An evolutionary model with best response and imitative rules

Lorenzo Cerboni Baiardi; Ahmad K. Naimzada

doi:10.1007/s10203-018-0219-y

Decisions in Economics and Finance

November 2018, Volume 41, Issue 2, pp 313–333 | Cite as

An evolutionary model with best response and imitative rules

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Lorenzo Cerboni Baiardi
Ahmad K. Naimzada

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First Online: 09 November 2018

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Abstract

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms.

Keywords

Imitation Heterogeneity Evolutionary game Logit dynamics Dynamic instability Dynamic systems

JEL Classification

C62 C63 C72 C73

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Notes

Acknowledgements

This work has been developed in the framework of the research project on “Dynamic Models for behavioural economics” financed by DESP, University of Urbino. The authors thank two anonymous referees for their useful comments.

Appendix

Proof of Proposition 1

Stationary states of map T are the solutions of the following algebraic system of equation:

$$\begin{aligned} \left\{ \begin{array}{l} q_1 = \max \left\{ 0,\dfrac{a-c}{2b} - \dfrac{1}{2}(N-1)\left( (1-\omega )q_1 + \omega q_2\right) \right\} \\ q_2 = \dfrac{(a-c-bN((1-\omega )q_1 + \omega q_2))(q_1^2 + q_2^2) - Cq_1}{ (a-c-bN((1-\omega )q_1 + \omega q_2))(q_1 + q_2) - C}\\ \omega = \dfrac{1}{1+e^{\beta \left\{ (a-c-bN((1-\omega )q_1 + \omega q_2))(q_1 - q_2)-C\right\} }} \end{array}.\right. \end{aligned}$$

(6.1)

We observe that the second equation in (6.1) can be re-expressed as

$$\begin{aligned} q_2 = \dfrac{\pi _1-C}{\pi _1+\pi _2-C}q_1+\dfrac{\pi _2}{\pi _1+\pi _2-C}q_2 \end{aligned}$$

which is equivalent to

$$\begin{aligned} (q_2-q_1)(\pi _1-C)=0. \end{aligned}$$

This equation is satisfied if either $q_1=q_2$ or $\pi _1-C=0$. In the former case, the condition $q_1=q_2$, together with the first equation in (6.1), implies either $q_1=q_2=q^{*}$ or $q_1=q_2=0$. Moreover, since the condition $q_1=q_2$ implies $\pi _1 = \pi _2$, the stationary share of imitators satisfying the third equation in (6.1) is fixed at the level $\omega ^{*} = 1/(1 + e^{-\beta C})$. However, only the point $(q^{*},q^{*},\omega ^{*})$ is a feasible stationary state of map T. Indeed, provided that both the representative best responder and imitator set null productions, namely $q_1=q_2=0$, the imitation rule (2.6) is not defined (see Remark 2).

Let us consider the latter occurrence in which $\pi _1-C=0$. In this case, the condition $q_1(t+1)=q_1(t)$ can be rewritten to express the stationary value $q_2$ in terms of $q_1$ as

$$\begin{aligned} q_2 = \dfrac{2}{(N-1)\omega }\left( \dfrac{a-c}{2b}-q_1\left( 1+\dfrac{1}{2}(N-1)(1-\omega )\right) \right) . \end{aligned}$$

(6.2)

Hence, the condition $\pi _1-C=0$ turns to a second-order polynomial in the variable $q_1$:

$$\begin{aligned} \pi _1-C = \left( 2b\dfrac{N}{N-1}q_1-\dfrac{a-c}{N-1}\right) q_1-C=0 \end{aligned}$$

(6.3)

whose positive root is $q_1^0$. Then, by substituting the value $q_1^0$ in Eq. (6.2), the value $q_2^0$ is obtained. We observe that, provided that condition (2.4) holds, $q_2^0$ is positive. Indeed,

$$\begin{aligned} q_2^0&=\dfrac{1}{4bN(N-1)\omega }\bigg ((a-c)(3N-1+\omega (N-1))\\&-(N+1-\omega (N-1))\sqrt{(a-c)^2+8bN(N-1)C}\bigg )\\&>\dfrac{1}{4bN(N-1)\omega }\bigg ((a-c)(3N-1+\omega (N-1))\\&\quad -(N+1-\omega (N-1))\sqrt{(a-c)^2+8N(N-1)\left( \dfrac{a-c}{N+1}\right) ^2}\bigg )\\&=\dfrac{a-c}{4bN(N-1)\omega }\bigg (2N+(N+1)\left( 1-\sqrt{1+\dfrac{8N(N-1)}{(N+1)^2}}\right) \\&\quad +\omega (N-1)\left( 1+\sqrt{1+\dfrac{8N(N-1)}{(N+1)^2}}\right) \bigg ) \\&=\dfrac{a-c}{4bN(N-1)\omega }\bigg (2+\omega (N-1)\left( 1+\dfrac{3N-1}{N+1}\right) \bigg )>0. \end{aligned}$$

