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
{"title":"Existence of equilibrium in a dynamic supply chain game with vertical coordination, horizontal competition, and complementary goods","authors":"Bertrand Crettez ,&nbsp;Naila Hayek ,&nbsp;Guiomar Martín-Herrán","doi":"10.1016/j.ejor.2024.09.027","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":55161,"journal":{"name":"European Journal of Operational Research","volume":null,"pages":null},"PeriodicalIF":6.0000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Operational Research","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S037722172400729X","RegionNum":2,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
引用次数: 0

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.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
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.
期刊最新文献
Prelim p. 2; First issue - Editorial Board Editorial Board An exact method for the two-echelon split-delivery vehicle routing problem for liquefied natural gas delivery with the boil-off phenomenon The demand for hedging of oil producers: A tale of risk and regret Data-driven dynamic police patrolling: An efficient Monte Carlo tree search
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1