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On Cournot–Bertrand competition with differentiated products

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Abstract

In this paper, we present a game theoretic framework for Cournot–Bertrand competition based on a nonlinear price function. The competition is between two firms and is assumed to take place in terms of pricing decision and quantity produced. However, the proposed objective function has not been used in literature before, yet the throughput obtained in this paper generalizes some of the existing results in literature. The competitive interaction between firms is described and analyzed using best-reply reaction, proposed adaptive adjustment and bounded rationality approach. The condition of stability of Nash equilibrium (NE) is induced by these approaches. Interestingly, we prove that there exists exactly a unique NE. Furthermore, it is noticed that when firms adopt best-reply and the proposed adaptive adjustment, the firms’ strategic variables become asymptotically stable. On the contrary, when the bounded rationality is used both quantity and price behave chaotically due to bifurcation occurred.

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Acknowledgments

This work was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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Authors and Affiliations

  1. Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455 , Riyadh, 11451, Saudi Arabia

    S. S. Askar

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura , 35516, Egypt

    S. S. Askar

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  1. S. S. Askar
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Correspondence to S. S. Askar.

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Askar, S.S. On Cournot–Bertrand competition with differentiated products. Ann Oper Res 223, 81–93 (2014). https://doi.org/10.1007/s10479-014-1612-8

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  • DOI: https://doi.org/10.1007/s10479-014-1612-8

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