Generic Structure to Simulate Acceptance Dynamics

Abstract

Social behaviour patterns and social equilibrium states are often guided by stable values such as social norms. However, changing environmental conditions (e.g. climate warming, resource scarcity) may require behavioural change and the acceptance of new technologies. Antecedents of aggregate behavioural change are value changes that predetermine when new behaviour patterns emerge and a new social equilibrium state can be reached. The paper addresses these phenomena. First, based on a waste recycling model, we explain these phenomena and develop a simple, generic mathematical model describing the basic traits of acceptance-rejection dynamics. Second, we propose a generic model structure for the simulation of acceptance-rejection behaviour that represents the dynamical characteristics of paradigm change processes. We show that a fourth-order potential function is a sine qua non for an adequate representation of a paradigm change. Third, we also explain why the well-known Bass model, is unable to capture acceptance and rejection dynamics.

Reprinted from Ulli-Beer et al (2010) Generic structure to simulate acceptance dynamics. Syst Dyn Rev 26(2):89–116. With kind permission from John Wiley & Sons, Inc., Nov. 2012.

Appendix

1.1 Generic Structure of the Waste Recycling Model

The 50 parameters and 33 nonlinear functions of the original waste recycling model have been reduced to the two parameters P and τ_R, and the two functions f(x) and g(y) with the following meanings:

• P = overall population

• X = number of adopters

• Y = number of non-adopters

• τ_R = time to adjust

• f(x) = influence of the adopters’ norm on non-adopters

• g(y) = influence of the non-adopters’ norm on adopters

The dynamical equations of the simplified model are

$$ \begin{array}{l}\frac{ dx}{ dt}=p\left(x,y\right)-q\left(x,y\right)\\ {}\frac{ dy}{ dt}=q\left(x,y\right)-p\left(x,y\right)\end{array} $$

(8.6)

with the condition

$$ P=x+y $$

(8.7)

and the two functions being defined by

$$ \begin{array}{l}p\left(x,y\right)=\frac{f(x)/P}{\tau_R}y\\ {}q\left(x,y\right)=\frac{g(y)/P}{\tau_R}x\end{array} $$

(8.8)

The two dynamical Eq. 8.6 for the two population groups x and y, together with the condition Eq. 8.7, can be expressed by one dynamical equation for x, describing the balance of the adoption rate p(x,y) and the frustration rate q(x,y). These two rates are defined symmetrically with the functions f(x) and g(y), and involve the time constant τ_R Eq. 8.8. The influence f(x) of the adopters’ norm on non-adopters vanishes for x = 0, because there is no adopters’ norm established without adopters. With only a few adopters, their influence is still negligible, suggesting a horizontal tangent f’(0) = 0. With an increasing number of adopters, however, their influence becomes important. The most simple functions f(x), and analogously g(y), which fulfil these three conditions, are quadratic polynomials:

$$ \begin{array}{l}f(x)={\nu}_1\cdot {x}^2\\ {}g(y)={\nu}_2\cdot {y}^2\end{array} $$

(8.9)

The two new parameters ν₁ and ν₂ describe the strength of the effect of the adopters’ norm and the non-adopters’ norm, respectively. We apply the following normalisation to further simplify the equations:

$$ \begin{array}{l}{x}^{\prime }=\frac{x}{P}\\ {}{y}^{\prime }=\frac{y}{P}=1-{x}^{\prime}\\ {}{\nu}_1^{\prime }=P\cdot {\nu}_1\\ {}{\nu}_2^{\prime }=P\cdot {\nu}_2\end{array} $$

(8.10)

With the substitutions Eq. 8.10, the dynamical Eq. 8.6 take a simple form. For the sake of convenience, the dashes are omitted in the following:

$$ \frac{ dx}{ dt}=\frac{\nu_1+{\nu}_2}{\tau_R}x\;\left(1-x\right)\;\left(x-\frac{\nu_2}{\nu_1+{\nu}_2}\;\right) $$

(8.11)

Equation 8.11 can be transformed into the following elegant form including a potential V(x):

$$ \frac{ dx}{ dt}=-\frac{ dV(x)}{ dx} $$

(8.12)

