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

2026-01-26

AI Summary

This study addresses the efficiency loss in decentralized pricing within multi-product firms, which arises from ignoring inter-product demand interactions. The authors analyze the Nash equilibrium of independent product pricing under a linear demand model that captures both substitution and complementarity effects. Under a diagonal dominance condition on the demand interaction matrix, they integrate game-theoretic analysis, Price of Anarchy concepts, and spectral matrix theory to establish, for the first time, a tight lower bound on the ratio of decentralized revenue to centralized optimal revenue: \(4(1-\mu)/(2-\mu)^2\), where \(\mu\) quantifies the strength of cross-price effects. This bound depends solely on \(\mu\), is shown to be tight via a symmetric market construction, and enables instance-specific characterization of efficiency loss through the spectral properties of the demand interaction matrix.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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 $\mu$, 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 $4(1-\mu)/(2-\mu)^2$, 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.

Problem

• Decentralized pricing
• Price of Anarchy
• Multi-product firms
• Cross-product demand interactions
• Nash Equilibrium

Innovation

• Price of Anarchy
• Nash Equilibrium
• Diagonal Dominance
• Decentralized Pricing
• Demand Interaction