Chapter 1

Deterministic and Stochastic Models of

Innovation Diffusion: An Overview

1. Introduction

The study of innovation diffusion in a social group is a phenomenon of

considerable interest and has a long history; it has been studied across wide

ranging disciplines to explain the dissemination of new ideas, rumors, news,

practices and new products throughout a social system. The term innovation

implies 'an idea, practice or object that is perceived as new by an individual or

other unit of adoption' (Rogers 1995). Diffusion has been defined as 'the

process by which an innovation is communicated through various channels over

time among the members of a social system', involving some mechanism of

information transfer or contagion, like spread of disease (Carrillo 2002).

The process of innovation diffusion (ID) consists of adoption of the

innovation through communication channels by which messages get from one

individual to another in a social system. Mass media and inter-personal

communication channels play an important role in determining the speed and

shape of diffusion patterns in social system. There have been numerous studies

based on a range of assumptions regarding social structure, population

characteristics and influence coefficients leading to a variety of diffusion

models. (Bernhardt and Mackenzie 1972). Modeling the phenomenon of

innovation diffusion is primarily concerned with the relationship between the

growth of number of adopters and channels of communication. Mathematical

models that describe innovation diffusion (ID) are variants of simple epidemic

models. The innovation spread in the population is analogous to epidemic,

where adopters influence non-adopters by contact leading to an eventual

adoption. In the context of new product diffusion, there is independent buying

due to 'innovators' and imitative buying due to 'imitators'. The impetus to

modeling came largely from the marketing literature which could provide large

and voluminous data sets for validation of 10 models.

It may be remarked that models can be classified as deterministic and

stochastic. A deterministic model is one wh.ose response to a certain stimulus

can be predicted with certainty, whereas for a stochastic model the response can

only be expressed in probabilistic terms. For a more realistic description of the

innovation process we may emphasize the need for stochastic modeling as the

underlying mechanisms are stochastic in nature. For instance the underlying

mechanism whether a person comes in contact with the mass mediated service

and adopts the innovation is stochastic. Similarly, the interpers9nal i

communication based on interactive process between adopters and non adopters

is also stochastic. Even though innovation diffusion is essentially stochastic in

nature, much of modeling efforts employ deterministic framework. The reason

being that mathematics of nonlinear stochastic models is intractable and

accordingly one generally falls back on deterministic framework to make

progress. Bartholomew (1982) observes: 'The deterministic model will be

regarded as an approximation and models will always be formulated

stochastically in the first instance.' Andersson and Britton (2000) mention that

an important issue in the context of stochastic model is to obtain: the I

deterministic version to which the stochastic model converges. This provides

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conditions under which a deterministic version would be a valid one.

Andersson and Britton (2000) make a strong case for developing stochastic

model for estimation purposes as it will provide uncertainty in the estimates

which in its absence is not of much use.

2. Deterministic Models A common feature of diffusion studies, carried in a large number of disciplines

is that the temporal pattern of diffusion of cumulative adoptions over 'time

follows an S-shaped curve. In the context of marketing, the models describe the

S-shaped curve which is related to life cycle dynamics of new product. These

models, besides giving insight into diffusion process, are also useful in

forecasting and analysis of descriptive-hypothesis testing issues. Several real

business applications based on these models have been documented

demonstrating their practical utility (Mahajan et al 2000).

2.1 Mixed Influence Model

Model which include both direct effects (mass media) and social interaction

(word of mouth) is referred to as mixed influence model which in the marketing

context was first used by Bass (1969). The time variation of non-cumulative

number of adopters gives product life cycle (PLC), which is a description of the

evolution of unit sales over the entire life span of a product (Bayes 1994). One

can verify that PLC for this model is a unimodal curve. Starting with initial

number of adopters no , the growth dynamics for the rate of adoption at time t is

governed by the differential equation

d:;t) = ( arM -n(l)] + fJ [nZ)}M -n(t)]J ' n(1 = 0) = no ' (1)