With the potential V(x) according to Eq. 8.13, the generalised Eq. 8.12 becomes identical to the special dynamics Eq. 8.11:

$$ V(x)=\frac{1}{12{\tau}_R}{x}^2\;\left\{6{\nu}_2-4\;\left({\nu}_1+2{\nu}_2\right)\;x+3\;\left({\nu}_1+{\nu}_2\right)\;{x}^2\right\} $$

(8.13)

This double-well potential (a polynomial of fourth order) has the following extremes:

$$ \begin{array}{c}V(0)=0\\ {}V(1)=\frac{\nu_2-{\nu}_1}{12{\tau}_R}\\ {}V\left(\frac{\nu_2}{\nu_1+{\nu}_2}\right)=\frac{1}{12{\tau}_R}{\left(\frac{\nu_2}{\nu_1+{\nu}_2}\right)}^3\left(2{\nu}_1+{\nu}_2\right)\end{array} $$

(8.14)

The first two extremes at x = 0 and x = 1 are stable minima and the third is an unstable maximum in between. In general, the potential Eq. 8.13 is asymmetric, because the minimum at x = 1 is above or below the x-axis, if ν₂ is larger or smaller than ν₁, respectively. For the special symmetric case ν₁ = ν₂ = ν, the extremes Eq. 8.14 of the potential Eq. 8.13 simplify to:

$$ \begin{array}{c}V(0)=0\\ {}V(1)=0\\ {}V\left(\frac{1}{2}\right)=\frac{1}{32{\tau}_R/\nu}\end{array} $$

(8.15)

The graph of this symmetric double-well potential is shown in Fig. 8.4 in the main text. Within the framework of the waste recycling model, the variable x (percentage of adopters) is confined to the interval between 0 and 1. To make the stable minima better visible, the potential V(x) is given for an extended x-range.

Our analysis shows that at least one of the two functions f(x) and g(1 − x), describing the effect of the perceived social norms, must be nonlinear to be able to lead to two simultaneously stable minima of the potential V(x). If both functions are assumed to be linear, the respective potential is a third-order polynomial with only one global minimum at x = 0 or at x = 1 (y = 1 − x) for the slope of f being smaller or larger than the slope of g, respectively. For this case with linear functions f and g, the basic character of the system would be different: As soon as the effect of the adopters’ norm had a larger slope than the effect of the non-adopters’ norm, the system would undergo a transient from x = 0 (not recycling) to x = 1 (recycling), without any external force (garbage bag charge) needed. For the case that the slope of the effect of the adopters’ norm would be smaller compared to that of the non-adopters’ norm, a garbage bag charge would push the system towards x = 1, but no paradigm change would occur, stabilising this state: As soon as the charge were relieved, the system would fall back to x = 0. This analysis demonstrates that one of the most important decisions during the modelling process is the choice of the shape of the norm-functions f and g. In the model validation process, observational evidence suggesting linear or nonlinear norm functions would therefore be of prime importance.

Another remark concerns the discrepancy of the numbers of parameters and functions between the full model and the simplified generic model. In every model useful for practical purposes, a large number of parameters are needed, because the important effective parameters (in our case τ_R, ν₁ and ν₂) must be related to practically relevant input parameters. The strength of the generic model, however, is not its application to simulate observed processes, but to help us understand its basic behaviour and to give us an idea of its solution manifold. It contains only a very limited number of effective parameters and functions, and thus shows us the relevant combinations of parameters and functions defining the trajectories to be expected.

1.2 The Light-Ball Metaphor

We consider a light air-inflated plastic ball with mass m moving downhill with velocity u. Its dynamics can be formulated with the notion of potential energy V(x) (in the physical literature, V(x) is called gravitational potential) in the following way:

$$ \begin{array}{c}V(x)=m\cdot g\cdot h(x)\\ {}m\cdot \frac{ du}{ dt}\approx -\frac{ dV}{ dx}-\alpha \cdot u\;\\ {}u\equiv \frac{ dx}{ dt}\;\end{array} $$

(8.16)

where g is the gravitational acceleration, t is time, and h(x) describes the height of a graph of the potential function V(x) with one horizontal dimension x. Multiplication of the slope dh/dx by -mg gives the force -dV/dx accelerating the ball downhill.³ The term -α · u describes the frictional braking force of the air (according to Stokes’ law, this frictional force is proportional to velocity u). Our experience tells us that such a light ball, after a short initial acceleration phase, rolls downhill at a constant speed, only depending on the slope. To simplify our dynamics Eq. 8.16, we therefore neglect the inertia term by setting

$$ m\cdot \frac{ du}{ dt}=0 $$

(8.17)

and find the approximate dynamics:

$$ u=-\frac{ dV}{ dx} $$

(8.18)

