Geodesic
Online-offline competition with heterogeneous consumers: an example for no existence of pure strategy Nash equilibrium
Contributions to game theory and management, Tome 15 (2022), pp. 41-50

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Existing literature on competition between online and offline firms has focused on market conditions that guarantee the existence of a pure strategy Nash equilibrium. In this note, by constructing a concrete example, we provide a first attempt to show that the equilibrium existence result does not necessarily hold when consumers' preferences are heterogeneous. Specifically, we consider the competition between one online firm and several offline firms in a market organized as a Salop model, where consumers' preferences have a binary distribution. We identify a boundary scenario where the type distribution is binary with one type of consumer loyal to online shopping and the other type loyal to offline shopping. We show that there is no pure strategy Nash equilibrium for this boundary scenario, which indicates that the market may not be stable under such conditions. Our study contributes to a better understanding of the equilibrium existence conditions for the online versus offline retail competition.
Keywords: online versus offline, price competition, salop model, heterogeneous consumers, existence of equilibrium. 
@article{CGTM_2022_15_a4,
     author = {Weixing Ford and Haipeng Han and Jie Zheng},
     title = {Online-offline competition with heterogeneous consumers: an example for no existence of pure strategy {Nash} equilibrium},
     journal = {Contributions to game theory and management},
     pages = {41--50},
     publisher = {mathdoc},
     volume = {15},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CGTM_2022_15_a4/}
}
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Weixing Ford; Haipeng Han; Jie Zheng. Online-offline competition with heterogeneous consumers: an example for no existence of pure strategy Nash equilibrium. Contributions to game theory and management, Tome 15 (2022), pp. 41-50. http://geodesic.mathdoc.fr/item/CGTM_2022_15_a4/