In equation (1), parameters a and ~ are coefficients of innovation (mass media)

and word of mouth (WOM) respectively, while parameter M is total ceiling of

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adopters. The first term in the above equation denotes adoptions due to mass

media and second term corresponds to adoptions due to interactions between

adopters and non-adopters. The first term is linear as it depends only on the

number of non-adopters in the system, while the second term is non-linear in

net). The temporal evolution of number of adopters is given as:

net) = M - a(M -no) exp( -(a+ j3M)(t-to ))

a+ j3no

Equation (2) describes the pattern of cumulative adoption over time, with exact

shape of the trajectory determined by parameter values. For a particular choice

of parameters, a=O.OOl, ~=0.400, a schematic description of evolution of

cumulative adopters is given in figure la.

1000

Evolution of cumulative 900 adopters

800

700

600

n(l) 500

400

300

200

100

0L-----~----~----~------~----~ o 10 15 20 25

Figure la. Mixed influence (Bass) model: Evolution of cumulative adopters.

When WOM mechanism due to ~ dominates, the sigmoid curve is obtained for

net). In that case (a=O) equation (1) gives the well known Verhulst or logistic

equation which has been used in marketing to illustrate diffusion as a pure

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imitation process. In this case equation (1) describe internal influence model,

the life cycle curve is for this case is shown in figure lb.

90

Non-cumulative 80 adopters

70

60

dn/dt 50

40

Figure lb. non cumulative adopters .a=O.020,~=0.300. 1. life cycle curve 2. adoptions due to WOM mechanism, 3. adoptions due to mass media.

Mansfield (1961) applied it to explain diffusion of technologies through

imitation. It has also been applied to innovations, infrastructure, and energy

consumption (Marchetti 1980,1986). Victor and Ausubel (2002) have used the

logistic equation to study global dynamics of generations of dynamic random ,

access memory (DRAM) with a view to forecast the next generations. Carrillo

(2002) has applied it to electricity consumption in US. In absence of WOM

effect (P=O), equation (1) reduces to external influence model stressing the

effect of direct marketing (mass media) as shown in figure lb. The model has

been applied by Coleman et al (1966) to study the diffusion pattern of a new

drug.

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Bass model and its variants have been established as an important tool in

the marketing research area. Several research articles have appeared,

successfully validating the model for a large number of new products both in

developed and developing countries (Nakicenovic and Grubler 1991, Grubler

1990, Gatignon et al 1989, Sharma and Bhargava 1996). Research on the model

includes studies examining the market penetration of products using the basic

formulations with extensions and refinements. These are in terms of

incorporating marketing mix variables such as price (Jain and Rao 1980,

Robinson and Lakhani 1975), advertising (Horsky and Simon 1983; Simon and

Sebastian 1987) and dynamic potential population (Mahajan and Peterson 1978).

Some studies have attempted to incorporate demand for the product (Bayes

1993) and optimal level of sampling (Jain, Mahajan, and Muller 1995). Various

estimation procedures have been described (¥ahajan 1986). Time varying and

Bayesian estimation procedures are designed to update parameters as new data

become available (Mahajan, Muller and Bass 1993).

2.2 Flexible diffusion models

Many extensions of the Bass model have been suggested, one of the most

commonly used flexible model being non-uniform influence (NUl) model. This

model allows the imitation effect to grow or decline rather than stay constant as

the diffusion process unfolds. The model is given as (Easingwood et a11983)

d:;t) = [ arM -n(l)] + f3 [ n~) r}M -n(t)], n(t = 0) = n,

In equation (3), second term represents the non-uniform interaction effect·

through parameter 8.

