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Marketplace Operators Can Induce Competitive Pricing

Tiffany Ding, Dominique Perrault-Joncas, Orit Ronen, Michael I. Jordan, Dirk Bergemann, Dean Foster, Omer Gottesman

TL;DR

This paper analyzes a price-quantity Stackelberg duopoly where a marketplace operator $\mathtt{M}$ can also sell directly, paying a referral fee $\alpha$ to the independent seller $\mathtt{I}$ and enjoying a consumer-experience benefit $k$ per unit. Using a linear demand $Q(p)=\theta-p$ and explicit residual-demand rules (intensity as default, with proportional as robustness) it derives the subgame-perfect Nash equilibrium, showing that pricing $p_{\mathtt{M}}$ low enough and inventory $q_{\mathtt{M}}$ sufficient can induce $\mathtt{I}$ to price competitively even when $\mathtt{I}$ would otherwise monopolize. The analysis demonstrates that $\mathtt{M}$’s entry as a seller transfers surplus from $\mathtt{I}$ to consumers and often increases consumer surplus, with welfare enhancements robust to variations in rationing and substitutability (including one-directional imperfect substitutability via a parameter $\gamma$). The results offer practical insights for platform design: operators can foster healthier competition without heavy-handed regulation by maintaining credible inventory and carefully chosen pricing, with phase boundaries depending on costs $c_{\mathtt{M}}, c_{\mathtt{I}}$, commission $\alpha$, and the customer-experience parameter $k$.

Abstract

As e-commerce marketplaces continue to grow in popularity, it has become increasingly important to understand the role and impact of marketplace operators on competition and social welfare. We model a marketplace operator as an entity that not only facilitates third-party sales but can also choose to directly participate in the market as a competing seller. We formalize this market structure as a price-quantity Stackelberg duopoly in which the leader is a marketplace operator and the follower is an independent seller who shares a fraction of their revenue with the marketplace operator for the privilege of selling on the platform. The objective of the marketplace operator is to maximize a weighted sum of profit and a term capturing positive customer experience, whereas the independent seller seeks solely to maximize their own profit. We derive the subgame-perfect Nash equilibrium and find that it is often optimal for the marketplace operator to induce competition by offering the product at a low price to incentivize the independent seller to match their price.

Marketplace Operators Can Induce Competitive Pricing

TL;DR

This paper analyzes a price-quantity Stackelberg duopoly where a marketplace operator can also sell directly, paying a referral fee to the independent seller and enjoying a consumer-experience benefit per unit. Using a linear demand and explicit residual-demand rules (intensity as default, with proportional as robustness) it derives the subgame-perfect Nash equilibrium, showing that pricing low enough and inventory sufficient can induce to price competitively even when would otherwise monopolize. The analysis demonstrates that ’s entry as a seller transfers surplus from to consumers and often increases consumer surplus, with welfare enhancements robust to variations in rationing and substitutability (including one-directional imperfect substitutability via a parameter ). The results offer practical insights for platform design: operators can foster healthier competition without heavy-handed regulation by maintaining credible inventory and carefully chosen pricing, with phase boundaries depending on costs , commission , and the customer-experience parameter .

Abstract

As e-commerce marketplaces continue to grow in popularity, it has become increasingly important to understand the role and impact of marketplace operators on competition and social welfare. We model a marketplace operator as an entity that not only facilitates third-party sales but can also choose to directly participate in the market as a competing seller. We formalize this market structure as a price-quantity Stackelberg duopoly in which the leader is a marketplace operator and the follower is an independent seller who shares a fraction of their revenue with the marketplace operator for the privilege of selling on the platform. The objective of the marketplace operator is to maximize a weighted sum of profit and a term capturing positive customer experience, whereas the independent seller seeks solely to maximize their own profit. We derive the subgame-perfect Nash equilibrium and find that it is often optimal for the marketplace operator to induce competition by offering the product at a low price to incentivize the independent seller to match their price.

Paper Structure

This paper contains 32 sections, 13 theorems, 37 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Non-simultaneous duopolies.
  4. Capacity constraints and rationing rules.
  5. Mixed oligopolies.
  6. Marketplaces.
  7. Model
  8. Best Response and Competitive Pricing
  9. Preliminaries
  10. $\mathtt{I}$'s best response
  11. When Should the Marketplace Operator Induce Competition?
  12. Implications for Consumers
  13. Robustness of Equilibrium Analysis
  14. Proportional rationing
  15. Imperfect substitutability
  16. ...and 17 more sections

Key Result

Proposition 1

Whenever $p_0 > \theta$, the equilibrium solution is that the independent seller does not sell ($q_{\mathtt{I}}=0$) and the marketplace operator sets price $p^{\star}_{\mathtt{M}}$ and quantity $Q(p^{\star}_{\mathtt{M}})$, where

Figures (14)

  • Figure 1: Visualization of $R(p_{\mathtt{I}})$ for $\theta=4$, $q_{\mathtt{M}}=1$, and $p_{\mathtt{M}}=2$ under intensity rationing.
  • Figure 2: Visualization of $\mathtt{I}$'s best response to different $q_{\mathtt{M}}$ (x-axis) and $p_{\mathtt{M}}$ (y-axis) combinations. Stripes are used to denote regions where $\mathtt{I}$ is de-monopolized. Game parameters are set to $\theta=10$, $c_{\mathtt{I}}=2$, and $\alpha=0.2$ (the value of $c_{\mathtt{M}}$ does not affect $\mathtt{I}$'s best response).
  • Figure 3: Equilibrium characterization for each combination ($c_{\mathtt{M}}, c_{\mathtt{I}}$) when $\alpha=0.2$ and $k=2$. In the left plot, red denotes cost combinations where it is optimal for $\mathtt{M}$ to induce $\mathtt{I}$ to compete; blue regions are where $\mathtt{M}$ sets a price that causes $\mathtt{I}$ to abstain, yellow regions are where $\mathtt{M}$ finds it optimal to induce $\mathtt{I}$ to wait; and green is used to denote regions where $\mathtt{M}$ chooses to abstain, but no such regions exist in the plots. The set of plots on the right show the corresponding equilibrium prices and quantities.
  • Figure 4: Residual demand function, $\mathtt{I}$'s best response, and equilibrium characterization under proportional rationing. The game parameters in Figures \ref{['fig:best_response_proportional']} and \ref{['fig:cA_cB_phase_diagram_proportional']} are set to the same values as Figures \ref{['fig:best_response_intensity']} and \ref{['fig:cA_cB_phase_diagram']}. The stripes in Figure \ref{['fig:best_response_proportional']} denote regions where $\mathtt{I}$ is de-monopolized.
  • Figure 5: Equilibrium strategies under one-directional imperfect substitutability with $\gamma=0.5$ for different $c_{\mathtt{M}}, c_{\mathtt{I}}$ combinations in the same setting as Figures \ref{['fig:cA_cB_phase_diagram']} and \ref{['fig:cA_cB_phase_diagram_proportional']}.
  • ...and 9 more figures

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Proposition 1: Trivial solution for large $p_0$
  • Proposition 2: $\mathtt{I}$ meets demand
  • Lemma 1: Best response under intensity rationing
  • Lemma 2: Optimal $q_{\mathtt{M}}$ given $p_{\mathtt{M}}$
  • Theorem 1: Equilibrium
  • Remark 1
  • Remark 2
  • Lemma 3: Surplus transfer
  • ...and 17 more