chase and flight: new product diffusion with social attraction and

Chase and Flight: New Product Diffusion with SocialAttraction and Repulsion

Nitin BakshiLondon Business School, London, UK NW1 4SA, [email protected]

Kartik HosanagarThe Wharton School, University of Pennsylvania, Philadelphia, PA 19104-6340, [email protected]

Christophe Van den BulteThe Wharton School, University of Pennsylvania, Philadelphia, PA 19104-6340, [email protected]

Asymmetric attraction and repulsion between consumer segments has quickly become a major topic among

marketers and consumer researchers. For example, middle class consumers often emulate the adoption deci-

sions of the elite whereas the latter prefer to use products that help distinguish them from the middle class.

This can result in a chase-and-flight pattern where the elite starts disadopting once the product or practice

has become too popular with the outsiders emulating it. We present a mathematical model of new product

diffusion featuring asymmetric attraction and repulsion between customer segments and provide an analyti-

cal and visual characterization of the chase-and-flight diffusion dynamics. We identify potential equilibrium

outcomes and specify conditions under which the product can achieve full market potential in both market

segments. We also identify which early diffusion trajectories are likely to lead to complete acceptance by

both segments versus complete rejection by one segment. Finally, we provide insights into the effectiveness

of several marketing policies to counter the repulsion effect. A number of accepted notions about optimal

market seeding do not always apply in markets with joint attraction/repulsion.

Subject Classifications: New products; customer segmentation

Area of Review: Marketing

Key words : social contagion; word-of-mouth; adoption of innovations; chase-and-flight; new product

diffusion; marketing strategy; market seeding

History : July 2, 2013

1. Introduction

People use products as both bridges and fences (Douglas and Isherwood 1979). Consumers imitate

the choices of those they are attracted to and would like to be identified with, but also choose

products and brands to distinguish themselves from those they do not want to be confused with.

These two forces can be at odds with each other. For instance, middle-class consumers often like to

emulate the choices of the elite, but the latter dislike this and prefer products that help distinguish

them from those below them (e.g., Bourdieu 1984; Simmel 1904). Such attraction/repulsion dynam-

ics create what McCracken (1988) called a “chase-and-flight” pattern: as one group chases another

1

Bakshi, Hosanagar and Van den Bulte: Chase and Flight2 Article submitted to Operations Research; manuscript no.

group and flocks to products and practices popular with it, the second group starts deserting

these very same products and practices. So, attraction/repulsion and the resulting chase-and-flight

dynamic create a challenge for firms. They need their new products to be successful among the

elite to stimulate demand among other consumers, but success in the mass market will drive the

elite away from the product.

Understanding and managing new product growth with asymmetric attraction/repulsion

between segments is especially important for status-oriented brands. Burberry, for instance, lost

both sales and stature among its traditional upper-class customer base when English hooligans

and other “riff-raff” started wearing the iconic plaid in the late 1990s (Moon 2004). Asymmetric

attraction/repulsion dynamics may also occur in more mundane product categories and within

rather than across economic strata, especially for products and brands associated with particu-

lar lifestyles or consumer identities. Black & Decker, for instance, saw professional craftsmen and

construction workers veer away from its power tools once they became “too popular” among the

casual do-it-yourself amateurs. Porsche, the famed maker of high-performance sports cars like the

iconic 911, faced a similar problem with the introduction of the Cayenne, its first sports utility

vehicle (SUV). Market research showed that well-heeled potential buyers of SUVs were favorably

disposed towards the Porsche brand, but having it become associated with soccer moms was bound

to displease the core segment of Porsche sports-car enthusiasts (Joshi et al. 2009). Diesel Jeans,

which nurtured a very “edgy” and “alternative” image appealing to contrarians, faced similar prob-

lems when it became successful among mainstream customers (Grigorian and Chandon 2004). The

risk and deleterious consequences of such “brand hijacking” by a particular segment are especially

pronounced for products with asymmetric attraction/repulsion between segments, regardless of

whether consumers are driven by “vertical” status or “horizontal” social identity considerations.

These complex dynamics have long been discussed in sociology in general conceptual terms

(Bourdieu 1984; Simmel 1904). More recently, a wealth of laboratory evidence has appeared doc-

umenting how attraction and repulsion among customer groups affects customer preferences (e.g.,

Berger and Heath 2008; Berger and Le Mens 2009; Han et al. 2010; White and Dahl 2006). Formal

analytical work has focused on two-period settings (e.g., Amaldoss and Jain 2008), generated only

very vague results about possible diffusion trajectories (Miller et al. 1993), or has modeled the

process without allowing attractive customers to disadopt the product, hence curtailing the force

of repulsion (Joshi et al. 2009) and a central feature of the chase-and-flight phenomenon and theory

(Simmel 1904). Though the question of how attraction/repulsion between segments affects new

product acceptance is of great interest to both managers and researchers, current work offers only

limited analytic results about new product diffusion patterns in such market settings (Van den

Bulte et al. 2010).

Bakshi, Hosanagar and Van den Bulte: Chase and FlightArticle submitted to Operations Research; manuscript no. 3

Repulsion results in flight from potential adopters, but does it also lead prior adopters to flee

and disadopt? It very often does, the one exception being situations with very high switching costs.

Men may become less inclined to buy a Porsche Cayenne as the car becomes more popular among

women, but current owners—though disgruntled—may not be willing or able to simply ditch their

Cayenne car and buy a luxury SUV. So, for very expensive durable goods, repulsion need not

lead to disadoption and it is quite appropriate not to allow the first segment’s hazard rate to

turn negative (Joshi et al. 2009). Many products, services and practices, however, are either not

as expensive or durable, so repulsion can and does lead flight by prior adopters or disadoption.

The classic example is of course fashion apparel. Others include relatively inexpensive durables

that people can easily decide not to use anymore. Once earbuds become popular with mainstream

consumers, for instance, hipsters and hardcore fans of street music reverted back to old-school

over-the-ear headphones. “Cool” students have been observed to stop wearing bicycling helmets or

yellow Livestrong wristbands once they felt too may geeks had started doing so (Berger and Heath

2008). Of course, disadoption is also quite easy for services. Some pundits have voiced the concern

that, as Facebook becomes increasingly popular with adults and even elderly, teenagers will find

it attractive to migrate to other networking platforms like Twitter and Tumblr. Frequenting vs.

avoiding particular bars, fitness clubs, or restaurants is another social arena where we see the chase-

and-flight dynamic between famous, cool or other elite people and their wannabes. Yogi Berra once

explained why he no longer went to Ruggeri’s, a St. Louis restaurant, by noting: “Nobody goes

there anymore. It’s too crowded” (Berra meant, obviously, nobody who is anybody). So, a very

large class of products, services and practices, exhibit both chase and flight, both adoption and

disadoption.

We provide new insights about how a new product diffuses through such a market and about

some interventions a firm may undertake to increase its products success. To this end, we use the

following system of interlinked differential equations:

dN1

dt= (a1 + b1N1 + c1N2)(m1−N1); (1a)

dN2

dt= (a2 + b2N2 + c2N1)(m2−N2), (1b)

where the subscript i refers to one of two segments, and ai > 0, bi > 0, mi > 0, 0≤Ni ≤mi. Each

equation represents the change in user base of a new product in each of two segments (Table 1).

The c parameters capture the cross-segment influence. When c1 = c2 = 0, the model reduces to

two independent Bass (1969) model equations. When c1 = 0, c2 > 0, the model reduces to the

Asymmetric Influence Model studied by Van den Bulte and Joshi (2007). That model provides

valuable insights into new product diffusion with distinct segments of leaders and followers, but


cannot capture the attraction/repulsion effects which have gained much attention among both

practitioners and consumer researchers since their work appeared. When c1 < 0, c2 > 0, the model

represents new product growth with asymmetric attraction/repulsion between segments. Though

this model is inspired by Simmel (1904), c1 and c2 map into Veblen’s (1899) distinction between

“invidious comparison” (attempts by the high class to distinguish themselves from the lower class)

and “pecuniary emulation” (attempts by the lower class to be perceived as members of the high

class).

