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.

翻译：在多产品企业中，去中心化决策可能导致效率损失，因为自主决策者未能将跨产品需求互动内部化。本文通过分析定价博弈中的无政府状态代价，量化了此类损失的规模。在该博弈中，每个决策者独立设定价格以最大化其产品级收益。我们采用线性系统建模需求，该系统同时捕捉产品间的替代效应与互补效应。我们首先在经济标准对角占优条件下证明了纯策略纳什均衡的存在性与唯一性。我们的主要贡献在于推导了去中心化收益与最优中心化收益比率的一个紧致最坏情况下限。研究表明，该效率损失由单一标量参数$μ$决定，该参数衡量了交叉价格效应相对于自身价格敏感性的总体强度。具体而言，我们证明收益比率的下界为$4(1-μ)/(2-μ)^2$，并通过构建对称市场拓扑结构（该下界在其中被精确达到）证明了该界限的紧致性。我们进一步基于需求交互矩阵的谱特性，提供了效率损失的实例精确表征，从而完善了分析框架。这些结果共同构成了评估多产品企业中心化定价与去中心化自主性之间权衡的量化框架。