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Article Excerpt
We model the diffusion of innovations in markets with two segments: influentials who are more in touch with new developments and who affect another segment of imitators whose own adoptions do not affect the influentials. This two-segment structure with asymmetric influence is consistent with several theories in sociology and diffusion research, as well as many "viral" or "network" marketing strategies. We have four main results. (1) Diffusion in a mixture of influentials and imitators can exhibit a clip or "chasm" between the early and later parts of the diffusion curve. (2) The proportion of adoptions stemming from influentials need not decrease monotonically, but may first decrease and then increase. (3) Erroneously specifying a mixed-influence model to a mixture process where influentials act independently from each other can generate systematic changes in the parameter values reported in earlier research. (4) Empirical analysis of 33 different data series indicates that the two-segment model fits better than the standard mixed-influence, the Gamma/Shifted Gompertz, and the Weibull-Gamma models, especially in cases where a two-segment structure is likely to exist. Also, the two-segment model fits about as well as the Karmeshu-Goswami mixed-influence model, in which the coefficients of innovation and imitation vary across potential adopters in a continuous fashion.
Key words: asymmetric influence; diffusion of innovations; innovation; market segments; social contagion; social structure
1. Introduction
Under pressure to increase their return on marketing investment through more astute targeting of resources, marketers are rediscovering the importance of social contagion. Recent "viral" and "network" marketing strategies often share two key assumptions: (1) some customers are more in touch with new developments than others, and (2) some (often, the same) customers' adoptions and opinions have a disproportionate influence on others' adoptions (e.g., Gladwell 2000, Moore 1995, Rosen 2000, Slywotzky and Shapiro 1993). Targeting those influential prospects who are more in touch with new developments and converting them into customers, the logic goes, allows marketers to benefit from a social multiplier effect on their marketing efforts. The two assumptions are quite reasonable, as they are consistent with several theories and a large body of empirical research (e.g., Katz and Lazarsfeld 1955, Rogers 2003, Weimann 1994), and the social multiplier logic cannot be faulted either (e.g., Case et al. 1993, Valente et al. 2003). However, marketing science provides little or no additional theoretical or descriptive insight into how new products diffuse in such markets. The reason is that the great majority of marketing diffusion models assume homogeneity rather than heterogeneity in the tendency to be in tune with new developments and the tendency to influence (or be influenced by) others. We address this gap between theory and emerging practice on the one hand, and marketing diffusion models on the other. Specifically, we model the aggregate-level diffusion path of a new product when the set of ultimate adopters is not homogeneous, but consists of two segments: influentials who are more in touch with new developments and who affect another segment of imitators whose own adoptions do not affect the influentials. We allow for the presence or absence of contagion among influentials and among imitators.
Many diffusion models incorporate the dual drivers of independent decision making affected by being in touch with new developments, and of imitation driven by others' prior adoptions, but they do so under the assumption that all potential adopters are ex ante equally affected by both factors. Taga and Isii (1959) in statistics; Mansfield (1961), Pyatt (1964), and Williams (1972) in economics; Coleman (1964) in sociology; and Bass (1969) and Massy et al. (1970) in marketing, all advanced a model specifying the rate at which actors who have not adopted yet do so at time t as h(t)= p + qF(t), where F(t) is the proportion of ultimate adopters that have already adopted, parameter q captures social contagion, and parameter p captures the time-invariant tendency to adopt early affected by consumer characteristics, the innovation's appeal, and efforts of change agents. (1) Because the proportion that adopts at time t can be written as f(t) = dF(t)/dt = h(t)[1 - F(t)], one obtains:
f(t) = dF(t)/dt = [p + qF(t)][1 - F(t)]. (1)
The solution of this differential equation can be written as:
F(t) = [[1 - [e.sup.-g-(p+q)t]]/1 + (q/p)[e.sup.-8-(p+q)t]], (2)
where g acts as a location parameter fixing the curve on the time axis (e.g., Mansfield 1961). When t = corresponds to the actual launch time such that F(0) = 0, then g = and Equation (2) reduces to the solution popular in marketing.
The rate is influenced by both the intrinsic tendency to adopt (p) and social contagion (q) at all times except at t = when qF(0) = 0. To reflect this dual influence, Mahajan and Peterson (1985) refer to the model as the mixed-influence model. Because the rate contains no contagion pressure at t = 0, those adopting at that time are sometimes referred to as innovators and contrasted against all others adopting later, who are called imitators (e.g., Bass 1969). However, this terminology can only be used ex post, and the model does not represent a diffusion process in an ex ante mixture of two segments, the first adopting independently at rate p and the second adopting because of social contagion at rate qF(t) (Bemmaor 1994, Jeuland 1981, Lekvall and Wahlbin 1973, Manfredi et al. 1998, Steffens and Murthy 1992, Tanny and Derzko 1988).
