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Bundling and Price-Matching in Competitive Complementary Goods Markets

Esmat Sangari, Rajni Kant Bansal

TL;DR

This work addresses how mixing bundling and price-matching guarantees interact in a duopoly selling complementary goods to heterogeneous customers. It builds a two-player game in which Retailer 1 may offer mixed bundling ($B ∈ {0,1}$) or not and Retailer 2 sells a bundle, with both firms potentially adopting PMGs; customers are partitioned into Loyal price-unaware, Loyal price-aware, and Non-loyal price-aware, driving complex demand responses. The authors derive closed-form Nash equilibria for all bundling-PMG configurations and show that mixed bundling strictly dominates no bundling when equilibria exist, with the PMG decision under bundling reflecting a trade-off between capturing strategic demand and margins on loyal customers. Numerically, bundling profitability is amplified by strong bundle-level complementarity and weaker item-level complementarity, while PMG adoption depends on the relative price-sensitivities of strategic and loyal segments, offering practical guidance for retailers facing bundle-only rivals or PMG considerations.

Abstract

We study mixed bundling and competitive price-matching guarantees (PMGs) in a duopoly selling complementary products to heterogeneous customers. One retailer offers mixed bundling while the rival sells only a bundle. We characterize unique pure-strategy Nash equilibria across subgames and compare them to a no-bundling benchmark. Mixed bundling strictly dominates whenever an equilibrium exists. Conditional on bundling, PMG adoption trades off strategic demand capture against margin losses on loyal customers and varies systematically with relative demand responsiveness to prices and complementarities.

Bundling and Price-Matching in Competitive Complementary Goods Markets

TL;DR

This work addresses how mixing bundling and price-matching guarantees interact in a duopoly selling complementary goods to heterogeneous customers. It builds a two-player game in which Retailer 1 may offer mixed bundling () or not and Retailer 2 sells a bundle, with both firms potentially adopting PMGs; customers are partitioned into Loyal price-unaware, Loyal price-aware, and Non-loyal price-aware, driving complex demand responses. The authors derive closed-form Nash equilibria for all bundling-PMG configurations and show that mixed bundling strictly dominates no bundling when equilibria exist, with the PMG decision under bundling reflecting a trade-off between capturing strategic demand and margins on loyal customers. Numerically, bundling profitability is amplified by strong bundle-level complementarity and weaker item-level complementarity, while PMG adoption depends on the relative price-sensitivities of strategic and loyal segments, offering practical guidance for retailers facing bundle-only rivals or PMG considerations.

Abstract

We study mixed bundling and competitive price-matching guarantees (PMGs) in a duopoly selling complementary products to heterogeneous customers. One retailer offers mixed bundling while the rival sells only a bundle. We characterize unique pure-strategy Nash equilibria across subgames and compare them to a no-bundling benchmark. Mixed bundling strictly dominates whenever an equilibrium exists. Conditional on bundling, PMG adoption trades off strategic demand capture against margin losses on loyal customers and varies systematically with relative demand responsiveness to prices and complementarities.
Paper Structure (6 sections, 5 theorems, 8 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model Setup
  4. Equilibrium Analysis
  5. Numerical Experiments
  6. Conclusions

Key Result

Theorem 1

Under Sufficient Condition Set A (see Appendix A), the profit functions $\pi_{r1}(p_{r1}^{i1},p_{r1}^{i2},p_{r1}^b,p_{r2}^{b})$ and $\pi_{r2}(p_{r1}^b,p_{r2}^{b})$ are jointly concave in subgames $(\text{CM},\text{CM})$ and $(\text{CM},\overline{\text{CM}})$ when $r_1$ adopts mixed bundling (i.e., $

Figures (3)

  • Figure 1: Sequence of Decisions and Interactions in the Duopoly Game
  • Figure 2: $r_1$’s profit from bundling relative to no bundling, measured by $\Delta \pi_{r1}^B$, as a function of $\lambda_l$ and $\theta_l$, for varying $b_l$ and $b_s$. Shaded regions indicate the optimal PMG regime under bundling.
  • Figure 3: $r_1$’s profit from bundling relative to no bundling, measured by $\Delta \pi_{r1}^B$, as a function of $b_l$ and $b_s$, for varying $\lambda_l$ and $\theta_l$. Shaded regions indicate the optimal PMG regime under bundling.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5