Table 1 Glossary of Terms

Ni number of users in segment i at time t (shorthand for Ni(t))

mi market potential for segment i

ai coefficient of innovation for segment i

bi coefficient of imitation or contagion within segment i

ci coefficient of imitation or contagion from segment j to segment i

When estimating this model to the Porsche Cayenne sales, Joshi et al. (2009) found the values

of both c1 and c2 to be significant, with the backlash or repulsion effect c1 being nearly six times

stronger than the attraction effect c2. In addition to allowing c1 to be negative, a key feature of our

analysis is that we allow the first segments hazard rate (a1 + b1N1 + c1N2) to be negative, i.e., in

accordance with Simmel’s (1904) theory of fashion of chase-and-flight, we allow the repulsion effect

to be so large that it leads members of the imitated segment to actually disadopt the product rather

than to merely refrain from further adopting it. In recent experiments, Berger and Heath (2008)

show that repulsion can indeed lead to disadoption. In one of their studies, “cool” people stopped

wearing a wristband when “geeks” started wearing them. The authors concluded that “people may

abandon tastes adopted by other social groups to avoid signaling their characteristics”. So, allowing

flight or disadoption is important for providing a more truthful mathematical representation of

sociological and behavioral theory. As we demonstrate later in this paper, accounting for the

complete process of chase and flight also impacts the equilibrium outcomes.1

Even though the system described by (1a) and (1b) does not have a closed form solution, we still

obtain important insights about a new product’s success in markets with attraction and repulsion

between segments. First, we identify possible equilibrium outcomes. Specifically, we identify under

what circumstances the “elite” or “cool” segment stays with the product rather than dropping it

once the product becomes popular with the “wannabe” segment. Simmel’s influential theory of

fashion is predicated on this dynamic, but does it always have to happen? Also, what levels of

penetration can one obtain in each segment? Is it possible to end up in a situation where no one

in the imitated segment uses the product yet all people in the wannabe segment do? We find that


irrespective of the strength of attraction and repulsion, the stable equilibrium outcomes belong

to the set {(0,m2), (m1,m2)}, where m1 and m2 are the market potential for segments 1 (“elite”)

and 2 (“wannabe”) respectively. Notably, equilibrium outcomes with intermediate values of market

penetration do arise, but these are always unstable.

Second, we identify which early diffusion trajectories are likely to lead to complete acceptance

by both segments versus complete rejection by one segment. Phase-plane analysis enables one to

clearly demarcate regions within which all trajectories converge to a particular equilibrium point,

which in turn can be used to identify the need for early managerial interventions. The boundaries

of these regions are referred to as separatrices. We describe a computational approach to plot these

separatrices. This is valuable because, in the vicinity of the separatrix, small shifts in the installed

base can lead to dramatically different equilibrium outcomes – a feature that is popularly described

as the “butterfly effect” in the study of dynamical systems and chaos theory. This underscores the

need for careful computation, but more important, it calls for careful managerial oversight in the

region proximate to the separatrix.

Finally, we discuss several marketing policies that firms can use to counter the detrimental effects

of repulsion, and investigate the effectiveness of seeding in detail. Seeding consists of converting a

small number of customers before launch to support the diffusion of the product in the imitated,

attractive segment. This jump-starts the diffusion in that segment and makes it robust to repulsion.

The analysis reveals several counter-intuitive results and demonstrates that ignoring joint attraction

and repulsion can generate incorrect policy recommendations. For example, while prior research

has revealed that the optimal seeding level is non-decreasing with the coefficient of imitation in the

seeded segment, we find that the optimal seeding level is decreasing in the coefficient of imitation

when there is a strong impeding influence from the other segment.

Our analytical results enrich current verbal theories of a phenomenon of great interest to both

managers and consumer researchers (Miller et al. 1993); show that scenarios where attraction

and repulsion perfectly cancel each other out are possible but not stable equilibria; and provide

qualitative guidance to managers about when seeding may be beneficial to overcome repulsion

effects.

2. Related Literature

Marketers have long recognized that word-of-mouth and other social influence processes affect how

quickly new products gain market acceptance. The pressure to boost the return on marketing

spending by leveraging word-of-mouth and the surge in popularity of social networking sites and

social media has put such “social contagion” dynamics at the top of the agenda of many marketing

practitioners and academics.


The great majority of new product diffusion research to date assumes that social influence

operates homogenously through the population. The logistic and Bass models, as well as most

of their extensions, assume perfect homogenous social mixing such that social contagion operates

equally across all dyads of customers (e.g., Niu 2006; Kumar and Swaminathan 2003; Ho et al.

2002). Almost all analyses also assume that the contagion effect is positive. More than half a

century ago, Leibenstein (1950) noted that the effect may be negative instead, which he referred to

as the snob effect. However, he too assumed that this “need for uniqueness” applied equally across

all customers.

Behavioral research and field studies, however, show that both susceptibility to social influence

and need for uniqueness vary across customers (e.g., Bearden et al. 1989; Iyengar et al. 2011; Tian

et al. 2001). Several strands of theory and recent empirical work suggest that social contagion may

not only vary across people (the nodes of a network), but also across pairs of people (dyads in a

network). Given practitioners’ interest in capitalizing on influential opinion leaders, several models

have been developed to investigate diffusion dynamics in markets consisting of two segments with

asymmetric influence, where opinion leaders influence followers but not the other way round (Ho

et al. 2012; Lehmann and Esteban-Bravo 2006; Muller and Yogev 2006; Van den Bulte and Joshi

2007). In these models, the cross-segment influence is positive and unidirectional—a pattern of

cross-segment influence that can be denoted as +/0. While these models allow for positive cross-

segment influence, they shed no light on the impact of repulsion or negative social influence. The

same holds for the work by Fibich and Gibori (2010) that moves beyond a simple two-segment

structure to analytically study the effect of spatial structure on diffusion dynamics.

More recently, behavioral marketing researchers have become very interested in patterns of

attraction versus repulsion. One segment aspires to look like the other segment and therefore

imitates its consumption behaviors. However, the second segment dislikes this muddling of social

identities and wants to keep itself clearly differentiated from the first segment. Hence, a product

popular in the first segment becomes less attractive to members of the second segment. The lat-

ter may therefore not want to adopt it anymore and may even start disadopting it. This pattern

cross-segment influence can be denoted as +/−, reminiscent of predator-prey models in biology.

More than a century ago, Simmel (1904) proposed such a dynamic between social classes to

explain fashion cycles: Particular styles first become popular among high status groups, which

makes them more popular among lower status groups, which in turn leads the high status groups

to move away from that that style and seek new ways to maintain its social differentiation. Many

recent behavioral studies in marketing have documented that people buy and consume particular

products both to assimilate or build bridges with others (Loveland et al. 2010; Mead et al. 2011)

and to differentiate or build fences against others (He et al. 2010; Irmak et al. 2010; White and


Dahl 2006). The fact that people can experience discomfort from loss of uniqueness (Arsel and

Thompson 2011; White and Argo 2009) leads to interesting patterns in what products they like

most. Specifically, consistent with the motive to maintain social differentiation and limit imitation,

people concerned with maintaining a distinct identity have been shown to (i) favor limited-edition

or even unpopular products (Amaldoss and Jain 2010; Berger and Le Mens 2009; Gierl and Huettl

2010), (ii) favor small rather than large brand logos and other subtle signals that only the “right”

people with sufficient cultural capital can recognize (Berger and Ward 2010; Han et al. 2010), and

even (iii) purposely curtail their own word-of-mouth about the products they bought (Cheema and

Kaikati 2010).

Practitioners are clearly aware of the marketing challenges presented by markets with asymmetric

attraction and repulsion between segments. Managing the growth of brands with strong iconic

content subject to such dynamics like Burberry, Diesel Jeans, Porsche, Red Bull, Vans, and Tommy

Hilfiger has proven especially tricky. Accounts of these brands’ struggles by Grigorian and Chandon

(2004), Kumar et al. (2004), Moon and Kiron (2002) and Moon (2004) are eerily similar to the

scenario Simmel presented a century ago. These brands first become popular among one set of

customers. That success in the original core segment subsequently spills over to other customers

who use the latter as reference or aspiration group. The success among such aspiring “wannabes”,

however, detracts from the appeal of the brand among the “originals” and may ultimately lead all of

them to drop the brand and choose another, less “overexposed” and still “authentic” iconic brand

to signify their identity (e.g., Berger and Heath 2008; Clunas 2004; Thornton 1996). As aspirants

imitate originals who seek to protect their distinctiveness, the latter disadopt the products, brands,

and cultural practices they helped make popular in the first place (Bourdieu 1984; Simmel 1904).