The objective of this study is to mathematically formalize prior theoretical arguments and research findings on social structure and diffusion, and to use this formalization to generate more refined theoretical insights on new product diffusion in a population of influentials and imitators. This is important because marketing practitioners increasingly deploy strategies assuming such a market structure, and because marketing researchers increasingly incorporate social structure into their diffusion investigations (e.g., Bronnenberg and Mela 2004, Frenzen and Nakamoto 1993, Garber et al. 2004, Godes and Mayzlin 2004, Putsis et al. 1997, Van den Bulte and Lilien 2001).
Our results offer formalized insights into current substantive and methodological research questions. First, diffusion in a mixture of influentials and imitators can exhibit a dip between the early and later parts of the diffusion curve. In contrast to what Moore (1991) claims, our model shows that it need not always be necessary for firms to change their product to gain traction among later adopters and the adoption curve to swing up again. (2) Like Steffens and Murthy (1992) and Karmeshu and Goswami (2001), but unlike Goldenberg et al. (2002), we obtain this result from a closed-form solution, and unlike those prior analyses, we show that a dip can occur even when influentials act independently from each other. Second, the proportion of adoptions
stemming from influentials need not decrease monotonically, but may first decrease and then increase. The management implication is that while it may make sense to shift the focus of one's marketing efforts from influentials to imitators shortly after launch, as shown by Mahajan and Muller (1998) using a two-period model, one may want to revert one's focus back to influentials later in the process. Third, erroneously specifying a mixed-influence model to a two-segment process can generate the systematic changes in the parameter values over time reported in several studies (e.g., Van den Bulte and Lilien 1997, Venkatesan et al. 2004). This analytical result is a specific formalization of Van den Bulte and Lilien's (1997) more general but qualitative argument that unaccounted heterogeneity in p or q can generate changes in these parameters' estimates as one extends the data window. Our result also complements Bemmaor and Lee's (2002) simulation analysis because we consider heterogeneity in a process with genuine contagion rather than in a Gamma/Shifted Gompertz process without contagion.
We also perform an empirical analysis and assess the descriptive performance of the two-segment model compared to that of the mixed-influence model and of three diffusion models incorporating heterogeneity in the form of a continuous rather than a discrete mixture. Given the difficulty of unambiguously identifying causal processes from aggregate diffusion data (Bemmaor 1994, Hernes 1976, Lekvall and Wahlbin 1973, Lilien et al. 1981, Van den Bulte and Stremersch 2004), the objective of this empirical analysis is not to conclusively demonstrate the validity of any model. Rather, it is to assess whether the differences between the discrete mixture and other models are sufficiently important to lead to differences in descriptive performance when applied to data of interest to marketing researchers. The two-segment model fits better than the mixed-influence, Gamma/Shifted Gompertz (Bemmaor 1994), and Weibull-Gamma models (Hardie et al. 1998, Massy et al. 1970, Narayanan 1992), especially in cases where a two-segment structure is likely (or even known) to exist, and fits about as well as a recently advanced mixed-influence model where p and q vary across potential adopters in a continuous fashion (Karmeshu and Goswami 2001).
We proceed by first outlining our model setting, and within that context, discuss five theories and frameworks that suggest the existence of ex ante influentials and imitators. Next, we develop a macro-level model of innovation diffusion in such a setting. Subsequently, we discuss how this model relates to the familiar mixed-influence model and to prior work on two-segment models. Finally, we report on the descriptive performance of the influential-imitator model compared to that of the mixed-influence and continuous-mixture models.
