💰 quantitative finance

Bundling and Price-Matching in Competitive Complementary Goods Markets

This paper analyzes a duopoly selling complementary goods to demonstrate that mixed bundling strictly dominates a no-bundling benchmark and that the strategic value of price-matching guarantees depends on the trade-off between capturing demand and protecting margins on loyal customers.

Original authors: Esmat Sangari, Rajni Kant Bansal

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Esmat Sangari, Rajni Kant Bansal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine two competing grocery stores in a neighborhood, let's call them Store A and Store B. They both sell a "perfect pair" of items that go great together, like a printer and ink cartridges, or a laptop and a mouse.

This paper is a mathematical story about how these two stores fight for customers when they use two specific weapons: Bundles (selling the pair together) and Price-Matching (promising to match a competitor's lower price).

Here is the breakdown of the story using simple analogies:

1. The Setup: The Two Stores

Store A (The Flexible Shop): This store is versatile. It can sell the printer and mouse separately, or it can sell them as a "Combo Deal." It can also choose to offer a "Price Match Guarantee" (if you find it cheaper at Store B, we'll match it).
Store B (The Bundle-Only Shop): This store is rigid. It only sells the printer and mouse as a pre-packed bundle. It cannot sell them separately. It can also choose to offer a "Price Match Guarantee."

2. The Customers: Three Types of Shoppers

The researchers imagine three types of shoppers walking into this neighborhood:

The Loyal Blind: These people love Store A or Store B and never look at the other store's prices. They just buy what they need.
The Loyal Smart: These people love their specific store, but if that store offers a price match, they will happily use it to get a better deal. They are loyal but opportunistic.
The Shoppers (Strategic): These people have no loyalty. They walk into both stores, compare the total price of the "Combo," and buy from whoever is cheapest.

3. The Big Discovery: Bundling Wins

The paper's first major finding is simple: If you can bundle, you should.
Store A (the flexible one) almost always makes more money if it offers a "Combo Deal" alongside selling items separately, rather than just selling items alone. It's like a restaurant that offers both a "Burger & Fries" combo and individual items; the combo usually brings in more profit because it captures more types of customers.

4. The Price-Matching Trap

The second part of the story is about the Price-Matching Guarantee (PMG). The researchers found that offering to match prices is a double-edged sword.

When it works: If the "Shoppers" (the price-sensitive ones) are very picky, and the "Loyal" customers aren't too sensitive to price changes, offering a price match can help Store A steal customers from Store B.
When it fails: If the "Loyal" customers are the ones who care most about price, offering a price match is a disaster. Why? Because you end up giving a discount to everyone who was already going to buy from you anyway, just to look competitive. It's like a store lowering prices for everyone just to look like they are fighting a war, but they end up losing money on their best customers.

5. The "Tug-of-War" of Pricing

The most interesting mathematical insight is how the prices are set:

If Store A offers a Price Match: Store A's prices become dependent on Store B. If Store B changes its bundle price, Store A must adjust its own prices to keep the math working. It's like a game of tug-of-war where Store A is tied to Store B's rope.
If Store A does NOT offer a Price Match: Store A is free. It sets its prices based only on its own customers and costs, ignoring Store B's specific bundle price. It's like playing a solo game.

6. The "Pure Bundle" Rival

Because Store B only sells bundles, it has a unique advantage and disadvantage.

Advantage: It doesn't have to worry about customers buying just one item.
Disadvantage: Its price is set in a vacuum. It doesn't react to Store A's individual item prices in the same complex way Store A reacts to Store B.

The Bottom Line

The paper concludes that in a market where people buy things in pairs:

Mixing it up is best: The store that offers both "Combo Deals" and "Individual Items" usually beats the store that only offers "Combo Deals."
Price Matching is a gamble: You should only promise to match prices if your "loyal" customers aren't too price-sensitive, but your "shoppers" are. If you get this mix wrong, you will lose money.
The Rigid vs. The Flexible: The store that only sells bundles (Store B) sets its price based on its own costs. The flexible store (Store A) has to constantly watch Store B's price if it wants to offer a price match, making its strategy much more complicated.

In short, selling bundles is a winning strategy, but adding a "we'll match your price" sign is only a good idea if you know exactly which type of customer is walking through your door.

