# Imitation Dynamics in Oligopoly Games with Heterogeneous Players

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Set-Up

#### 2.1. Learning Heuristics

#### 2.2. Population Dynamics

## 3. Heterogeneity in Behavior in Cournot Oligopolies

#### 3.1. Cournot vs. Imitation Firms with Fixed Fractions

**Proposition**

**1.**

**Proof.**

#### 3.2. Rational vs. Imitation Firms with Fixed Fractions

**Proposition**

**2.**

**Proof.**

## 4. Evolutionary Competition between Two Heuristics

#### 4.1. Cournot vs. Imitation Firms with Endogenous Switching

**Proposition**

**3.**

**Proof.**

#### 4.2. Rational vs. Imitation Firms with Switching

**Proposition**

**4.**

**Proof.**

## 5. Rational vs. Cournot vs. Imitation with Switching Heuristics

**Proposition**

**5.**

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**3.**

**Proof**

**of**

**Proposition**

**4.**

**Proof**

**of**

**Proposition**

**5.**

## Notes

1 | Firms that display Cournot behaviour take the current period’s aggregate output of their competitors as a predictor for the next period competitors’ aggregate output and best-respond to that. |

2 | In models with heterogeneous expectations producers can have different heuristics to adjust their production. |

3 | Essentially, a Cournot adjustment process with long memory, introduced by [7]. |

4 | We thank an anonymous referee for pointing out this literature on preferences evolution as an alternative to modelling heuristics (rules) evolution in game dynamics. |

5 | Sufficient conditions for the existence and uniqueness of the interior Cournot-Nash equilibrium are that $P\left(\xb7\right)$ is twice continuously differentiable, nonincreasing and that $C\left(\xb7\right)$ is twice continuously differentiable, nondecreasing and convex, see [24]. |

6 | Ref. [6] provide an example of a Type II duopoly (linear demand, and marginal costs decreasing faster than the demand). Such Type II oligopolies display, in addition to the interior Cournot-Nash equilibrium, two boundary Nash equilibria with one of the firms producing the monopoly output and the other producing nothing. |

7 | Throughout the paper, a (locally) asymptotically fixed point (or a steady state) ${\mathbf{x}}^{*}$ is a fixed point of the dynamical system which attracts all initial conditions in a neighborhood of ${\mathbf{x}}^{*}.$ Technically, ${\mathbf{x}}^{*}$ is the $\omega -$ limit set of all initial conditions ${\mathbf{x}}_{\epsilon}\left(\mathbf{0}\right)$ that lie within $\epsilon $ distance from it. A fixed point is locally asympt. stable (a sink) if all eigenvalues of the Jacobi matrix, evaluated at the equilibrium, lie within the unit circle. If at least one eigenvalue lies outside the unit circle, the fixed point will be called unstable [25]. |

8 | Note that the population dynamics remains in the interior of the unit simplex for finite $\beta $. This implies that in each time period all behavior rules are present in the population and no behavioral rule will ever vanish (this is the so-called no-extinction condition). Furthermore, no new behavioral rules emerge from this model (this is the so-called no-creation condition). The property that the simplex is invariant under the $K(\xb7)$ dynamic also ensures that fractions, and therefore quantities played, remain bounded. |

9 | This rational choice solution exists and is unique for standard assumptions as strictly concave inverse demand functions $P(\xb7)$ and nondecreasing cost functions $C(\xb7).$ Outside this class of oligopolies, for the case of multiple (local) maxima, we make the additional assumption that rational players are able to coordinate on the global maximum of the profit function. |

10 | Notice that in the opposite case $C<0$ (i.e., Imitation is costlier than Cournot heuristic) and large selection pressure $\beta \to \infty $ equilibrium fraction of Cournot players ${\eta}^{*\phantom{\rule{4.pt}{0ex}}}$ approaches 1 and we recover [1] instability threshold with Cournot only players, $n=3$. |

11 | Economically, this threshold number of firms can only be an integer but mathematically the number of firms can be treated as a continuous variable. |

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**Figure 3.**Linear n-player Cournot competition between rational, Cournot and imitation firms with endogenous fraction dynamics.

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Lindeman, D.; Ochea, M.I.
Imitation Dynamics in Oligopoly Games with Heterogeneous Players. *Games* **2024**, *15*, 8.
https://doi.org/10.3390/g15020008

**AMA Style**

Lindeman D, Ochea MI.
Imitation Dynamics in Oligopoly Games with Heterogeneous Players. *Games*. 2024; 15(2):8.
https://doi.org/10.3390/g15020008

**Chicago/Turabian Style**

Lindeman, Daan, and Marius I. Ochea.
2024. "Imitation Dynamics in Oligopoly Games with Heterogeneous Players" *Games* 15, no. 2: 8.
https://doi.org/10.3390/g15020008