When new products and brands diffuse in markets characterized by the existence of multiple

segments with attraction/repulsion between them, then their diffusion dynamics and long-term

market acceptance may be much harder to predict than classical models suggest.

Attraction and repulsion between segments, though of great interest to both marketing practi-

tioners and behavioral researchers, has not been studied much using analytic models. In an early

effort to formalize trickle-down fashion theory, Miller et al. (1993) consider asymmetric attrac-

tion/repulsion. Their analysis, however, provides only a very vague result: markets characterized

by such a pattern of social influence “could display all types of fashion trends with the exception

of [discontinuous] fluctuations”(p.153). Amaldoss and Jain (2008, 2010) explicitly study asym-

metric attraction/repulsion between reference groups, but focus on identifying optimal price and

product strategy in a two-period game-theoretic setting rather than on characterizing adoption

dynamics over time. Further, they assume that the repulsive imitator segment can purchase only

in the second period, and the attractive leader segment only in the first period based on rational


expectations about the adoption behavior of the other segment. Our model, in contrast, allows for

concurrent product adoption in the two segments, tracks how diffusion evolves in continuous time,

and also allows for within-segment contagion. Joshi et al. (2009) analyze the optimal timing of

entry in two segments with asymmetric attraction/repulsion between them, which contrasts with

our focus on identifying possible equilibrium outcomes and the paths that lead to each outcome.

Another important distinction is that their model does not allow for disadoption, a restriction

which we relax as abandoning overly popular items is critical to sociological theories of distinc-

tion and chase-and-flight cultural dynamics increasingly applied to marketing (e.g., Berger and

Heath 2008; Berger and Le Mens 2009; McCracken 2008; Simmel 1904). As we demonstrate in this

paper, disadoption significantly influences equilibrium outcomes and affects managerial takeaways

regarding the impact of asymmetric attraction and repulsion between segments.

In short, how joint attraction and repulsion between segments affects new product acceptance is

of great interest to both managers and researchers. Yet, little analytic insight exists so far about

the nature of diffusion trajectories in such market settings.

3. Methodology

The diffusion model characterized by equations (1a) and (1b) does not have a closed-form solution

when both cross-effects are at work (Peterson and Mahajan 1978). Therefore, we study its behavior

by identifying the possible equilibrium outcomes and trajectories of N1 and N2. We do so using

phase plane analysis – a popular technique for studying dynamical systems (Hubbard and West

1995). The key tool in such analysis is the phase diagram, which is the path of the product growth

process in the (t,N1,N2) space projected onto the N1 −N2 plane. This section provides a brief

overview of the technique and terminology we use in our analysis. There are five main steps in the

phase plane analysis (Hubbard and West 1995) required for our purposes:

1. Identify the isoclines, i.e., the curves in the phase plane representing points at which dN1dt

=

N1′ = 0 or dN2

dt=N2

′ = 0. The isoclines for N1 are identified by the condition N1′ = 0. Substi-

tuting this into equations (1a), we get

N1 =m1, (2)

or

a1 + b1N1 + c1N2 = 0. (3)

Since c1 < 0, equation (3) can be rewritten as

N2 = (a1 + b1N1)/ |c1| . (4)

The isoclines for N2 are identified by the condition N2′ = 0. Since c2 ≥ 0, the only isocline

satisfying the required condition is N2 =m2.


2. Sketch the trajectories to visually illustrate the behavior of the dynamical system. We adopt

the convention that N1 is measured along the horizontal axis and N2 on the vertical axis of

the phase diagram. Any point on the isoclines represented by equations (2) and (4) cannot

undergo a change in N1 (since N1′ = 0). Thus, from any point on those isoclines, the N1N2

trajectory must either remain at that point (if N2′ = 0), move up (if N2

′ > 0), or move down

(if N2′ < 0) in the phase plane. Further, since c2 ≥ 0, it follows that N2

′ > 0, for all N2 <m2.

Thus, from any point on the isoclines represented by (2) and (4), the N1N2 trajectory must

move upwards as long as N2 <m2, as shown in Figure 1. Points on the isocline N2′ = 0 can

only move to the right or to the left, depending on whether N1′ > 0 or N1

′ < 0 respectively.

3. Identify the singular points, i.e., the points where the isoclines N1′ = 0 and N2

′ = 0 cross each

other. Such points are of special interest as they may, but need not, be stable equilibrium

outcomes. The two singular points in our system are (m1,m2) and (m2|c1|−a1b1

,m2). The latter

can lie within the feasible region, or outside. For the purposes of this illustrative discussion,

we focus our attention on the case in which the point lies within the feasible region (0 <m2|c1|−a1

b1<m1).We consider both cases in Section 4. The two singular points are represented

as points A and B respectively in Figure 1. Even though the boundary point (0,m2) is not

a singular point because it does not lie at the intersection of isoclines, it can nevertheless

be a candidate equilibrium outcome because of the impact of the constraint N1 ≥ 0, which

effectively renders N1′ = 0 whenever N1 = 0 and N1

′ < 0.

4. Formally characterize the stability properties of the relevant singular and boundary points.

One achieves this by slightly perturbing the system in the neighborhood of the point of

interest, and then checking the sign of the derivatives N1′ and N2

′ to determine if the system

returns to the same point (stable), or not (unstable). For the cases dealt with in this paper,

a singular point is either a stable sink or an unstable saddle point. Sinks are stable singular

points into which infinitely many trajectories converge. A sink represents a stable equilibrium

because a small perturbation will cause the system to return to the same equilibrium. Saddle

points, in contrast, are unstable singular points into which precisely two trajectories converge.

A saddle point represents an unstable equilibrium because a small perturbation can change

the resulting equilibrium.

5. Segment the phase plane into regions within which all trajectories converge to the same stable

equilibrium outcome. We provide an algorithm to plot the boundary between any two such

adjacent regions (also known as the separatrix ) in Appendix B. The following is a heuristic

explanation for the observed convergence behavior. Consider the isoclineN2 = (a1+b1N1)/ |c1|.

Points on this line have N1′ = 0. We know that N2

′ > 0 for all N2 <m2. Thus, from points

on this line, the N1N2 curve will move vertically upwards as shown by the arrows in Figure


1. Points to the left of this isocline have N1′ < 0 and N2

′ > 0 (see step 2) and thus the N1N2

curve passing through points in this region will move left and upwards as shown by the arrows.

Thus, if we start at any point in this region, then N1 will keep decreasing and N2 increasing

until we reach the boundary (0,m2). Thus, all trajectories passing through points to the left of

the isocline N2 = (a1 + b1N1)/ |c1| or on the isocline itself, will eventually converge to (0,m2).

For points to the right of the isocline we have N1′ > 0 and N2

′ > 0. Thus, the trajectory will

move right and upwards as shown by the arrows. Points that are close to point A will converge

to (m1,m2). However, that cannot be said of all points to the right of the isocline because a

trajectory can potentially cross the isocline. The curve which separates the trajectories into

two regions, depending on the final equilibrium outcome, is the separatrix (see Figure 1).

We note that the framework of phase plane analysis is quite flexible and can be easily

extended to handle multiple (> 2) consumer segments. Another setting that we have analyzed

using this framework, but do not report in order to keep the exposition simple, is the case of

mutually impeding influence, i.e., c1 < 0, c2 < 0. For example, cellular service providers often

observe such negative interaction between the mainstream and youth segments (Maier 2003).

If the brand is primarily perceived as one for teenagers, it draws many of the mainstream

customers, especially business customers, away from the service. Similarly, increased adoption

by business users impedes its diffusion among the teenager segment. The analysis and results

for the mutually impeding case are available from the authors upon request.