Technical Summary: Bundling and Price-Matching in Competitive Complementary Goods Markets

Problem Statement
This study investigates the strategic interaction between mixed bundling and competitive price-matching guarantees (PMGs) in a duopoly market selling complementary products to heterogeneous customers. While bundling and PMGs are widely used in practice (e.g., in electronics, meal kits, and travel), existing literature has largely analyzed these strategies in isolation. The bundling literature typically ignores PMGs, while PMG literature focuses on single-product settings. There is a gap in understanding how these strategies interact when retailers compete with asymmetric formats: one retailer (Retailer 1) offers mixed bundling (individual items and bundles), while the rival (Retailer 2) sells exclusively as a bundle. The central question is how PMG adoption influences the profitability of bundling and how bundling affects the efficacy of PMGs under varying customer behaviors.

Methodology
The authors model a two-retailer duopoly game involving two complementary products with costs $c_1$ and $c_2$ .

Firm Decisions: Retailer 1 chooses between mixed bundling ( $B=1$ ) or selling only stand-alone items ( $B=0$ ). Both retailers simultaneously decide whether to adopt a PMG and set prices. Retailer 2 is constrained to selling only bundles.
Customer Segmentation: The market is divided into three distinct segments based on loyalty and price awareness:
1. Loyal, price-unaware: Purchase from a preferred retailer without comparing prices.
2. Loyal, price-aware: Committed to a retailer but utilize PMGs if available.
3. Non-loyal, price-aware (Strategic): Actively compare the effective price of obtaining both products across retailers and purchase from the lowest-priced option.
Demand Structure: The model employs linear demand functions incorporating own-price sensitivity ( $b_l$ ), complementarity parameters ( $\theta_l$ between items and $\lambda_l$ between the bundle and the sum of parts), and segment-specific demand bases. A key constraint ensures the bundle price does not exceed the sum of individual item prices.
Equilibrium Analysis: The authors derive unique pure-strategy Nash equilibria for four strategic subgames (combinations of PMG adoption by both firms) under mixed bundling, as well as a benchmark case where Retailer 1 sells only stand-alone items. They utilize Karush-Kuhn-Tucker (KKT) conditions to solve the firms' profit-maximization problems and establish sufficient conditions for joint concavity of profit functions.

Key Results

Dominance of Mixed Bundling: Mixed bundling strictly dominates stand-alone selling whenever an equilibrium exists. The profitability gains are most significant when bundle complementarity is high and stand-alone interaction effects are relatively weak.
Price Interdependence and Asymmetry:
- When Retailer 1 adopts a PMG under mixed bundling, its optimal bundle price (and consequently its individual item prices) depends on Retailer 2's bundle-demand parameters. This occurs because the effective price faced by price-aware customers is tied to the rival's lower price.
- In contrast, Retailer 2's bundle price is determined solely by its own demand parameters, regardless of Retailer 1's pricing strategy.
- If Retailer 1 does not adopt a PMG, or if its equilibrium bundle price is naturally lower than Retailer 2's, its prices depend only on its own demand parameters.
PMG Adoption Trade-offs: The adoption of PMGs involves a trade-off between capturing strategic demand and losing margins on loyal customers.
- PMGs are unattractive when strategic customers are highly price-sensitive while loyal customers are relatively price-insensitive.
- PMGs become optimal when both customer groups exhibit medium-to-high price sensitivity.
Equilibrium Robustness: Equilibrium existence is less likely under strong asymmetries in price sensitivity across customer groups. However, the equilibrium is more robust when bundle complementarity is high and stand-alone interaction effects are limited.

Significance and Contributions
The paper claims to provide the first integrated characterization of how bundling and PMGs jointly shape competition for complementary goods in a setting with heterogeneous customers and asymmetric retail formats. By linking these two strategic levers within a common game-theoretic framework, the study offers:

Theoretical Integration: It bridges the gap between the bundling and PMG literature streams, which have previously evolved in isolation.
Strategic Insight: It clarifies the specific conditions under which the interaction between mixed bundling and PMGs transmits competitive conditions across retailers through effective bundle prices.
Managerial Guidance: The findings offer actionable guidance to retailers facing bundle-only competitors or those strategically considering PMG adoption, highlighting that the profitability of these strategies is contingent on the specific mix of customer price sensitivities and product complementarity.