrival Monitoring
  • Theorem 5.1
  • Proposition 5.2
  • Theorem 5.3
  • ...and 16 more