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(3)

1000 Cumulative

800 adopters 0=0.8 0=1.2

net) 600

400

dnldt

200

oL-~--~====~L-----~----~------~----~ o 5 10 15 20 25 30

Figure 2a

200 0= 8 life cycle curve

150 0=1.2

100

50

o~---=d=====~--~~~----~------~==--~ o 5 10 15 20 25 30

Figure 2b

Figure 2 a. Number of cumulative adopters for NUl model for two values of parameter 8=0.8 and 8=1.2 b. life cycle curves

In the model, the internal influence may either increase or decrease with time

besides staying constant. The model gives systematically time varying nature of

internal influence, and thus provides flexibility to accommodate many diffusion

patterns with varying point of inflection. Figure 2 (a,b) shows the cumulative

number of adopters net) and life cycle curve for NUl model. It may be remarked

that flexible models help to develop taxonomy of diffusion patterns, since they

reproduce the S-curve conforming to the data rather than force the data to

conform to a given shape (Mahajan and Peterson 1985).

2.3 Parameter variation in ID models

Changes during unfolding of the ID process such as advertising quality, taste

and income of the population are likely to cause parameters vary over time. In

some variants of innovation diffusion models, internal influence can only

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decrease with time (Floyd 1962, Sharif and Kabir 1976). However, some studies

state that internal influence should increase with time since late adopters are in a

better position to assess the innovation than earlier ones (Bundagaard-Nielsen

1976). Kotler (1971) mentions: 'The coefficient of imitation should decline

with time, rather than stay constant because the remaining potential adopters are

less responsive to the product and associated communications'. Hernes (1976)

also argues that the parameters should be time dependent. It is consistent with

the arguments given by Bretchneider (1980), that due to change In

characteristics of adopter population and other factors, the parameters of

diffusion model are likely to change over time. Understanding parameter

variation i~ important since the form of the variation can provide insight into the

nature of the diffusion process (Putsis 2000).

2.4 Spatio-temporal aspects of innovation diffusion

Mathematical modeling of ID has largely been concerned with the study of the

temporal aspects of the problem to the neglect of spatial effects. Haynes,

Mahajan and White (1977) and Lal, Karmeshu and Puri (1998) have modeled

innovation diffusion in a two dimensional space using the deterministic model

a IfJ ~~' t) = (a + f31f/(r, t) )O-If/(r. t) ) + D \l21f/(r, t), (4)

where IfJ (r, t) represents the fraction of adopters at time t and a and P have the

usual meaning as in Bass model. The partial differential equation (p.d.e.) (4) is

known as reaction diffusion equation which may be responsible for traveling

wave characteristics.

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3. Development and Applications of Innovation Diffusion

Models: Real life Applications The success of ID Models in describing the growth trajectories and explaining

the life cycle patterns have encouraged several researchers to adopt, the

mathematical framework to more challenging issues in real life. The ID'

framework is so versatile that it has been adopted to deal with issues concerning

pre and post launch strategic decisions in connection with new product

diffusion. For the purpose of illustration, we discuss some case studies based

on ID models.

3.1 Modelling legal and shadow diffusion in presence of software piracy

Givon, Mahajan and Muller (1995) extended Bass model to explain the prQblem

of software piracy. They separately model ·legal and shadow diffusion, with

influence from both legal buyers and pirates. The diffusion model is formulated

as a coupled set of differential equations, which read as:

dx(t) _ [ bl x(t) + b2 yet)] [N( ) ( ) ( )] --- a+a t -xt -yt dt N(t)

(5)

dy(t) = [(1- a) bl x(t) + b2 yet)] [N(t) - x(t) - yet)] dt N(t)

(6)

where N(t), x(t) and yet) denote number of total microcomputer owners,

software buyers and pirates respectively. Givon et al point to an interesting

finding on the basis of estimation of parameters for UK data, that in late 1,980's,

for each buyer who purchased software there were about six pirates who were

also using the software.

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3.2 Timing Diffusion and Substitution of Successive Generations of

Technological Innovations.

Norton and Bass (1987) have attempted to integrate the diffusion of successive

generation of technologies and their substitution within the context of Bass

model. Employing this framework, Mahajan and Muller (1996) proposed a

generalized model that simultaneously captured substitution pattern for each

successive generation of technological innovation. They develop the model for

four successive generations of IBM mainframe computers for addressing issues

with regard to new product launch strategy, risk associated with pre-mature

delayed product introduction and optimal level for new product introduction.

