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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
2
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
3
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
4
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.
5
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.
6
(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
7
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.
8
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.
9
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.
10
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,
12
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).
13
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.
14
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
16
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.
18
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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