Figure 1 Phase Diagram

SEPARATRIX


An additional concept helpful in classifying the different types of inter-segment interaction is the

limiting hazard rate (LHRi) for product i, where i, j ∈ {1,2} and i 6= j:

LHRi = limNi→mi

Nj→mj

1

(mi−Ni)

dNi

dt= lim

Ni→mi

Nj→mj

(ai + biNi + ciNj).

LHRi is the hazard rate of the adoption process at the upper extreme of the feasible region, i.e.,

(Ni,Nj)→ (mi,mj). Although LHRi is defined only in the limit, intuitively speaking, a negative

value captures a very strong negative influence of segment j on segment i, while a positive LHRi

indicates a mild negative influence of segment j. In all future reference to LHRi, it is implied that

the term is defined in the limit.

4. Equilibrium Analysis

We analyze the various equilibrium outcomes that might be attained and identify conditions under

which both segments can achieve their full market potential.

There are two singular points, identified by determining the points of intersection of the isoclines

as represented by N1 =m1;a1 + b1N1 + c1N2 = 0;N2 =m2. These singular points are (m1,m2) and

(m2|c1|−a1b1

,m2). These points are possible equilibrium or steady state outcomes since they satisfy

the condition that dN1dt

= dN2dt

= 0. When a1 + b1m1 + c1m2 > 0, which corresponds to mild repulsion

by segment 2 on segment 1, then m2|c1|−a1b1

<m1, so the singular point (m2|c1|−a1b1

,m2) lies within the

feasible region. Conversely, when a1 + b1m1 + c1m2 < 0, which corresponds to strong repulsion by

segment 2 on segment 1, the point lies outside the feasible region. Accordingly, we consider these

two cases separately below. The case in which m2|c1|−a1b1

= m1 leads to a continuum of unstable

equilibria, and is not considered further.

4.1. Mild Repulsion (Positive LHR1)

When 0< m2|c1|−a1b1

<m1, there are two singular points within the feasible region. These are labeled

A and B in Figure 1. As explained in Section 3, the boundary point (0,m2) is of interest as well

because the constraint N1 ≥ 0 can reset N1′ = 0 whenever equation (1a) implies N1

′ < 0 at N1 = 0.

The singular point (m1,m2) is a sink or a stable equilibrium, whereas (m2|c1|−a1b1

,m2) is a saddle

point, which is unstable. (0,m2) is also a stable equilibrium because any perturbance within the

feasible region in the vicinity of this point results in the trajectory converging back to (0,m2).

Therefore, the system has only two long-term stable outcomes, namely (0,m2) and (m1,m2). This

result is formally stated in Proposition 1.

Proposition 1. For the Attraction/Repulsion model with a positive value for LHR1, the only

stable equilibrium outcomes are (0,m2) and (m1,m2).


Proof: See Appendix A.

So, with weak repulsion, all diffusion trajectories converge to either (0,m2) or (m1,m2). The

product gains full penetration in the repulsive segment, and either zero or full penetration in the

attractive segment. Intermediate levels of penetration are not a stable equilibrium—which is a

sharper result than that obtained by Miller et al. (1993).

We next investigate which type of trajectory results in what equilibrium. We do so by iden-

tifying two regions in the phase plane, one within which all trajectories converge to (0,m2) and

another within which all trajectories converge to (m1,m2). A powerful result to this end is that

the trajectories that converge to the saddle point, the so-called separatrices, separate the phase

plane into regions demonstrating different convergence behavior (Hubbard and West 1995). These

separatrices can be computed numerically for any system as described in detail in Appendix B.

A spreadsheet implementation of the algorithm for computation of the separatrix for the scenario

described in this section is also available upon request.

Figure 2 shows a plot of the phase diagram with the separatrix and several sample trajectories

for the following system with 0≤N1 ≤ 100 and 0≤N2 ≤ 100:

dN1

dt= (0.02 + 0.002N1− 0.001N2)(100−N1);

dN2

dt= (0.03 + 0.002N2 + 0.001N1)(100−N2). (5)

The order of magnitude of the parameters in (5) is consistent with prior research (e.g., Bucklin and

Sengupta 1993; Van den Bulte and Stremersch 2004). For trajectories to the left of the separatrix,

Figure 2 Phase Diagram with Separatrix for the System in (5)

SEPARATRIX

ISOCLINE


the relative size of the installed base in segment 2 to that in segment 1 is so large that the negative

influence exerted by segment 2 dominates segment 1’s growth rate. As a result, the users in segment

1 will eventually disadopt, leading to a final equilibrium outcome (0,m2). Conversely, all trajectories

to the right of the separatrix converge to (m1,m2), resulting in full penetration in both segments.

Identifying the stable equilibrium outcomes and the separatrix allows one to answer several

questions. For example, starting from (0,0), can the system converge to (m1,m2)? For the system

in (5), the location of the separatrix in Figure 2 shows that this is impossible.

4.2. Strong Repulsion (Negative LHR1)

When m2|c1|−a1b1

>m1, which corresponds to strong repulsive influence by segment 2 on segment1,

the singular point (m2|c1|−a1b1

,m2) is outside the feasible region and thus not of interest. The only

singular point in the feasible region is (m1,m2). A small perturbation of the system around the

singular point (m1,m2) will result in a negative value for dN1/dt (by virtue of negative LHR1),

and thus the trajectory will not return to (m1,m2). This suggests that the singular point (m1,m2)

may not be a stable equilibrium point.

The phase diagram for this case can be constructed following the steps described in Section 3,

and is shown in Figure 3. Since N2′ > 0 for all N2 <m2, a trajectory starting from any point in

the feasible region will eventually be above the isocline N2 = (a1 + b1N1)/|c1|. Once the trajectory

is above the isocline, we have N1′ < 0 and N1 decreases until it reaches zero. As a result, the only

possible long-term outcome is (0,m2). In the presence of strong repulsion, the long-term outcome

is that the product has no penetration whatsoever in the attractive segment yet full penetration

in the segment that tries to imitate them. This is stated formally in our next result.

Proposition 2. For the Attraction/Repulsion model with a negative value for LHR1, the only

stable equilibrium outcomes is (0,m2) .

Proof: See Appendix A.

4.3. Implications of Equilibrium Analysis

The results show that, in markets with joint attraction and repulsion between segments, the new

product gains full penetration in the repulsive segment, and either zero or full penetration in the

attractive segment. Intermediate levels of penetration cannot be a stable equilibrium. Also, full

penetration in both segments is possible only if the repulsion is sufficiently weak. This confirms

the concerns of managers at companies operating in such markets, like Burberry and Diesel.

Marketing analysts can use the technique described above to identify scenarios in which the

product will achieve its full market potential in both segments or not. If the system parameters


Figure 3 Phase Diagram with Separatrix for Cases with Strong Repulsion

result in a phase diagram as in Figure 3, then full market penetration (m1,m2) will never mate-

rialize. If, in contrast, the situation is as in Figure 2, full-market-penetration is possible but not

guaranteed.

The analysis may also prevent managers becoming lulled in a false sense of security. Consider

the diffusion of the product captured in Figure 3. For a trajectory in the region below the isocline

(where N1′ < 0), the diffusion within segment 1 may seem to be proceeding quite smoothly early

on, but it will reverse once adoption in segment 2 is sufficiently high. Phase plane analysis may help

protect managers from putting too much faith in early success and believing that the penetration

level will continue to grow.

Finally, the analysis can also be used to identify conditions that managers may want to create in

order to realize the product’s full potential. Lets turn again to Figure 2. A diffusion process that

starts at its “natural” initial state (0,0) will not reach (m1,m2). Nor will one that starts at (2,0).

But one that starts at (10,0) will reach (m1,m2) without any other intervention. So, phase plane

analysis can determine whether or not some initial intervention, such as seeding the process with

initial converts, may actually help achieve full market penetration.

So, equilibrium analysis can help marketing analysts not only determine the extent to which

repulsion is a cause for concern but also assess how strong a marketing intervention is necessary

to steer the process towards full market penetration.