Figure 3 is adapted from Mahajan and Muller (1996) depicting diffusion and

substitution of IBM mainframe computers.

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1955 1958 1961 1964 1967 1970 1973 1976

Figure 3. Diffusion and Substitution: IBM Mainframe Systems in Use (in thousands)

Source: Mahajan and Muller (1996).

To illustrate the model, we briefly describe the differential equations

corresponding to each generation. Assuming the period of first generation TIs

tsT 2 the growth equation is akin to Bass model and reads as

dx~;t) = ( a +b, [ X~)]}N, -x, (/)], (7a)

The second generation begins at time epoch T 2 and in this case there are two

competing generations. Denoting by U2 the fraction of those who decide to

adopt the base technology will purchase the generation and fraction l-u2 will

adopt the earlier generation. The two resulting differential equations are

dx1 / dt = (1- a 2 ) (a 2 + (b1x1 + b2x2 )/ NJ(N2 - x) + a 2 (a; + b;X2 / N 2 )xl'

T2 S t s T3 (7 b)

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It may be remarked that in the third generation, a leapfrogging phenomenon

when adopters of first generation adopt the latest technology skipping the

second generation altogether. The governing differential equations are

dx3 / dt = a 3(a3 + I,b;x; / N 3)(N3 - x) + a3/33(a~ + b;X3 / N 3)x,

+ a3(a~ + b~X3 / N 3)X2

dx2 / dt = (1- a 3)!33(a3 + I,b;x; / N 3)(N3 - x) + a 3/33(a; + b~X3 / N 3)X2

+ a 3(1- /33)(a~ + b~X3 / N3)x, ,

dx, / dt = (1- a 3) (1- !33(a3 + I,b;x; / N 3)(N3 - x) + a3/33(a~ + b~X3 / N 3)x,

T3 ~ t ~ T4 (7 c )

Here symbols have their usual meanings. For details of the meaning of various

parameters, one may refer to work of Mahajan and Muller (1996). They

hypothesize that introduction of a new generation has an effect on the primary

demand, and there is a sharp drop in the sales of the first generation when the

second generation is introduced. The study also suggest that a firm should ,either

introduce a new generation as soon as it is available or else delay its introduction

until the maturity stage in the life-cycle of the current generation.

4. Incorporating heterogeneity in ID models Demographic and socio-economic factors like rate of population growth, degree

of urbanization, life styles and preferences for new products, purchasing power

and income influence the consumer's preferences choices in innovation

diffusion (Wind 1981). The relation of economic conditions, e.g. purchasing

power and demographic change to diffusion propensity has recently been

examined by Bulte (2000). The process of adoption of an innovation by an

adopter is influenced by several factors e.g. by hislher taste, perception,

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preference and demographic and environmental factors. These factors may vary

from one individual to another across the population, introducing heterogeneity

with regard to parameters of the model. In most cases heterogeneity is not

planned and hence cannot be regarded as a fixed effect that is built into the

design and analysis of an evaluation. Plewis (2002) points out. that any impact

of an intervention will also vary across individuals and contexts. Population

heterogeneity is one aspect which gives rise to variations. in parameters.

Coefficients of influence for early and late adopters may be diff~rent across high

and low income households (Horsky 1990). Accordingly, an equation

describing the ID process for the whole population is not likely to give a true

picture of the process, since the groups can have varying income and may

adopt the innovation on different time scales with or without iagged response

between the sub-groups.

As mentioned by Roberts (2000), it is desirable to have diffusion models

that segment the population and allow for specific targeting of individual

members. Different segments have varying status with respect i10 the adoption

process, with diffusion taking place within each segment with or without

interaction across the segments. The assumptions of homogeneous mixing has

to be abandoned and the model describing the ID process should allow for I

heterogeneity across the target population. The incorporation of population

heterogeneity in the diffusion model allows many flexible and asymmetric I

shapes of patterns for new product diffusion. The need to examine the effect of i

heterogeneity in ID has been emphasized by researchers. leuland (1981a) and

Kalish (1985) take up the aspect of population heterogeneity at the individual

level. One way to incorporate population heterogeneity is to treat the

parameters as random variables (Chatterjee and Eliashberg 1990, leuland

1981a, Karmeshu and Goswami 2001).

