Quantity and price competition in a differentiated triopoly: static and dynamic investigations

• Published in: Nonlinear dynamics 2018-02, Vol.91 (3), p.1963-1975
• Main Authors: Askar, S. S., Abouhawwash, M.
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Classical Mechanics; Computer simulation; Control; Convexity; Dynamical Systems; Economic models; Engineering; Equilibrium; Evolutionary algorithms; Mathematical models; Mechanical Engineering; Original Paper; Quadratic forms; Rationality; Stability criteria; Static models; Vibration
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-017-3994-z

The game among three competing firms, which is called a triopolistic game, is formulated and studied in this paper. The adopted prices by those firms are derived from a quadratic form of utility that is quasi-concave and has a strict convexity to the origin level curves. The paper is divided into three parts: the first part analyzes a static model on which firms use different strategic variables. The stability condition of the equilibrium points of that model is investigated. In the second part, it is assumed that the three firms may conjecture that their opponents’ current step will remain as it is in the next time step and therefore an instantaneous adjustment model is considered. Schur–Cohn criterion of stability is adopted to detect the stability condition of the equilibrium points of that model. The fractional bounded rationality is introduced in the third part. We introduce in this part a dynamical model of three competing firms whose strategic variables are quantities. As in the previous models, the stability condition of the equilibrium points of the model is studied. The main interesting observation we obtain in this part is that for the firms to stay stable for a long time in the market they should play with fractional bounded rationality rather than the traditional bounded rationality. This important observation is confirmed using numerical simulation for some cases under the same values of parameters. Furthermore, for the models considered in this paper some numerical experiments are carried out to satisfy our theoretical results. Moreover, Karush–Kuhn–Tucker (KKT) proximity measure based on evolutionary algorithm is performed in order to confirm the obtained results.