Equilibrium selection of a homogenous duopoly with extrapolative foresight

Zhang, Yanfang; Gao, Xing

doi:10.1016/j.cnsns.2018.07.028

Cited by 12 publications

(3 citation statements)

References 20 publications

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“…Moreover, Cánovas and Muñoz-Guillermo [4] analytically proved the existence of chaos if the firms in the model of Kopel are homogeneous. Furthermore, Zhang and Gao [17] extended the model by assuming that each firm could forecast its rival's output through a straightforward extrapolative foresight technology. Elsadany and Awad [5] modified Kopel's game by assuming one player uses a naive expectation whereas the other employs an adaptive technique and applied the feedback control method to control the chaotic behavior.…”

Section: Introductionmentioning

confidence: 99%

Equilibria and their stability in an asymmetric duopoly model of Kopel

Li¹,

Kongyan²

2023

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In this paper, we investigate the equilibria and their stability in an asymmetric duopoly model of Kopel by using several tools based on symbolic computations. We explore the possible positions of the equilibria in Kopel's model. We discuss the possibility of the existence of multiple positive equilibria and establish a necessary and sufficient condition for a given number of equilibria to exist. Furthermore, if the two duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the existence of distinct numbers of positive equilibria for the first time.

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Section: Introductionmentioning

confidence: 99%

Equilibria and their stability in an asymmetric duopoly model of Kopel

Li¹,

Kongyan²

2023

Preprint

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“…Detailed bifurcation analyses were also conducted in, e.g., [20,43], which may help in understanding the occurrence and the structure of bifurcation cascades. Furthermore, Zhang and Gao [44] extended the model by assuming that each firm could forecast its rival's output through a straightforward extrapolative foresight technology. Elsadany and Awad [11] modified Kopel's game by assuming one player uses a naive expectation whereas the other employs an adaptive technique and applied the feedback control method to control the chaotic behavior.…”

Section: Introductionmentioning

confidence: 99%

Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations

Kongyan

Wei

et al. 2023

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Since Kopel's duopoly model was proposed about three decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish necessary and sufficient conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel's model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel's model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective is to prove the existence of snapback repellers in Kopel's map, which implies the existence of chaos in the sense of Li–Yorke according to Marotto's theorem.

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“…Detailed bifurcation analyses were also conducted in, e.g., [20,41], which may help in understanding the occurrence and the structure of bifurcation cascades. Furthermore, Zhang and Gao [42] extended the model by assuming that each firm could forecast its rival's output through a straightforward extrapolative foresight technology. Elsadany and Awad [11] modified Kopel's game by assuming one player uses a naive expectation whereas the other employs an adaptive technique and applied the feedback control method to control the chaotic behavior.…”

Section: Introductionmentioning

confidence: 99%