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5. Bimodal and Multimodallife cycle patterns It is possible to explain asymmetric shapes of the diffusion patterns by

incorporating heterogeneity into diffusion models. Most life cycle curves

exhibit a uni-modal pattern. The other types of patterns depicting bi-modal and

multi-modal shapes are also observed and have been documented. There has

been a recent perspective in the marketing studies which gives different

attention to early and main markets. The view is that early marke.t adopters

differ in their inclination or reluctance to adopt the innovation (Rogers 1995).

It has also been pointed out by Moore (1991) that there is a discontinuity in the

diffusion process between early market and late market adopters. Moore ,

identifies a discontinuity after about 16% of the populatioil adopts the

innovation. A dual-market phenomenon is observed by Goldenberg ei al (2002)

who differentiate between early and main market adopters as different markets.

They employ a model assuming the market to be made of early market and main

market to explain the intermediate decline in sales. Goldenberg et ~l employ

cellular automata approach to model bimodal life cycle patterns. As pointed out I

in Goldenberg et al (2002): ' ... a saddle phenomenon is observable if the growth

of sales in the main market begins late; that is, the main market takes off shortly

after the sales in the early market reach their peak .... .'. This phenomenon is

depicted in figure 4.

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140,-----.--.------.------.-------,---.------,----.-----..

bimodal life cycle curve

120

100

80

60

40

2 4 6 8 10 12 14 16 18 YEAR

Figure 4. Bimodal pattern for life cycle curve in a heterogeneous population divided in two segments 1 and 2.

A market may consist of many segments with different adopting

propensity for each segment depending upon socio-economic and demographic

factors. Karmeshu and Goswami (2001) have provided a rationale for such a

phenomenon by introducing parameter variability random diffusion (PVRD)

model to discuss stochastic evolution of adopters due to population

heterogeneity. The framework adopted by Karmeshu and Goswami (2001) is

capable of segmenting the population to yield unimodal as well as multimodal

life cycle patterns.

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6. Stochastic considerations in ID models

A basic issue concerns the use of a stochastic framework for realistic modeling

of the ID phenomenon. Bartholomew (1982) outlines the need of a stochastic

framework by highlighting the role of chance mechanisms operating at various

levels. He observes: 'Whether or not a person hears the information will

depend on (a) his/her coming into contact with the source or a spreader and (b)

the information being transmitted when contact is being established. In less

rigidly organized systems neither (a) nor (b) is a certain event and so the

development of the process is unpredictable. Hence it can only be described

stochastically. '

Stochastic diffusion models can be classified on the basis of level of

aggregation and the type of stochasticity. . There can be aggregate level or

individual (disaggregate) level stochasticity (Kalish and Sen 1986). For

aggregate level models, there are two types of stochasticties viz intrinsic

(structural) and environmental or parametric stochasticity (Karmes~u and

Pathria 1980b). Intrinsic stochasticity arises due to discreteness of the variables

in the problem and relative fluctuations in this case fall rapidly as the size of the

system. It is described in terms of transition probabilities for the system in a

small interval of time. Environmental stochasticity is caused by random

changes in the social, economic and political environment in which the system is

embedded. As a result of fluctuations in the environment, the parameters of the

problem are subject to stochastic fluctuation. The resulting stochasticity,

referred to as environmental stochasticity, is generally studied through stochastic

differential equations with random parameters. In contrast to the results of

intrinsic stochasticity, the effects of environmental stochasticity are independent

of the size of the system (Karmeshu and Pathria 1980a). In modeling

parametric stochasticity, a stochastic error term is added to the deterministic

formulation describing the state of the system. Jain and Raman (1982) develop

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and analyze a stochastic version of the Bass model using perturbation expansion

method.

