Existence of equilibrium in a dynamic supply chain game with vertical coordination, horizontal competition, and complementary goods

IF 6 2区 管理学 Q1 OPERATIONS RESEARCH & MANAGEMENT SCIENCE European Journal of Operational Research Pub Date : 2024-09-19 DOI:10.1016/j.ejor.2024.09.027
Bertrand Crettez , Naila Hayek , Guiomar Martín-Herrán
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Abstract

We consider supply chain competition and vertical coordination in a linear–quadratic differential game setting. In this setting, supply chains produce complementary goods and each of them includes a single manufacturer and a single retailer who coordinate their decisions through a revenue-sharing contract with a wholesale price and a fixed sales revenue share. We study a multiple leader-follower Stackelberg game where the manufacturers are the leaders and the retailers are the followers. Competition occurs at both levels of the supply chains. Retailers play Nash and compete in price; manufacturers also play Nash but they compete in choosing their production capacities by exploiting the equilibrium price decisions made by the retailers. We show that open-loop Nash equilibria exist when the manufacturers only receive a wholesale price (there are no longer exploiting the equilibrium price decision made by the retailers, however). When the manufacturers receive both a wholesale-price and a share of the retailers’ sales revenues, equilibria generally no longer exist. The non-existence of an equilibrium stems from the fact that the manufacturers’ instant profits are discontinuous functions of their production capacities. This discontinuity leads to a major technical difficulty in that one cannot apply standard optimal control approaches to study the equilibria of the dynamic game. Our results illustrate the possibility that competition between supply chains might not be sustainable when they sell complementary products and rely on a revenue-sharing agreement.
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具有纵向协调、横向竞争和互补商品的动态供应链博弈中均衡的存在性
我们考虑的是线性-二次微分博弈环境下的供应链竞争和纵向协调。在这种情况下,供应链生产互补商品,每条供应链都包括一个制造商和一个零售商,他们通过一份收入共享合同协调决策,该合同包含批发价格和固定的销售收入份额。我们研究的是一个多领导者--追随者的斯塔克尔伯格博弈,其中制造商是领导者,零售商是追随者。竞争发生在供应链的两个层面。零售商参与纳什博弈,在价格上展开竞争;制造商也参与纳什博弈,但他们利用零售商做出的均衡价格决策,在选择生产能力上展开竞争。我们的研究表明,当制造商只收到批发价时,存在开环纳什均衡(但不再利用零售商的均衡价格决策)。当制造商同时获得批发价和零售商的销售收入分成时,均衡一般就不存在了。不存在均衡的原因在于,制造商的即时利润是其生产能力的不连续函数。这种不连续性导致了一个重大的技术难题,即我们无法运用标准的最优控制方法来研究动态博弈的均衡。我们的研究结果表明,当供应链销售互补产品并依赖于收入共享协议时,它们之间的竞争可能无法持续。
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来源期刊
European Journal of Operational Research
European Journal of Operational Research 管理科学-运筹学与管理科学
CiteScore
11.90
自引率
9.40%
发文量
786
审稿时长
8.2 months
期刊介绍: The European Journal of Operational Research (EJOR) publishes high quality, original papers that contribute to the methodology of operational research (OR) and to the practice of decision making.
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