Finally, the condition $\omega (t+1) = \omega (t)$ computed at $q_1 = q_1^0$ and $q_2 = q_2^0$ leads to the equation $G(\omega ) = 0$, where

$$\begin{aligned} G(\omega )&=\left( a-c-bN((1-\omega ) q_1^0+\omega q_2^0)\right) (q_1^0-q_2^0)-C- \dfrac{1}{\beta }\ln \left( \dfrac{1}{\omega }-1\right) . \end{aligned}$$

By Eq. (6.3), $G(\omega )$ can be simplified as follows:

$$\begin{aligned} G(\omega )&= -\dfrac{C}{q_1^0}q_2^0 - \dfrac{1}{\beta }\ln \left( \dfrac{1}{\omega }-1\right) . \end{aligned}$$

Equation $G(\omega ) = 0$ has a unique root $\omega ^0$ within the interval [0, 1] such that $\omega ^0\in (0,1)$ provided that condition 2.4 holds. Indeed, in this case, there holds

$$\begin{aligned} \lim _{\omega \rightarrow 0^+}G(\omega )&= -\infty ,\;\; \lim _{\omega \rightarrow 1^-}G(\omega ) = +\infty \end{aligned}$$

and $G'(\omega )>0$ for all $\omega \in (0,1)$.

Proof of Proposition 2

The Jacobian matrix of map T computed at $E^{*}$ is given by

$$\begin{aligned} J(E^{*})&=\left( \begin{array}{cccc} -(N-1)(1-\omega ^{*})/2&{} &{}-(N-1)\omega ^{*}/2&{}0\\ \dfrac{\pi ^{*}-C}{2\pi ^{*}-C}&{} &{}\dfrac{\pi ^{*}}{2\pi ^{*}-C}&{}0\\ -\beta N\cdot \dfrac{a-c}{N+1}\cdot \dfrac{e^{-\beta C}}{(1+e^{-\beta C})^2}&{} &{} \beta N\cdot \dfrac{a-c}{N+1}\cdot \dfrac{e^{-\beta C}}{(1+e^{-\beta C})^2}&{}0 \end{array}\right) . \end{aligned}$$

The Jacobian matrix $J(E^{*})$ has a vanishing column, and its characteristic polynomial can be factorized as $P(\lambda ) = -\lambda {\hat{P}}(\lambda )$, where

$$\begin{aligned} {\hat{P}}(\lambda )&= \lambda ^2 - \lambda \left( -\dfrac{N-1}{2}(1-\omega ^{*})+ \dfrac{\pi ^{*}}{2\pi ^{*}-C}\right) \\&\quad -\dfrac{N-1}{2}\cdot \dfrac{\pi ^{*}}{2\pi ^{*}-C}(1-\omega ^{*}) + \dfrac{N-1}{2}\cdot \dfrac{\pi ^{*}-C}{2\pi ^{*}-C}\omega ^{*} \end{aligned}$$

is the characteristic polynomial of the $2\times 2$ matrix ${\hat{J}}$ representing the Jacobian matrix related to the first two recurrences of map T computed at $q_1 = q_2 = q^{*}$. Hence, the stability conditions for $E^{*}$ are the Jury’s conditions for the stability of equilibria in two-dimensional discrete-time maps and read as

$$\begin{aligned} {\hat{P}}(1)&>0\;\Longrightarrow \; N+1>0\text { (always satisfied)}\\ {\hat{P}}(-1)&>0\;\Longrightarrow \; \omega ^{*}>\omega _f:=\dfrac{1}{2}\cdot \dfrac{3\pi ^{*}-C}{2\pi ^{*}-C}\cdot \dfrac{N-3}{N-1}\\ \text {det}{\hat{J}}&<1\;\Longrightarrow \;\omega ^{*}<\omega _{ns}:=\dfrac{2}{N-1}+\dfrac{\pi ^{*}}{2\pi ^{*}-C} \end{aligned}$$

and the thesis follows.

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Lorenzo Cerboni Baiardi
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Ahmad K. Naimzada
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1.Department of Economics, Quantitative Methods and ManagementUniversity of Milano - BicoccaMilanoItaly

An evolutionary model with best response and imitative rules

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