5. Optimal Seeding with Attraction/Repulsion

Marketers increasingly try to leverage the power of word-of-mouth to boost the return on their

marketing expenditures. As a result, the idea of “seeding” or converting a small number of cus-

tomers before launch to help achieve deeper or faster market penetration is currently receiving

much attention from marketers. The practice has a long history in the pharmaceutical and med-

ical devices industries, and has become popular in consumer goods and services, as reflected in

the emergence of “viral-for-hire” companies like BzzAgent, SheSpeaks, and Tremor. Seeding has

also become the subject of quite a bit of academic research (e.g., Aral et al. 2011; Bakshy et al.

2011; Berger and Schwartz 2011; Godes and Mayzlin 2009; Goldenberg et al. 2009; Hinz et al.

2011; Iyengar et al. 2011; Jain et al. 1995; Kitsak et al. 2010; Lehmann and Esteban-Bravo 2006;

Toubia et al. 2010). While much of this work focuses on identifying the most influential seeding

points in network structures more complex than the two-segment work we analyze, research to

date has ignored the possibility of attraction/repulsion dynamics between segments. As the results

presented here show, the presence of repulsion dynamics can greatly affect not only the optimal

level of seeding but even whether one wants to engage in seeding at all.

An important consequence of repulsion is that seeding can affect a firm’s profits not only by

accelerating the diffusion of the new product but also by affecting the odds of reaching full market

penetration. To illustrate, let’s turn again to Figure 2. A diffusion process that starts at its “natural”

initial state (0,0) will not reach full market penetration (m1,m2) but a seeded process starting at

any point right of the separatrix, like (10,0), will.

Jain et al. (1995) investigated the optimal seeding level in a single-segment setting and found

that optimal levels are high when the coefficient of innovation is low or that of imitation is high.

Lehmann and Esteban-Bravo (2006) investigated a two-segment setting with asymmetric attraction

(+/0) influence, and found that the optimal seeding in the attractive segment decreases as its

coefficient of innovation increases and that it first decreases and then increases as its coefficient of

imitation increases. We extend these analyses by investigating optimal seeding for a two-segment

setting with attraction/repulsion between the segments.

We assume that the product is launched in both segments simultaneously. Seeding occurs in

segment 1whose propensity to adopt is impeded by the penetration in segment 2. The purpose of

seeding, obviously, is to jump start the process in segment 1 so that its own positive within-segment

imitation effect is big enough to withstand the negative repulsion effect. We restrict our analysis

to the commonly studied situation where the firm seeds the market only at time 0 and does not

intervene with additional seeding later in the diffusion process. The decision problem in a discrete

time period analysis is:


maxNS

1

π= maxNS

1

∞∑t=1

δt−1(p1−u1)(dN1/dt)dt+∞∑t=1

δt−1(p2−u2)(dN2/dt)dt− (s1N1S), (6)

where π is the net present value (NPV), δ is a discount factor, pi is the price charged to segment i,

ui is the unit cost of manufacturing the product, s1 is the unit cost of generating a seed in segment

1, and NS1 is the number of seeded customers in segment 1. This formulation is the same as in Jain

et al. (1995), with the exception that there are two consumer segments in (6). We investigate the

solution numerically using parameters within the same range as those used in Jain et al. (1995).

In order to focus on the impact of repulsion c1, we assume that c2 = 0. The discount factor is

δ = 1/1.08 = 0.926, and s1 = ui = 0 so pi can be interpreted as profit margin instead of price.

Note that seeding is costly even when s1 = 0, because of the lost revenue p1NS1 from the seeded

customers. We evaluate the value of dN1/dt and dN2/dt using the Euler approximation with step

size 1.2 The parameter settings are summarized in Table 2.

Table 2 Parameter Settings

Parameter Segment 1 Segment 2

Innovation (a) 0.005:0.03 0.03

Within-segment Contagion (b) 0.002 0.0018

Cross-segment Contagion (c) 0:-0.002 0

Market Potential (m) 100 100

Price (p) 1 1

Unit Cost (u) 0 0

Table 3 provides the results for the optimal seeding level for various values of a1 and c1. The

shaded cells indicate that optimal seeding shifted the equilibrium outcome from (0,m2) to full

penetration (m1,m2). For cells to the left of the shaded area, the equilibrium was (m1,m2) without

seeding and remained so after optimal seeding. For cells to the right, the equilibrium remained

(0,m2). So, seeding can improve profits both by accelerating diffusion and by shifting the equilib-

rium from zero to full penetration in the attractive segment.

Reading the table from left to right provides insights into the effect of repulsion on seeding.

For a specific coefficient of innovation (a1), the level of optimal seeding initially increases as c1

becomes more negative. This is because more seeding points are needed to help accelerate diffusion

in the face of repulsion. In the shaded area, this acceleration results in a shift of equilibrium to

full penetration. Of particular interest is the decrease in seeding once c1 becomes very negative.

Consider the top right corner of the Table where the optimal level of seeding is nil. When the

attractive segment has a low intrinsic tendency to adopt (low a1) and is severely repulsed by the


Table 3 Optimal Seeding Levels

Parametersc1

0 -0.0005 -0.001 -0.0015 -0.0020.005 6 11 16 26 00.006 6 10 16 25 00.007 5 10 15 24 0

0.00766 5 9 15 24 00.00767 5 9 15 24 21

a1 0.008 5 9 15 24 210.009 4 9 14 23 210.01 4 8 14 22 210.015 1 6 11 19 190.02 0 3 8 16 170.025 0 1 6 13 150.03 0 0 3 10 14

Note: In the shaded cells, optimal seeding shifted the equilibrium outcome from (0,m2) to (m1,m2). For cells to theleft of the shaded area, the equilibrium was (m1,m2) both before and after optimal seeding. For cells to the right,the equilibrium was (0,m2) both before and after optimal seeding.

other segment (highly negative c1), then effective seeding may become so expensive that the firm

prefers to forego seeding entirely. This explains the zero values in the top right entries in Table 3.

Reading the table from top to bottom provides insights into how the tendency to adopt regardless

of social dynamics (a1) affects optimal seeding. For c1 = 0, the optimal seeding levels decrease

with the coefficient of innovation, as reported in Jain et al. (1995). The reason is that seeding is a

substitute for the tendency to adopt spontaneously. The results are qualitatively similar for mild

repulsion, i.e., low negative values of c1 . However, for highly negative values of c1 shown in the

far right column of Table 3, optimal seeding levels may be zero when the coefficient of innovation

is low, then jump quite markedly to a high level beyond a threshold value for the coefficient of

innovation, and finally decline again as the coefficient of innovation increases further. The zero

entries occur because a very large number of seeds may be needed in order to sufficiently accelerate

diffusion when a1 is very small and c1 very negative. As a result, seeding is prohibitively expensive.

However, beyond a certain threshold value for a1, it becomes profitable to generate seeding points

among consumers in segment 1. In summary, unlike the findings reported in Jain et al. (1995)

in the single segment case and Lehmann and Lehmann and Esteban-Bravo (2006) in the (+/0)

case, optimal seeding need not always decrease with the coefficient of innovation in settings with

negative interaction.

Next, we investigate the impact of the coefficient of within-segment imitation (b1) on the optimal

level of seeding. The parameters for the simulation are reported in Table 4 and the results in Table

5. The shaded cells in the latter indicate that optimal seeding shifted the equilibrium outcome from

(0,m2) to full penetration (m1,m2). For cells to the left of the shaded area, the equilibrium remained

(m1,m2) regardless of optimal seeding, and for the cells to the right it remained (0,m2). Here


again, seeding can improve profits both by accelerating diffusion and by shifting the equilibrium

from zero to full penetration in the attractive segment.

Reading the table from top to bottom shows the impact of within-segment imitation. In the

absence of repulsion (c1 = 0), the optimal seeding level is non-decreasing with the coefficient of

imitation, as reported by Jain et al. (1995). This is because WOM complements seeding: The

higher the imitation, the higher the return obtained from the seeding. However, when there is an

impeding influence from segment 2 (c1 < 0), the optimal number of seeding points is not increasing

but typically decreasing in the coefficient of imitation. This reversal occurs because, the stronger

the endogenous feedback is within the attractive segment, the less the firm needs to engage in costly

seeding to protect the diffusion process within that segment from being harmed by the negative

cross-segment effect. However, there is an exception to this pattern. When the within-segment

imitation is weak but the cross-segment repulsion is strong, which corresponds to the top right area

in Table 5, very large number of seeds may be needed in order to change the equilibrium outcomes.