It may be remarked that besides intrinsic and environmental

stochasticity, there is another type of stochasticity which emanates from

population heterogeneity. One way to account for heterogeneity is to treat the

parameters as random variables. Accordingly, for investigating the resulting

diffusion patterns, one has to study differential equation with random parameters

(Soong 1973). Recently (Karmeshu and Goswami 2001) proposed a theoretical

framework for studying diffusion patterns emerging on account of randomly

distributed parameters, which are modeled in terms of two point distribution

(2PD). For specification of 2PD, they require statistical information about the

parameters prescribed through first three moments. The advantage of 2PD

framework is that it renders the mathematical analysis tractable. However, in

reality if we assume that parameters are distributed according to a specified

distribution, the advantage gained by 2PD framework is lost and one does not

get closed form expression for moments of number of adopters. In such a

situation, one has to resort to simulation approach based on Monte Carlo

techniques for studying the behavior of the system in a dynamic setting.

7. Organization of the thesis

The purpose of the thesis is to develop both deterministic and stochastic models

of innovation diffusion taking into account the heterogeneity of the population.

The thesis is organized in six chapters. Chapter 2 deals with the study of time­

dependent behavior of mean and variance of number of adopters when intrinsic

stochasticity is incorporated. The resulting model is described in terms of, non­

linear birth process. For analysis of the model, system-size expansion (SSE) is

employed which requires for its validity, an asymptotically large population.

However in practice, even for small popUlation size, SSE gives fairly accurate

17

results. It is found that deterministic approximation overestimates the mean

evolution of adoption over the entire life span of the innovation~ An entropy

framework is employed to provide evolution of uncertainty. Effect of explicit

time-dependence on stochastic evolution of number of adopters is examined.

The effects of finite size correction in small population bring out new qualitative

features.

In chapter 3, we present a hierarchy of moment equations which emerges

on account of nonlinearity of the problem. For gaining insight into evolution of

moments of number of adopters, one has to truncate the hierarchy, so as to

obtain a closed set of differential equations. We have employed some known

truncation procedures and find that, for large population size, the results are in

fairly good agreement with those obtainec\ through SSE. New truncation

procedures based on two point and three point distributions are proposed and we

find they also yield fairly accurate results. To illustrate the utility of truncation

procedures, two non-linear stochastic models corresponding to Bass and a

special case of NUl model (8=2) are considered. Bass and NUl models contain . ,

quadratic and cubic non-linearity respectively.

Chapter 4 is devoted to study the effects of population heterogeneity

and intrinsic stochasticity using Monte Carlo simulation technique. This

requires the study of non-linear differential equation with random parameters.

To this end, Monte Carlo techniques are adapted to examine the dynamic

aspects of innovation diffusion. Statistical analysis of simulation runs for Bass

and NUl Model giving the confidence interval for mean evolution of number

adopters is done. Further attempt has been made to investigate the combined

effects of intrinsic stochasticity and population heterogeneity.

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Chapter 5 addresses the diffusion of innovation in heterogeneous

population segmented into sub-groups. Stochastic model is developed to

explain the emergence the multi-model life cycle patterns. System

characteristics describing dependence among segment though entropy

framework is outlined. Stochastic evolution of total number of adopters in

population is governed by Ornstein-Uhlenbeck process and explicit computation

for the stochastic analysis of two and three interacting segments are carried out.

The emergence of unimodal and bimodal life curves through superposition of

curves in segments is illustrated for two interacting segments.

The last Chapter is concerned with validation of innovation diffusion in

segmented population. The theoretical framework with regard to innovation

diffusion in homogeneous segments has been examined to explain bimodal and

multi modal life cycle patterns using different time scales for each segment. For

validation, we have used data of TV adoption (B&W and color) in India. The

data sets exhibit bimodal life cycle patterns which are explained on the basis of

population segmented into two subgroups. It is found that the model captures

the observed patterns and fits turn out to be quite good.

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