Decentralized Multi-product Pricing: Diagonal Dominance, Nash Equilibrium, and Price of Anarchy

TL;DR

This paper analyzes efficiency losses from decentralized multi-product pricing when cross-product demand interactions exist. It adopts a linear demand model with symmetric cross-elasticities and strict diagonal dominance, establishing existence and uniqueness of a pure-strategy Nash equilibrium and closed-form centralized and decentralized prices. A tight worst-case Price of Anarchy bound, depending on the diagonal-dominance parameter μ, shows decentralization can be nearly optimal when cross-effects are weak, yet significantly coercive when products are tightly coupled. The authors deepen the insight with an exact spectral characterization using a normalized interaction matrix, enabling instance-specific PoA guarantees that reflect market topology, such as hub-and-spoke networks, and providing managerial guidance on when centralized pricing is essential versus when decentralized autonomy suffices.

Abstract

Decentralized decision making in multi--product firms can lead to efficiency losses when autonomous decision makers fail to internalize cross--product demand interactions. This paper quantifies the magnitude of such losses by analyzing the Price of Anarchy in a pricing game in which each decision maker independently sets prices to maximize its own product--level revenue. We model demand using a linear system that captures both substitution and complementarity effects across products. We first establish existence and uniqueness of a pure--strategy Nash equilibrium under economically standard diagonal dominance conditions. Our main contribution is the derivation of a tight worst--case lower bound on the ratio between decentralized revenue and the optimal centralized revenue. We show that this efficiency loss is governed by a single scalar parameter, denoted by $μ$, which measures the aggregate strength of cross--price effects relative to own--price sensitivities. In particular, we prove that the revenue ratio is bounded below by $4(1-μ)/(2-μ)^2$, and we demonstrate the tightness of this bound by constructing a symmetric market topology in which the bound is exactly attained. We further refine the analysis by providing an instance--exact characterization of efficiency loss based on the spectral properties of the demand interaction matrix. Together, these results offer a quantitative framework for assessing the trade--off between centralized pricing and decentralized autonomy in multi--product firms.

Figures (1)

• Figure 1: The values of the bound $4(1-\mu)/(2-\mu)^2$ for different $\mu$.

Theorems & Definitions (43)

• Remark 1: Role of $\mu$ in the analysis
• Lemma 1: Negative definiteness of $\bm{B}$
• Lemma 2
• Definition 1: Decentralized pricing game
• Definition 2: Pure--strategy Nash equilibrium
• Proposition 1: First--order conditions at a Nash equilibrium
• Definition 3: Nash equilibrium matrix
• Lemma 3
• Theorem 1: Existence, uniqueness, and characterization of the Nash equilibrium
• Remark 2