Such a high level of seeding can be prohibitively expensive, so the optimum is not to use seeding at

all. This result is consistent with the results in Table 3. So, unlike the results reported by Lehmann

and Esteban-Bravo (2006) for the (+/0) case without repulsion, we find that the optimal level

of seeding may be decreasing smoothly, or show abrupt upward jump points followed by smooth

declines, as the coefficient of imitation increases.


Parameter Product 1 Product 2Innovation (a) 0.02 0.03Within-segment Contagion (b) 0.001: 0.005 0.002Cross-segment Contagion (c) 0:-0.002 0Market Potential (m) 100 100Price (p) 1 1Unit Cost (u) 0 0

Seeding counters the detrimental effect of repulsion and by doing so often shifts the long term

equilibrium. As a result, the optimal level of seeding can be much higher in markets with inter-

segment attraction and repulsion than in markets with homogeneous consumers. For example, Jain

et al. (1995) report that the maximum seeding level observed in their analysis was never higher

than 9%. However, for similar parameters, we find that the maximum seeding levels in the presence

of strong inter-segment interaction can exceed 20%. This is because the role of seeding is not only

to accelerate diffusion, but also to mitigate cross-segment repulsion.


Table 5 Optimal Seeding Levels

Parametersc1

0 -0.0005 -0.001 -0.0015 -0.0020.001 0 7 0 0 00.0015 0 4 15 0 00.002 0 3 9 17 170.0025 0 3 6 10 16

b1 0.003 1 3 5 8 110.0035 1 3 4 6 80.004 2 3 4 5 60.0045 2 3 4 4 50.005 2 3 3 4 5

Note: In the shaded cells, optimal seeding shifted the equilibrium outcome from (0,m2) to (m1,m2). For cells to theleft of the shaded area, the equilibrium was (m1,m2) both before and after optimal seeding. For cells to the right,the equilibrium was (0,m2) both before and after optimal seeding.

6. Other Policies to Manage Repulsion

As we have shown, seeding can be used to counter the negative effect of repulsion. While this

particular marketing policy is receiving quite a bit of attention, there are also other policies that

can help marketers counter repulsion profitably.

The first policy is to weaken the repulsive effect itself. Decreasing the extent to which success

in the repulsive segment detracts from the product’s appeal in the attractive segment amounts

to making the c2 parameter less negative. Many companies attempt to achieve such “segment

decoupling” by targeting distinct product designs to different segments. Cheaper lines featuring

large brand logos are targeted towards the mass market, whereas expensive designs featuring much

more subtle signals are targeted towards the attractive segment (Berger and Ward 2010; Han et al.

2010). Our analysis suggests two additional policies that could be used effectively.

The second policy is to launch the product sequentially, first in the attractive segment and only

later in the repulsive segment. The idea is similar to that of seeding: Build an early installed base

in the attractive segment that can withstand the negative effect from the other segment. Like

seeding, delaying the launch in segment 2 allows the diffusion process in segment 1 to gain enough

momentum such that it does not get diverted away from full penetration by the repulsion effect.

How entry timing can optimally balance attraction and repulsion in order to avoid disadoption was

already studied by Joshi et al. (2009).

The third policy consists of purposely limiting the market size of the customer segment harming

product adoption in the other segment. This third alternative warrants a bit more discussion,

since it has not been evaluated in the literature. In Section 4.2 we showed that when LHR1 is

negative, the only stable equilibrium outcome is (0,m2). In such cases, seeding will not be helpful in

achieving (m1,m2) as a stable outcome. Also, when the repulsive effect is very high, mere product


versioning may not be effective in reducing c2 sufficiently. Limiting the market size of the segment

exerting the negative influence can still be effective in such extreme cases. Restricting m2 to m′2,

where m′2 < m2, can make (m1,m′2) a stable equilibrium outcome that is more attractive than

(0,m2). Such a policy can be implemented in several ways, including limiting the distribution of

the product to select channels rarely patronized by the impeding segment. For instance, Diesel

and Burberry have limited the effective access of their products to “wannabes” by restricting

distribution to certain exclusive channels. Such selective distribution is known to be practiced by

other luxury goods retailers as well; see Buettner et al. (2009a,b) for a related discussion and

associated anti-competition concerns. Adding features with no functional utility to the product, or

offering limited editions are other ways to control the availability of the product to the impeding

segment (Amaldoss and Jain 2008).

We now provide a numerical illustration. Table 6 lists the parameter values for the two segments.

The discount factor is set to δ = 0.926. Table 7 reports the results. Segment 2 exerts so much

negative influence on segment 1 that, in the absence of any intervention, the only stable equilibrium

is (0,100). Reducing the market potential for segment 2 from 100 to 50 increases the total discounted

profit by 22.1% (from 54.42 to 66.46). The new equilibrium is (100,50) and thus the product reaches

and maintains full penetration in the attractive segment.


Parameter Segment 1 Segment 2Innovation (a) 0.02 0.03Within-segment Contagion (b) 0.003 0.001Cross-segment Contagion (c) -0.002 0.002Market Potential (m) 100 100Price (p) 1 1Unit Cost (u) 0 0

Table 7 Results for Reduced Market Size

Base Case Demand ControlMarket Potential (m2) 100 50

Is (m1,m2) stable? No YesProfit 54.42 66.46

7. Conclusion

Attraction and repulsion dynamics are quite important for status-oriented products and brands,

and have become of considerable interest to marketing scientists due to recent experimental and


empirical evidence (e.g., Amaldoss and Jain 2008; Berger and Heath 2008; Joshi et al. 2009). We

present several novel insights about new product growth in such markets.

(i) With weak repulsion, the product always gains full penetration in the repulsive segment, but

achieves either zero or full penetration in the attractive segment in stable equilibrium. Intermediate

levels of penetration are not a stable equilibrium. With strong repulsion, the product always gains

full penetration in the repulsive segment and zero penetration in the attractive segment in stable

equilibrium. This is a sharper result than that obtained by Miller et al. (1993) in their earlier

analysis of attraction and repulsion. It also corroborates marketers’ concerns about the detrimental

effect of repulsion.

(ii) One can compute separatrices to separate trajectories that converge to different equilibria.

Separatrix analysis allows analysts to identify which scenarios will evolve to full penetration in

both segments without special managerial intervention. Separatrices can be numerically computed

for any system, as described in detail in Appendix B. MATLAB code is available (e.g., Polking

and Arnold 2003; http://math.rice.edu/ polking/odesoft/), and the algorithm for computing the

separatrix can also be implemented in a spreadsheet. An Excel implementation for the scenario

described in section 4.1 is available upon request.

(iii) Seeding can be an effective marketing policy to counter the effects of repulsion.

(iv) Repulsion affects how the optimal level of seeding in the attractive segment changes with

the coefficient of innovation in that segment. In traditional markets without repulsion, it is optimal

to decrease seeding when the coefficient of innovation increases, so seeding is a substitute for spon-

taneous independent adoption. In markets with repulsion, this does not always hold. Specifically,

when the repulsion is very strong, it can be optimal not to engage in any seeding whatsoever even

as the coefficient of innovation increases, until it reaches a threshold after which the traditional

substitution pattern holds.

(v) Repulsion reverses how the optimal level of seeding in the attractive segment changes with

the coefficient of imitation. In traditional markets without repulsion, it is optimal to increase

seeding when the coefficient of imitation increases, so seeding is a complement to social contagion.

In markets with repulsion, the reverse is typically true. Provided the within-segment contagion

reaches a minimum threshold, higher contagion implies less rather than more seeding. The reason is

that, in markets with repulsion, the benefits of seeding lie less in leveraging contagion to accelerate

diffusion than in quickly building an installed base to counter repulsion.

Just as with the coefficient of innovation, there is a discontinuity. Specifically, when the repulsion

is very strong, it can be optimal not to engage in any seeding whatsoever unless the coefficient of

imitation reaches a threshold after which seeding and contagion act as substitutes.


