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Repeated Sales with Heterogeneous Buyer Sophistication

Rishi Patel, Emmanouil Pountourakis, Samuel Taggart

Abstract

This paper considers behavior-based price discrimination in the repeated sale of a non-durable good to a single long-lived buyer, by a seller without commitment power. We assume that there is a mixed population of forward-looking ``sophisticated'' buyers and myopic ``naive'' buyers. We investigate the impact of these dynamics on the seller's ability to learn about the buyer and exploit this learning for revenue. We obtain conclusions that differ dramatically with the time horizon of the interactions. To understand short time horizons, we analyze a two-period model, and find that the strategic demand reduction observed with fully sophisticated buyers is robust to the introduction of naive types. In fact, despite the inability of naive buyers to game the pricing algorithm, their introduction can further harm the seller's revenue, due to more intense demand reduction overall. For long horizons, we consider an infinite-horizon model with time discounting. We find that the extreme demand reduction predicted by previous work does not survive the introduction of naive buyers. Instead, we observe equilibria where the seller learns meaningfully despite the sophisticated buyers' demand reduction. We prove that for a natural family of such equilibria, the seller's revenue is not just high, but approximates the revenue attainable with commitment power, even when the fraction of naive types is vanishingly small.

Repeated Sales with Heterogeneous Buyer Sophistication

Abstract

This paper considers behavior-based price discrimination in the repeated sale of a non-durable good to a single long-lived buyer, by a seller without commitment power. We assume that there is a mixed population of forward-looking ``sophisticated'' buyers and myopic ``naive'' buyers. We investigate the impact of these dynamics on the seller's ability to learn about the buyer and exploit this learning for revenue. We obtain conclusions that differ dramatically with the time horizon of the interactions. To understand short time horizons, we analyze a two-period model, and find that the strategic demand reduction observed with fully sophisticated buyers is robust to the introduction of naive types. In fact, despite the inability of naive buyers to game the pricing algorithm, their introduction can further harm the seller's revenue, due to more intense demand reduction overall. For long horizons, we consider an infinite-horizon model with time discounting. We find that the extreme demand reduction predicted by previous work does not survive the introduction of naive buyers. Instead, we observe equilibria where the seller learns meaningfully despite the sophisticated buyers' demand reduction. We prove that for a natural family of such equilibria, the seller's revenue is not just high, but approximates the revenue attainable with commitment power, even when the fraction of naive types is vanishingly small.
Paper Structure (30 sections, 12 theorems, 10 equations, 3 figures)

This paper contains 30 sections, 12 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and Contributions
  3. Results: Short Time Horizon
  4. Results: Long Time Horizon
  5. Related Work
  6. Notation and Preliminaries
  7. Game Description
  8. Naive Buyers.
  9. Histories
  10. Equilibrium
  11. Threshold Equilibria
  12. Short-Horizon Setting: Notational Simplifications
  13. Short-Horizon Warmup: Equilibrium With Linear Demand
  14. Equilibrium Structure
  15. Naive-Focused Regime
  16. ...and 15 more sections

Key Result

lemma thmcounterlemma

Every first-round price $p_1$ has a valid implementation.

Figures (3)

  • Figure 1: On-path prices and threshold for the linear-demand case.
  • Figure 2: Illustration of the seller's pricing problem after first-round reject for $\mu=0.7$. The only way to satisfy the threshold equation $p_1=\mathbb E[p_2^\mathcal{R}]$ with a sequentially rational price is to balance the two optima of $R^\mathcal{R}$. Vertical dotted line: $p_1$. Horizontal dotted line: optimal second-round revenue.
  • Figure 3: Plot of the seller's equilibrium revenue.

Theorems & Definitions (15)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • ...and 5 more