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

Abstract

We rigorously prove the existence of chaotic dynamics for two nonlinear Cournot triopoly game models with heterogeneous players, for which in the existing literature the presence of complex phenomena and strange attractors has been shown via numerical simulations. In the first model that we analyze, costs are linear but the demand function is isoelastic, while, in the second model, the demand function is linear and production costs are quadratic. As concerns the decisional mechanisms adopted by the firms, in both models one firm adopts a myopic adjustment mechanism, considering the marginal profit of the last period; the second firm maximizes its own expected profit under the assumption that the competitors' production levels will not vary with respect to the previous period; the third firm acts adaptively, changing its output proportionally to the difference between its own output in the previous period and the naive expectation value. The topological method we employ in our analysis is the so-called "Stretching Along the Paths" technique, based on the Poincaré-Miranda Theorem and the properties of the cutting surfaces, which allows to prove the existence of a semi-conjugacy between the system under consideration and the Bernoulli shift, so that the former inherits from the latter several crucial chaotic features, among which a positive topological entropy.

Chaotic dynamics in the presence of medical malpractice litigation: a topological proof via linked twist maps for two evolutionary game theoretic contexts

“…Footnote 6). In regard to economic settings, we recall that the SAP method, on which the LTMs technique is based, has been recently used to prove the existence of chaotic dynamics for some discrete-time triopoly game models in [39,40], while in [31] the SAP technique had been applied to one-and two-dimensional discrete-time economic models, concerning overlapping generations and duopoly frameworks.…”

Proving chaos for a system of coupled logistic maps: A topological approach

“…to [9][10][11][34][35][36] for continuous-time planar applications of the SAP method, and to 37,38 for 3D continuous-time applications of it to non-Hamiltonian systems. On the other hand, to the best of our knowledge, the only applications of the SAP technique to discrete-time models can be found in [6][7][8] , where 1D 39 , 2D (in 6 ) and 3D (in 7,8 ) economic settings are considered. In more detail, in 7,8 different triopoly game models with heterogeneous players, taken respectively from 40 and 41,42 , are analyzed, while in 6 OLG models with and without production (taken e.g.…”

“…On the other hand, to the best of our knowledge, the only applications of the SAP technique to discrete-time models can be found in [6][7][8] , where 1D 39 , 2D (in 6 ) and 3D (in 7,8 ) economic settings are considered. In more detail, in 7,8 different triopoly game models with heterogeneous players, taken respectively from 40 and 41,42 , are analyzed, while in 6 OLG models with and without production (taken e.g. from 43,44 ), as well as the duopoly game model with heterogeneous players taken from 45 , are considered.…”

“…Following the approach employed in [6][7][8] , along the manuscript we show how to apply the SAP method to the model in 1 so as to detect chaotic dynamics therein. In particular, the possibility of finding a generalized rectangle for which the stretching relation is satisfied with respect to two disjoint compact subsets of it depends, in addition to the value assumed by the usual parameter present in the two logistic maps, on the sign and on the value of the parameters describing the link between the two economies.…”

“…Any such investigation must incorporate two different aspects, namely the global dynamics of diffeomorphisms and the theory of critical lines developed in Mira et al 5 ", we here follow, in view of showing the presence of chaos in that setting, a different approach which, being topological in nature, does not require any differentiability condition. Moreover, the SAP technique has proven to be successfully applicable both in discrete- [6][7][8] and in continuous-time contexts (see e.g. [9][10][11] ).…”