The parameter α has been set to unity because it does not affect the character of the solutions of Eq. 8.18. In the physical literature, this approximation is called overdamped limit, because the respective system approaches an equilibrium point gradually rather than with damped oscillations. This property can be demonstrated, e.g., for the equilibrium point in a quadratic potential V = x ² situated at x = 0. Introducing this most simple nonlinear potential into Eq. 8.18 gives

$$ u\equiv \frac{ dx}{ dt}=-\frac{d}{ dx}{x}^2=-2x $$

(8.19)

with the solution

$$ x(t)={x}_0{e}^{-2t} $$

(8.20)

where x₀ is the initial position of the ball and t is time. Equation 8.20 describes a trajectory approaching the equilibrium point x = 0 gradually, without oscillations. Mathematically, the ball would need an infinite amount of time to reach x = 0, but for practical applications, t = 3 is already sufficient, giving a distance to zero of less than 1 % of the initial value x₀.

For a multistable system, we need at least two stable equilibria, described by a double-well potential V(x). A simple form of such a potential is a polynomial of fourth order:

$$ \begin{array}{l}V(x)=a{x}^2\left\{{x}^2-2{\mu}^2\right\}\\ {}-\frac{ dV}{ dx}=-4 ax\left\{{x}^2-{\mu}^2\right\}\end{array} $$

(8.21)

To prevent the ball escaping to infinity, we assume a ≥ 0. At x = 0, we find an unstable equilibrium, and two locally stable equilibria are located at x = ±μ. Combined with Eq. 8.18, we get the following dynamics:

$$ \frac{ dx}{ dt}=-4 ax\left\{{x}^2-{\mu}^2\right\} $$

(8.22)

To assign simple meanings to the two parameters a and μ, we define two new parameters τ and η (their meanings will be explained below):

$$ \begin{array}{l}\tau =\frac{1}{8a{\mu}^2}\\ {}\eta =a{\mu}^4\end{array} $$

(8.23)

and write the dynamics Eqs. 8.21 and 8.22 with these new parameters:

$$ \begin{array}{l}\frac{ dx}{ dt}=\frac{x}{2\tau}\left\{1-\frac{x^2}{8\tau \eta}\right\}+F(t)\\ {}V(x)=\frac{x^2}{8\tau}\left\{\frac{x^2}{8\tau \eta}-2\right\}\end{array} $$

(8.24)

In addition, an external force F(t) has been introduced. The stable equilibria (with F = 0) are now located at

$$ {x}_s=\pm \sqrt{8\tau \eta} $$

The parameter η is the height of the “activation potential” (e.g. the unstable equilibrium) with its top at x_u = 0 lying in between the two stable equilibria at x_s, as can easily be verified:

$$ V\left({x}_u\right)-V\left({x}_s\right)=0-\frac{8\tau \eta}{8\tau}\left\{\frac{8\tau \eta}{8\tau \eta}-2\right\}=\eta $$

(8.25)

τ is the endogenous time constant of the system near its stable equilibria x_s. This can be verified by linearisation of the dynamics around x_s. To this aim, we replace x with the new coordinate ξ, being the distance from x_s:

$$ \xi =x-{x}_s $$

(8.26)

After introducing Eq. 8.26 into Eq. 8.24, we linearise the dynamics for small ξ and get the approximate differential equation for the trajectory in the neighbourhood of the stable equilibria

$$ \frac{ d\xi}{ d t}\approx -\frac{1}{\tau}\xi $$

(8.27)

with the solutions

$$ \xi (t)\approx {\xi}_0{e}^{-\frac{t}{\tau }} $$

(8.28)

ξ₀ is the initial position of the ball at t = 0. By definition, τ is the time constant for the relaxation of the system to its equilibrium point ξ = 0, which had to be shown.

For a graphical representation of the potential V(x) according to Eq. 8.24 for τ = η = 1, see Fig. 8.7 in the main text.

