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Bundling in Oligopoly: Revenue Maximization with Single-Item Competitors

Moshe Babaioff, Linda Cai, Brendan Lucier

TL;DR

The paper studies a multi-product principal facing single-item competitive sellers in an additive buyer setting. It derives an equilibrium-robust upper bound on the principal’s revenue given by the buyer’s expected truncated welfare, and shows that, under price-sensitivity and high-variance conditions, grand-bundle pricing achieves a constant-factor approximation to this bound in every equilibrium. The results also delineate limits: there are distributions where grand-bundle pricing is not a constant-approximation to the optimum, and partition-based menus can surpass grand-bundle revenue by a large factor in some instances. This work advances understanding of bundling as a revenue-extraction tool in oligopolistic markets and highlights when simple bundling strategies are near-optimal. It also opens avenues for extending the framework to more complex valuations, collusion possibilities, and alternative bundling schemes.

Abstract

We consider a principal seller with $m$ heterogeneous products to sell to an additive buyer over independent items. The principal can offer an arbitrary menu of product bundles, but faces competition from smaller and more agile single-item sellers. The single-item sellers choose their prices after the principal commits to a menu, potentially under-cutting the principal's offerings. We explore to what extent the principal can leverage the ability to bundle product together to extract revenue. Any choice of menu by the principal induces an oligopoly pricing game between the single-item sellers, which may have multiple equilibria. When there is only a single item this model reduces to Bertrand competition, for which the principal's revenue is $0$ at any equilibrium, so we assume that no single item's value is too dominant. We establish an upper bound on the principal's optimal revenue at every equilibrium: the expected welfare after truncating each item's value to its revenue-maximizing price. Under a technical condition on the value distributions -- that the monopolist's revenue is sufficiently sensitive to price -- we show that the principal seller can simply price the grand-bundle and ensure (in any equilibrium) a constant approximation to this bound (and hence to the optimal revenue). We also show that for some value distributions violating our conditions, grand-bundle pricing does not yield a constant approximation to the optimal revenue in any equilibrium.

Bundling in Oligopoly: Revenue Maximization with Single-Item Competitors

TL;DR

The paper studies a multi-product principal facing single-item competitive sellers in an additive buyer setting. It derives an equilibrium-robust upper bound on the principal’s revenue given by the buyer’s expected truncated welfare, and shows that, under price-sensitivity and high-variance conditions, grand-bundle pricing achieves a constant-factor approximation to this bound in every equilibrium. The results also delineate limits: there are distributions where grand-bundle pricing is not a constant-approximation to the optimum, and partition-based menus can surpass grand-bundle revenue by a large factor in some instances. This work advances understanding of bundling as a revenue-extraction tool in oligopolistic markets and highlights when simple bundling strategies are near-optimal. It also opens avenues for extending the framework to more complex valuations, collusion possibilities, and alternative bundling schemes.

Abstract

We consider a principal seller with heterogeneous products to sell to an additive buyer over independent items. The principal can offer an arbitrary menu of product bundles, but faces competition from smaller and more agile single-item sellers. The single-item sellers choose their prices after the principal commits to a menu, potentially under-cutting the principal's offerings. We explore to what extent the principal can leverage the ability to bundle product together to extract revenue. Any choice of menu by the principal induces an oligopoly pricing game between the single-item sellers, which may have multiple equilibria. When there is only a single item this model reduces to Bertrand competition, for which the principal's revenue is at any equilibrium, so we assume that no single item's value is too dominant. We establish an upper bound on the principal's optimal revenue at every equilibrium: the expected welfare after truncating each item's value to its revenue-maximizing price. Under a technical condition on the value distributions -- that the monopolist's revenue is sufficiently sensitive to price -- we show that the principal seller can simply price the grand-bundle and ensure (in any equilibrium) a constant approximation to this bound (and hence to the optimal revenue). We also show that for some value distributions violating our conditions, grand-bundle pricing does not yield a constant approximation to the optimal revenue in any equilibrium.
Paper Structure (27 sections, 39 theorems, 98 equations, 1 figure)

This paper contains 27 sections, 39 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Market Timing
  4. The Game and Equilibrium Concepts
  5. An Upper Bound on the Principal's Revenue
  6. Understanding the Buyer's Optimal Decision
  7. Item Seller Best-Responses
  8. An Upper Bound on the Principal's Revenue
  9. Approximating Optimal Revenue via Bundle Pricing
  10. Implications
  11. I.I.D. Valuations
  12. Non-I.I.D. Smooth and Revenue-Concave Valuations
  13. Outline of the Proof of Theorem \ref{['thm:generalPrincipalBundleRev']}
  14. Additional Notations.
  15. Preliminary: The Berry-Esseen Theorem
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose the buyer has independent item values $v_i \sim F_i$ and that $r_i \in \arg\max_r\{ r(1-F_i(r)) \}$ is the maximal revenue-maximizing price for item $i$. Then for any menu of the principal seller and any mixed Nash equilibrium among the item sellers, the expected revenue of the principal sel

Figures (1)

  • Figure 1: $(\lambda, C)$-Price Sensitive Revenue Curve. For distribution $F_i$, the revenue function $\mathsf{Rev}(q_i)= q_i\cdot (1-F_i(q_i))$, must be below the line with slope $\lambda$ that passes through $(r_i, \mathsf{Rev}(r_i))$ for every price below $C\cdot r_i$, where $r_i$ is the maximal Myerson price for $F_i$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2: Informal
  • Definition 2.1
  • Claim 3.0
  • Claim 3.0
  • Proposition 3.0
  • Lemma 3.1
  • proof
  • Definition 3.2: Expected Truncated Social Welfare
  • Theorem 3.3
  • ...and 73 more