Decentralized Multi-product Pricing: Diagonal Dominance, Nash Equilibrium, and Price of Anarchy
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917223474397184 |
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| author | Chen, Boxiao Jiang, Jiashuo Jasin, Stefanus |
| author_facet | Chen, Boxiao Jiang, Jiashuo Jasin, Stefanus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18117 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decentralized Multi-product Pricing: Diagonal Dominance, Nash Equilibrium, and Price of Anarchy Chen, Boxiao Jiang, Jiashuo Jasin, Stefanus Computer Science and Game Theory Optimization and Control 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. |
| title | Decentralized Multi-product Pricing: Diagonal Dominance, Nash Equilibrium, and Price of Anarchy |
| topic | Computer Science and Game Theory Optimization and Control |
| url | https://arxiv.org/abs/2601.18117 |