1.3 Light-Ball System Response to Transient Forces

1.3.1 Small Transient Forces

We apply a small constant force F₀ and ask for the deviation δ from the force-free equilibrium point at position x_s = −(8ητ)^1/2:

$$ \delta =x+\sqrt{8\tau \eta} $$

(8.29)

We substitute x in Eq. 8.24 by δ according to Eq. 8.29 and ask for the stationary solution by setting the time derivative to zero. This leads to the following relation between F₀ and δ:

$$ \delta \left\{{\delta}^2-3\delta \sqrt{8\tau \eta}+16\tau \eta \right\}=16{\tau}^2\eta {F}_0 $$

(8.30)

For small δ, the bracket in Eq. 8.30 reduces to the constant term and we get approximately:

$$ \delta \approx {F}_0\cdot \tau $$

(8.31)

An example for a dynamical simulation with F₀ = 0.1 is given in Fig. 8.10. Note that the bracket of Eq. 8.30 reads for τ = η = 1 and δ = 0.1: (0.01 − 0.85 + 16). Clearly, the first two terms are negligible compared to the constant third term!

Fig. 8.10

Trajectory beginning at the left equilibrium (xs = −2.828) for a constant external force F0 = 0.1 in the time interval t = 0…10. For t > 10, F0 = 0 is assumed. After a few τ have elapsed, a new equilibrium position is found with a distance δ from the stable force-free equilibrium point. The simulated δ is 0.105, as indicated by the double arrow. The approximation Eq. 8.31 gives δ = 0.100, only 5 % smaller than the simulated δ.

The internal time constant τ of the system is identical to the proportionality constant mediating between an applied small external force and the resulting deviation δ according to Eq. 8.31. To prove this generic result, we approximate an arbitrary potential V(x) around one of its minima by a second-order polynomial:

$$ V\left(\xi \right)=\frac{\xi^2}{2\tau } $$

(8.32)

with the coordinate ξ being the distance from the respective minimum. From the dynamic equation

$$ \frac{ d\xi}{ d t}=-\frac{ d V}{ d\xi}+{F}_0=-\frac{\xi }{\tau }+{F}_0 $$

(8.33)

with the solution

$$ \xi (t)={\xi}_0\cdot {e}^{-\frac{t}{\tau }} $$

(8.34)

for F₀ = 0, we immediately find the first meaning of τ being the system-internal relaxation time constant. From the same dynamical Eq. 8.33, we get the deviation δ of the equilibrium point for small forces F₀

$$ \frac{ d\xi}{ d t}=0=-\frac{\delta }{\tau }+{F}_0 $$

(8.35)

leading directly to Eq. 8.31, where τ has the second meaning as a proportionality constant between an applied small external force and the resulting deviation.

1.3.2 Large Transient Forces

To induce a transition from the left equilibrium point to the right, the transient force F₀ must be able to push the ball up the steepest slope of V(x). With the dynamic equation

$$ \begin{array}{l}\frac{ dx}{ dt}=-\frac{ dV}{ dx}+{F}_0\\ {}V(x)=\frac{x^2}{8\tau}\left\{\frac{x^2}{8\tau \eta}-2\right\}\end{array} $$

(8.36)

this first condition is equivalent to

$$ \frac{ dx}{ dt}=-\frac{ dV}{ dx}+{F}_0>0 $$

(8.37)

for the most positive slope of V(x) occurring at x_m with

$$ {\left.\frac{d^2V}{d{x}^2}\right|}_{x_m}=0 $$

(8.38)

From Eqs. 8.36 and 8.38, we find

$$ {x}_m=-\sqrt{\frac{8}{3}\tau \eta} $$

(8.39)

leading with Eq. 8.37 to the first necessary condition

$$ {F}_0>\sqrt{\frac{8}{27}\frac{\eta }{\tau }} $$

(8.40)

Another condition to induce a transition refers to the duration T of the force F₀. For the best case of a flat potential V(x) = 0, Eq. 8.37 defines a constant velocity dx/dt = u = F₀. With this velocity u, the system state must travel at least from the left equilibrium point to zero during T, giving the second necessary condition

$$ u\cdot T={F}_0\cdot T>\sqrt{8\tau \eta} $$

(8.41)

