Existence of Weakly Cooperative Equilibria for Infinite-Leader-Infinite-Follower Games
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Existence of Weakly Cooperative Equilibria for Infinite-Leader-Infinite-Follower Games Zhe Yang1,2 · Qing-Bin Gong1 Received: 30 March 2018 / Revised: 7 August 2018 / Accepted: 5 December 2018 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we first generalize Yang and Ju’s (J Glob Optim 65:563–573, 2016) result in Hausdorff topological vector spaces. Second, we introduce the model of leader-follower games with infinitely many leaders and followers, that is, infiniteleader-infinite-follower game. We next introduce the notion of weakly cooperative equilibria for infinite-leader-infinite-follower games and prove the existence result. Keywords (Weakly) cooperative equilibrium · Infinite-leader–infinite-follower game · Existence Mathematics Subject Classification 91A10 · 91A12 · 91A40
1 Introduction To study economic models with a multi-leader-follower framework, Pang and Fukushima [1] introduced a class of multi-leader-follower games whose equilibria were studied as generalized Nash equilibria. Later, Yu and Wang [2] applied the fixed point theorem to give a general proof of equilibria for multi-leader-follower games. Following Pang and Fukushima [1], Yu and Wang [2], the equilibrium existence result
This research was supported by the National Natural Science Foundation of China (No. 11501349) and Graduate Innovation Foundation sponsored by Shanghai University of Finance and Economics (No. CXJJ-2017-375).
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Zhe Yang [email protected] Qing-Bin Gong [email protected]
1
School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China
2
SUFE Key Laboratory of Mathematical Economics, Ministry of Education, Shanghai 200433, China
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Z. Yang, Q.-B. Gong
of multi-leader-follower games has been extended by Hu and Fukushima [3], Ding [4] and Jia et. al. [5]. By defining α-blocking concept, the cooperative solutions were first introduced by Aumann [6]. Scarf [7] proved the existence of cooperative solution for a normal-form game with continuous and quasiconcave payoff functions. Later, Scarf’s work [7] was extended to games with nonordered preferences (Kajii [8]), games with incomplete information (Askoura et. al. [9], Noguchi [10,11], Askoura [12]) and games with infinitely many players (Askoura [13,14], Yang [15,16]). Yang and Ju [17] first introduced the notion of cooperative equilibria for multileader-multi-follower games and established its existence theorem. Inspired by Yang and Ju [17], Yang [15], in this paper, we shall generalize Yang and Ju’s work [17] to leader-follower games with infinitely many leaders and followers, called by infiniteleader-infinite-follower game. Following definitions of Yang [15], we introduce the notion of weakly cooperative equilibria for infinite-leader-infinite-follower games. We first obtain the existence theorem of cooperative equilibria for multi-leader-multifollower games defined on Hausdorff topological vector space
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