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Cournot Duopoly Games: Models and Investigations

by S. S. Askar 1,2,*,† and A. Al-khedhairi 1,†
Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Department of Mathematics, College of Science, Mansoura University, Mansoura 35516, Egypt
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2019, 7(11), 1079;
Received: 21 October 2019 / Revised: 5 November 2019 / Accepted: 6 November 2019 / Published: 8 November 2019
(This article belongs to the Special Issue Mathematical Game Theory 2019)
This paper analyzes Cournot duopoly games that are constructed based on Cobb–Douglas preferences. We introduce here two models whose dynamic adjustments depend on bounded rationality, dynamic adjustment, and tit-for-tat mechanism. In the first model, we have two firms with limited information and due to that they adopt the bounded rationality mechanism. They update their productions based on the changing occurred in the marginal profit. For this model, its fixed point is obtained and its stability condition is calculated. In addition, we provide conditions by which this fixed point loses its stability due to flip and Neimark–Sacker bifurcations. Furthermore, numerical simulation shows that this model possesses some chaotic behaviors which are recovered due to corridor stability. In the second model, we handle two different mechanisms of cooperation. These mechanisms are dynamic adjustment process and tit-for-tat strategy. The players who use the dynamic adjustment increase their productions based on the cooperative output while, in tit-for-tat mechanism, they increase the productions based on the cooperative profit. The local stability analysis shows that adopting tit-for-tat makes the model unstable and then the system becomes chaotic for any values of the system’s parameters. The obtained results show that the dynamic adjustment makes the system’s fixed point stable for a certain interval of the adjustment parameter. View Full-Text
Keywords: bounded rationality; Puu’s incomplete information; tit-for-tat; stability; bifurcation bounded rationality; Puu’s incomplete information; tit-for-tat; stability; bifurcation
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Askar, S.S.; Al-khedhairi, A. Cournot Duopoly Games: Models and Investigations. Mathematics 2019, 7, 1079.

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