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

TL;DR

This paper provides a rigorous topological proof of chaotic dynamics in a triopoly game model with three heterogeneous firms using the 'Stretching Along the Paths' method, complementing previous numerical studies.

Contribution

It introduces a topological proof of chaos in a triopoly model with diverse decision mechanisms, expanding the understanding beyond numerical evidence.

Findings

Chaotic dynamics are rigorously proven in the triopoly model.

The model includes heterogeneous firms with different decision rules.

The 'Stretching Along the Paths' technique is effectively applied.

Abstract

We rigorously prove the existence of chaotic dynamics for the triopoly game model already studied, mainly from a numerical viewpoint, in the working paper by Naimzada and Tramontana [2011]. 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 Pireddu and Zanolin [2007], based on the Poincar\'e-Miranda Theorem and on the properties of the cutting surfaces.