Chaotic dynamics in three dimensions: A topological proof for a triopoly game model

Abstract

We rigorously prove the existence of chaotic dynamics for the triopoly game model already studied, mainly from a numerical viewpoint, in [15]. In the model considered, the three firms are heterogeneous and in fact each of them adopts a different decisional mechanism, i.e., linear approximation, best response and gradient mechanisms, respectively. The method we employ is the so-called "Stretching Along the Paths" (SAP) technique in [20], based on the Poincaré-Miranda Theorem and on the properties of the cutting surfaces.

About Chaotic Dynamics in the Twisted Horseshoe Map

“…With respect to the treatment of higher-dimensional maps we can take into account [Zgliczyński, 1999;Zgliczyński & Gidea, 2004;Pireddu, 2015;Ruiz-Herrera & Zanolin, 2015]. Since the tool behind the proof of Theorem 3 works also in an N -dimensional setting with one expansive direction (see [Medio et al, 2009]), these papers motivate us to deal with three-dimensional difference equations, instead of two-dimensional difference equations as the one in (4).…”

Complexity Analysis of a Triopoly Cooperation‐Competition Game Model in Convergence Product Market

“…And the basic assumption has been modified from complete rational to bounded rational [13,14]. In [15], the existence of chaotic dynamics for a triopoly game model with different decisional mechanism was rigorously proved by the "Stretching along the Paths (SAP)" method. However, for the third types of monopoly, the economic monopoly, which means firms acquire the monopoly status relying on powerful economic strength, enormous paten ownership, and successful marketing strategy, is always regarded as the product of free completion and progress in science and technology.…”

A topological proof of chaos for two nonlinear heterogeneous triopoly game models

“…19 to rigorously prove the presence of chaos for some discretetime one-and two-dimensional economic models of the classes of overlapping generations and duopoly games, as well as in Ref. 29 where the SAP method has been applied to the nonlinear Cournot triopoly game model with heterogeneous players introduced in Ref. 25.…”

“…25 and further analyzed in Ref. 29, but the frameworks in Refs. 25 and 41 differ in the behavioral assumptions on one of the three firms, which in Ref.…”

“…Following the line of research started in Ref. 29, in this paper we are going to complement the analysis performed in Refs. 14 and 41, where the presence of complex phenomena and strange attractors for two nonlinear Cournot triopoly game models with heterogeneous players has been shown via numerical simulations.…”