(vi) Firms operating in markets with attraction and repulsion may want to consider a fourth

marketing policy besides seeding, decoupling the two segments through product versioning

(Berger and Ward 2010; Han et al. 2010), and launching the product sequentially rather than

simultaneously (Joshi et al. 2009). Limiting the supply to the second segment, possibly through

channel selection or limited editions, offers another means to profitably reach full penetration in

the attractive segment even when strong repulsion makes seeding ineffective.

A key consideration driving our results is that our analysis allows for disadoption, consistent

with Simmel’s (1904) early theoretical insight and subsequent experimental evidence about joint

attraction and repulsion (e.g., Berger and Heath 2008). Our results suggest that the causes and

consequences of disadoption, a phenomenon hardly studied to date, merit more research. With-

out the possibility of decreases in the user base, growth trajectories subject to strong repulsion

would not revert back towards the zero penetration in the attractive segment and settle at (0,m2).

Instead, the trajectory in segment 1 would likely “freeze” at a particular level of N1. So, the pos-

sibility of disadoption drives the notable insight that partial market penetration is not a stable

equilibrium outcome. Also notable is that, unlike situations without joint attraction and repulsion,

the optimal level of seeding may show abrupt jumps followed by smooth declines as the coefficients

of innovation and imitation increase. The possibility of disadoption can be important here as well.

In situations with repulsion and disadoption, seeding not only smoothly accelerates diffusion but

also can more drastically redirect the diffusion trajectory from zero penetration to full penetration

in the attractive segment. How disadoption can lead to a stable equilibrium characterized by zero

penetration in the attractive segment also raises the question whether policies facilitating disadop-

tion, such as leasing rather than selling in the case of Porsche cars, may in some cases lead to less

desirable equilibrium outcomes.

Our study has limitations some of which present additional opportunities for future work. First,

there are other ways conceivable to model attraction/repulsion dynamics. The rationale for the

specification used here is that it combines the classic insight of Simmel (1904) on attraction and

repulsion with the classical new product diffusion modeling approach. This allows us to contrast

our seeding results with those for the classical model (Jain et al. 1995) and for its extensions incor-

porating asymmetric leader-follower attraction without repulsion (Lehmann and Esteban-Bravo

2006). Our model also is consistent with the empirically validated framework proposed in Joshi

et al. (2009). Finally, the framework we have introduced can be easily scaled to handle multi-

ple consumer segments (and can also be adapted to handle mutually impeding influence between

segments). The model and analysis, we believe, capture key features of attraction/repulsion. How-

ever, future research may fruitfully explore the robustness of our results using other mathematical

formulations.


All the marketing policies we studied or discussed (seeding, decoupling, sequential launch, limit-

ing supply) pertained to interventions at launch. Since our framework is amenable to the analysis

of dynamic intervention strategies, some researchers may be enticed to study time-varying policies,

such as controlling price or advertising continuously. Rigorously embedding such considerations

into a diffusion model requires applying optimal control theory. The absence of an analytical solu-

tion for the diffusion model with joint attraction/repulsion and our experience with trying to apply

control theory to the simpler asymmetric influence model without repulsion, however, make us pes-

simistic about the prospects of such an endeavor. Other approaches, such as approximate dynamic

programming, may prove fruitful in evaluating dynamic policies.

Finally, it could be interesting to study the situation where repulsion operates not only from

segment 2 to segment 1, but also within segment 2. Such a need for uniqueness within the low-

end segment, though not consistent with Simmel’s theory that motivated our work, can easily be

accommodated within our model by setting b2 < 0.

Joint attraction/repulsion dynamics are increasingly recognized as an important feature of con-

sumer behavior. Though noted more than a century ago by Simmel, it is only very recently that

consumer researchers have started to appreciate how important these dynamics are. Our analysis

shows that several accepted notions of long-run outcomes and effective marketing policies need not

hold in markets with attraction/repulsion dynamics. Practice as well as theory will benefit from

further investigations of this phenomenon.

Endnotes

1. Our model accommodates churn in segment 1 induced by popularity in segment 2. Let dN1/dt=

f(N1)− g(N1), where f represents the number of adoptions and g the number of disadoptions in

segment 1. Setting f(N1) = (a1 +b1N1−cN2)(m1−N1) and g(N1) =N1r(N1) =N1kN2(m1/N1−1)

leads to eq. (1a) with c1 =−(c+ k). So, the negative cross-segment impact of N2 on N1 operates

through both adoptions and disadoptions in segment 1. Note how the churn rate r has the desirable

property of increasing with N2 in such a fashion that higher levels of popularity among fellow

segment 1 members (N1) make one less sensitive to popularity among segment 2 members (N2).

2. More sophisticated numerical techniques do not change the qualitative nature of the results.

Some representative calculations in the parameter range of interest show that a 90% reduction in

step size lead to less than 1.75% change in undiscounted revenue.

Acknowledgments

The authors benefitted from discussions with Prof. Charles Epstein, Department of Mathematics, University

of Pennsylvania. Part of this research was conducted while the third author was Visiting Scholar at Columbia

University.


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Appendix A: Proofs

Lemma 1. For the dynamical systems described in (1a) and (1b), stability analyses of the singular

points yield the following:

(i) Point A, or (m1,m2), is a singular point for any set of values of the hazard rate parameters.

It is stable if and only if both LHR1 and LHR2 are positive, and LHR1 6=LHR2.

(ii) Point B, or (−a1+m2c1b1

,m2) with c1 < 0, is a singular point if and only if LHR1 is positive.

Furthermore, it is a saddle point, whenever it qualifies as a singular point.

(iii) Point C, or (m1,−a2+c2m1b2

) with c2 < 0, is a singular point if and only if LHR2 is positive.

Furthermore, it is a saddle point, whenever it qualifies as a singular point.

(iv) Point D, or ( |c1|a2+a1b2|c1||c2|−b1b2

, a1|c1|

+ b1|c1|

( |c1|a2+a1b2|c1||c2|−b1b2

)) with c1 < 0 and c2 < 0, is a singular point when-

ever it is within the feasible region (i.e.,0< |c1|a2+a1b2|c1||c2|−b1b2

<m1 and 0< a1|c1|

+ b1|c1|

( |c1|a2+a1b2|c1||c2|−b1b2

)<

m2). Furthermore, it is a saddle point, whenever it qualifies as a singular point.

NOTE: New product growth with asymmetric attraction/repulsion between segments implies c1 <

0, c2 > 0, so points C and D are not relevant for the analysis.

Proof: The system of equations (1a) and (1b) can be represented as:dN1

dt= f(N1,N2);

dN2

dt=

g(N1,N2); N1 ≥ 0;N2 ≥ 0. In order to study the stability properties of the singular points of this

non-linear system of equations, we will have to linearize it in the region around the singular point.

Then we can use standard techniques to analyze the stability of singularities in linear systems.

ssssssssss

Bakshi, Hosanagar and Van den Bulte: Chase and Flight

28 Article submitted to Operations Research; manuscript no.

ssssssssss

We do this as follows. Since f and g are twice continuously differentiable, we can expand them in

a Taylor polynomial around the singular point (N10,N2

0):

f(N1,N2) =∂f

∂N1

∣∣∣∣(N1

0,N20)

(N1−N10) +

∂f

∂N2

∣∣∣∣(N1

0,N20)

(N2−N20) +P (N1−N1

0,N2−N20);

g(N1,N2) =∂g

∂N1

∣∣∣∣(N1

0,N20)

(N1−N10) +

∂g

∂N2

∣∣∣∣(N1

0,N20)

(N2−N20) +Q(N1−N1

0,N2−N20),

where P (N1 −N10,N2 −N2

0) and Q(N1 −N10,N2 −N2

0) are functions that comprise terms that

are at least quadratic, or higher order, in the arguments (N1 −N10) and (N2 −N2

0). Near the

singular point, these terms will be negligibly small compared to the linear terms, and hence we

can study the behavior of the non-linear system near its singular points, by making the following

change of coordinates:

x1 =N1−N10;x2 =N2−N2

0,

to get

dx1

dt=

∂f

∂N1

∣∣∣∣(N1

0,N20)

x1 +∂f

∂N2

∣∣∣∣(N1

0,N20)

x2;

dx2

dt=

∂g

∂N1

∣∣∣∣(N1

0,N20)

x1 +∂g

∂N2

∣∣∣∣(N1

0,N20)

x2.

