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

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