# 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 lattermore »

- Authors:

- Department of Mathematics and Applications, University of Milano-Bicocca, U5 Building, Via Cozzi 55, 20125 Milano (Italy)

- Publication Date:

- OSTI Identifier:
- 22596514

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 8; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ATTRACTORS; CHAOS THEORY; COMPUTERIZED SIMULATION; ENTROPY; GAME THEORY; NONLINEAR PROBLEMS; TOPOLOGY

### Citation Formats

```
Pireddu, Marina, E-mail: marina.pireddu@unimib.it.
```*A topological proof of chaos for two nonlinear heterogeneous triopoly game models*. United States: N. p., 2016.
Web. doi:10.1063/1.4960387.

```
Pireddu, Marina, E-mail: marina.pireddu@unimib.it.
```*A topological proof of chaos for two nonlinear heterogeneous triopoly game models*. United States. doi:10.1063/1.4960387.

```
Pireddu, Marina, E-mail: marina.pireddu@unimib.it. Mon .
"A topological proof of chaos for two nonlinear heterogeneous triopoly game models". United States.
doi:10.1063/1.4960387.
```

```
@article{osti_22596514,
```

title = {A topological proof of chaos for two nonlinear heterogeneous triopoly game models},

author = {Pireddu, Marina, E-mail: marina.pireddu@unimib.it},

abstractNote = {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.},

doi = {10.1063/1.4960387},

journal = {Chaos (Woodbury, N. Y.)},

number = 8,

volume = 26,

place = {United States},

year = {Mon Aug 15 00:00:00 EDT 2016},

month = {Mon Aug 15 00:00:00 EDT 2016}

}