The standard form of a linear system of differential equations is dx/dt=Ax, for vector x. Then

A=

∂f

∂N1

∣∣∣∣(N1

0,N20)

∂f

∂N2

∣∣(N1

0,N20)

∂g

∂N1

∣∣∣∣(N1

0,N20)

∂g

∂N2

∣∣(N1

0,N20)

.

In the above matrix:

∂f

∂N1

∣∣∣∣(N1

0,N20)

= b1m1− a1− c1N20− 2b1N1

0;∂f

∂N2

∣∣∣∣(N1

0,N20)

= c1(m1−N10);

∂g

∂N1

∣∣∣∣(N1

0,N20)

= c2(m2−N20);

∂g

∂N2

∣∣∣∣(N1

0,N20)

= b2m2− a2− c2N10− 2b2N2

0.

Next, we compute the eigenvalues of the matrix A by solving Det(A−λI) = 0. This gives us the

quadratic equation:

λ2−λ[b1m1− a1 + b2m2− a2− (c2 + 2b1)N1

0− (c1 + 2b2)N20]

+ (b1m1− a1− c1N20− 2b1N1

0)(b2m2− a2− c2N10− 2b2N2

0)− c1c2(m1−N10)(m2−N2

0) = 0.


Article submitted to Operations Research; manuscript no. 29

Let the eigenvalues corresponding to a singular point be denoted by λ1 and λ2. A singular point is

a saddle point if the eigenvalues are of opposite sign. The singular point is a sink if the eigenvalues

are distinct and negative (Hubbard and West 1995). We now determine the eigenvalues associated

with each of the 4 identified singular points.

(i) Stability analysis of A, or (m1,m2):

Solving the quadratic equation at (N10,N2

0) = (m1,m2), we get the following eigenvalues:

λ1 =−(a1 + b1m1 + c1m2) =−LHR1;

λ2 =−(a2 + b2m2 + c2m1) =−LHR2.

Under the conditions 0< m2|c1|−a1b1

<m1 and 0< |c2|m1−a2b2

<m2, which are also implied by a

positive value for LHR1 and LHR2, we can easily see both eigenvalues are strictly negative,

and if we assume ssssssssss

them to be distinct then we can conclude that (m1,m2) is a sink. (Equal eigenvalues imply

a degenerate node.) At the same time, when either LHR1 or LHR2 is negative (at least one

eigenvalue is positive), then (m1,m2) is a saddle point.

(ii) Stability analysis of B, or (−a1 +m2c1b1

,m2):

When c1 > 0, this singular point does not lie in the feasible region and is thus not relevant.

We therefore focus on the case in which c1 < 0. The singular point can be rewritten as

(m2|c1|−a1b1

,m2). After solving the quadratic equation, the eigenvalues are:

λ1 = b1m1 + a1− |c1|m2;

λ2 =−[b2m2 + a2 +c2b1

(−a1 + |c1|m2)].

Under the condition 0< m2|c1|−a1b1

<m1 (i.e., singular point B is in the feasible region), we can

clearly see that λ2 < 0< λ1. The condition for B to lie in the feasible region is also implied

by a positive LHR1. Hence, the singular point, if it exists in the feasible region, is always a

saddle point.

Lemma 2. For the asymmetric attraction/repulsion model, if |c1|< (m1b1 +a1)/m2, then (m1,m2)

is a stable equilibrium point (a sink) and (m2|c1|−a1b1

,m2) is an unstable equilibrium point (a saddle

point) for the dynamic system represented by (1a) and (1b).

Proof: Follows from Lemma 1-(i) and Lemma 1-(ii). QED.

ssssssssss


30 Article submitted to Operations Research; manuscript no.

ssssssssss

ssssssssss

Lemma 3. For asymmetric attraction/repulsion between the segments (i.e., c1 < 0, c2 ≥ 0), if |c1|<

(m1b1 + a1)/m2, then limt→∞(N1(t),N2(t)) ∈ {(m1,m2), (0,m2), (m2|c1|−a1

b1,m2)}. Further, because

((m2 |c1| − a1)/b1,m2) is a saddle point, the only stable equilibrium outcomes are (0,m2) and

(m1,m2).

Proof: At (0,m2), and in its vicinity, the sign of dN1/dt is negative and that of dN2/dt is positive.

As a result, a slight perturbation in location from (0,m2) to a nearby location in within the

feasible region on the phase plane will result in the trajectory returning to (0,m2). So even though

(0,m2) is not a singular point, the constraints N1 ≥ 0 and N2 ≤m2, ensure that it behaves like

one. All trajectories in the neighborhood of (0,m2) will converge to it in the limit as t→∞. This

observation, along with Lemma 2 proves the result. QED.

Lemma 4. For asymmetric attraction/repulsion between the segments or products (i.e., c1 < 0, c2 ≥

0), if |c1|> (m1b1 +a1)/m2, then (m1,m2) is a saddle point for the system represented by (1a) and

(1b).

Proof: Follows from Lemma 1-(i). QED.

Lemma 5. For asymmetric attraction/repulsion between the segments or products (i.e., c1 < 0, c2 ≥

0), if |c1| > (m1b1 + a1)/m2, then limt→∞(N1(t),N2(t)) ∈ {(m1,m2), (0,m2)}. Further, because

(m1,m2) is a saddle point, the only stable equilibrium outcome is (0,m2).

Proof: All trajectories in neighborhood of (0,m2) will converge to it in the limit as t→∞, by

a reasoning similar to the one presented in the proof of Lemma 3. This observation, along with

Lemma 4, proves the result. QED.

Proof of Proposition 1

Follows immediately from Lemma 2 and Lemma 3. QED.

Proof of Proposition 2

Follows immediately from Lemma 4 and Lemma 5. QED


Appendix B: Numerical Computation of the Separatrix

Exactly two N1N2 trajectories pass through a saddle point, which is an unstable singular point.

The paths traced out by the incoming trajectories corresponding to the negative eigenvalues are

the Separatrices for the dynamical system under study. Below, we outline an algorithm to plot

the Separatrices associated with a given saddle point. We also illustrate the steps for the following

dynamical system corresponding to the Attraction/Repulsion Model described in Section 3.1:

dN1

dt= (0.02 + 0.002N1− 0.001N2)(100−N1);

dN2

dt= (0.03 + 0.002N2 + 0.001N1)(100−N2).

1. Calculate the eigenvalues associated with the saddle point, (N10,N2

0), being analyzed, using

the quadratic equation described in the proof of Lemma 1:

λ2−λ[b1m1− a1 + b2m2− a2− (c2 + 2b1)N1

0− (c1 + 2b2)N20]

+(b1m1−a1−c1N20−2b1N1

0)(b2m2−a2−c2N10−2b2N2

0)−c1c2(m1−N10)(m2−N2

0) = 0.

For the above system, the saddle point is (40,100). The two eigenvalues are 0.12 and -0.27

and the corresponding eigenvectors are (1,0) and (0.1,0.65).

2. Identify a ‘starting point’ which is slightly displaced from the saddle point in the direction

of the eigenvector corresponding to the negative eigenvalue. Recall that for a saddle point

λ2 < 0<λ1. For the above system, we identify a starting point which is slightly displaced from

(40,100) in the direction (0.1,0.65) while ensuring we are within the feasible region. This gives

us the starting point (39.9,99.35).

3. Using the discretized version of the system of equations described in (1a) and (1b), and the

‘starting point’ identified in step 2, trace out the path (N1N2 trajectory) obtained by moving

in positive time, as well as that obtained by moving in negative time. For the former, the

displacement of the starting point should be along the direction of the eigenvector obtained

in step 2, while for the latter the displacement should be in the opposite direction. The path

obtained is the Separatrix of interest.

The accompanying spreadsheet (available on request) demonstrates the detailed implementation

of these steps for the above dynamical system. MATLAB code is available as well (e.g., Polking

and Arnold 2003; http://math.rice.edu/ polking/odesoft/).

chase and flight: new product diffusion with social attraction and

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