1.4 Bass Dynamics with the Light-Ball Model

We define the system state x of the light-ball model as the number of items sold until time t and concentrate on the portion of the potential V(x) between x = 0 and x = x_s + F₀τ (x_s = positive stable equilibrium for F₀ = 0). The system dynamics with small external force F₀ > 0, according to Eqs. 8.2 or 8.24, is the following:

$$ \frac{ dx}{ dt}=\frac{x}{2\tau}\left\{1-\frac{x^2}{8\tau \eta}\right\}+{F}_0 $$

(8.42)

For small x, Eq. 8.42 reduces to dx/dt = F₀ leading to a linear growth of x. At larger x, for x ² < < 8τη and F₀ < < x/(2τ), we find approximately an exponential growth of x with time constant 2τ:

$$ x(t)\approx {x}_0{e}^{\frac{t}{2\tau }} $$

(8.43)

For x_m with maximum slope of V(x), F₀ can be neglected and we find:

$$ \begin{array}{l}{x}_m\approx \sqrt{\frac{8\tau \eta}{3}}\\ {}\frac{ dx}{ dt}\approx \sqrt{\frac{8\eta }{27\tau }}\end{array} $$

(8.44)

Finally, for x near the equilibrium x_s + F₀τ, we find an exponential approximation with time constant τ according to Eq. 8.27.

The Bass model is generally presented in the following form:

$$ \frac{f(t)}{1-F(t)}=p+ qF(t) $$

(8.45)

where f(t) stands for the sale rate of a product and F(t) for the total amount of items sold until time point t. p is called coefficient of innovation and q is the coefficient of imitation. If we use the relation f = dF/dt, we can write the Bass model in the form:

$$ \frac{ dF}{ dt}=\left(p+ qF\right)\left(1-F\right) $$

(8.46)

For small F (near t = 0), the dynamics reduces to

$$ \frac{ dF}{ dt}\equiv f\approx p $$

(8.47)

giving a constant sale rate f = p and a linear increase of the total amount F of items sold. For a time interval, where qF > > p and F << 1, the approximate dynamics are

$$ \frac{ dF}{ dt}\equiv f\approx qF $$

(8.48)

leading to an exponential growth of both, F and f with time constant 1/q:

$$ \begin{array}{l}F(t)\approx {F}_0{e}^{qt}\\ {}f(t)\approx q{F}_0{e}^{qt}\end{array} $$

(8.49)

The maximum slope of F (maximum selling rate f) is found from Eq. 8.46 at F_m lying near 50 % of the ultimate market potential (which is normalised to 1) for the majority of situations characterised by p << q:

$$ \begin{array}{l}{F}_m=\frac{1}{2}\left(1-\frac{p}{q}\right)\approx \frac{1}{2}\\ {}{f}_m\approx \frac{q}{4}\end{array} $$

(8.50)

For F near to unity, we find from the approximated dynamics

$$ \frac{ dF}{ dt}\approx q\left(1-F\right) $$

(8.51)

the trajectory

$$ F(t)\approx 1-{e}^{- qt} $$

(8.52)

i.e., an exponential approximation of the ultimate market potential with a time constant 1/q.

By comparing each of the three phases (initial linear growth, exponential growth, final exponential approximation of the ultimate market potential) described by the two different models, we find the following equivalences:

• By definition

  $$ \mathrm{F}=\mathrm{x} $$

• Following from the definition

  $$ \mathrm{f}=\mathrm{u}=\mathrm{dx}/\mathrm{dt} $$

• x_s + F₀τ = 1 and x_s ² = 8τη

  $$ \eta \approx \frac{{\left(1-{F}_0\tau \right)}^2}{8\tau } $$

• Equation D-1 for small x compared with Eq. 8.47

  $$ \mathrm{p}={\mathrm{F}}_0 $$

• Time constant τ from Eq. 8.27 compared with Eq. 8.52

  $$ \mathrm{q}=1/\uptau $$

It should be added here that the two models could be made exactly identical by replacing the fourth-order potential Eq. 8.24 of the ball model with the third-order potential

$$ V(x)=\frac{x}{\tau}\left\{\frac{x^2}{3}-\frac{x}{2}\left(1-{F}_0\tau \right)-{F}_0\tau \right\} $$

(8.53)

Here, F₀ would no longer be an external force, but an additional parameter shaping the potential V(x). Equation 8.53 is proven to be correct by the substitutions x → F, 1/τ → q, F₀ → p and comparing dF/dt = −dV(F)/dF with Eq. 8.46.