2. Theories Motivating a Two-Segment Structure of Influentials and Imitators
The situation we model is the following. The set of eventual adopters has a constant size M and consists of two a priori different types of actors, influentials and imitators. We use the subscripts I and 2 to denote each type, and the subscript m to denote the entire mixture population of adopters. We use [theta] to denote the proportion of Type 1 actors in the population of eventual adopters (0 [less than or equal to] [theta] [less than or equal to] 1), and F(t) to denote the cumulative penetration. Finally, w denotes the relative importance that imitators attach to influentials' versus other imitators' behavior (0 [less than or equal to] w [less than or equal to] 1). Each type's adoption behavior is then captured by the following hazard functions:
[h.sub.1](t) = [p.sub.1] + [q.sub.1] [F.sub.1]( t); (3)
[h.sub.2](t) = [p.sub.2] + [q.sub.2][w[F.sub.1](t) + (1 - w)[F.sub.2](t)]. (4)
Note the asymmetry in the influence process: Type 1 may influence Type 2, but the reverse is not true. Because, ex ante, anyone of Type 1 may influence anyone of Type 2, we label the former influentials and the latter imitators. When [p.sub.2] = 0, contagion from influentials to imitators ([wq.sub.2] > 0) is critical for the diffusion process among the latter to get started. Obviously, when [theta] = 1 or [theta] = 0, everyone falls into a single segment and the situation reduces to the mixed-influence model (MIM). When < [theta] < 1 but w = 0, the model reduces to two disconnected MIMs and, with further restrictions, to a model with two disconnected logistic or exponential functions (e.g., Moe and Fader 2001, Perrin 1994). Also, when imitators put equal weight on all prior adoptions regardless of origin, then we have [h.sub.2](t) = [p.sub.2] + [q.sub.2] [F.sub.m] (t), which implies w = [theta] (see [section] 3).
The distinction between influentials and imitators is based on what drives their adoption behavior, not on whether they adopt early or late. Hence, the distinction is different from that of innovators versus imitators in Bass (1969) and innovators versus early adopters versus early majority versus late majority versus laggards in Rogers (2003). Conceptually, causal drivers and time of adoption need not map one-to-one. Empirically, while those adopting early may act independently of others, and those adopting late may be subject to contagion, this is not always so: Many early adoptions may be driven by contagion, and the bulk of the late adoptions may stem from people not subject to social contagion (e.g., Becker 1970, Coleman et al. 1966).
Several theories and conceptual models suggest such a two-segment structure, although there is some disagreement on whether [q.sub.1] and [p.sub.2] may be larger than zero. We first describe sociological arguments focusing on social character, social status, and social norms. We then turn to the two-step flow hypothesis, which focuses on interest in new developments, and finally to the chasm idea, which focuses on enthusiasm for innovations versus risk aversion.
2.1. Social Character
In his classic treatise on the changing nature of modern society, Riesman (1950) distinguished three types of social character: autonomous, inner directed, and other directed. The first two have in common the presence of clear-cut internalized goals, but differ as to whether these are consciously chosen (autonomous) or inculcated during youth by elders (inner directed). Other-directed actors, in contrast, use their peers as their source of direction. The typology is in essence about conformity stemming from the need for approval and direction from others. Riesman worked on a broad social and cultural canvas and his typology is best used to refer to patterns of behavior found in a variety of specific contexts rather than to types of persons or personalities. However, his concepts have direct relevance for consumer behavior (e.g., Riesman 1950, Schor 1998). Some actors in some situations will exhibit autonomous or inner-directed adoption behavior independent from their peers (hence [q.sub.1] = 0), while others will exhibit other-directed behavior driven by social contagion from peers. Riesman did not narrowly specify who these peers are, and allowed them to be all of society (therefore w = [theta] being possible). 2.2. Status Competition and Maintenance
People buy and use products not only for functional purposes, but also to construct a social identity, and to confirm the existence and support the reproduction of social status differences (Bourdieu 1984). A long-held idea in diffusion theory is that people seek to emulate the consumption behavior of their superiors and aspiration groups (e.g., Simmel 1971), and also quickly pick up innovations adopted by others of similar status if they fear that such adoptions might undo the present status ordering (Burt 1987). In short, actors tend to imitate the adoptions of those of higher and similar social status.
Assuming one can divide the population into a high-status and a low-status group, status considerations suggest that both groups may exhibit contagion. Higher-status actors may imitate each other out of fear of falling behind ([q.sub.1] [greater than or equal to] 0), and lower-status actors imitate to catch up. Whose adoptions the imitators act upon is not clear a priori. If they care only about adoptions by the high-status influentials, then w [right arrow] 1. However, most authors follow Simmel and posit a finer-grained hierarchy with multiple strata (approximated imperfectly by a dichotomy) and a cascading pattern where all prior adoptions contribute equally to social contagion (w = [theta]). Finally, to the extent that status is maintained by adhering to social norms enforced among one's direct peers of similar position, imitators should care mostly about fellow imitators (w [right arrow] 0).
2.3. Middle-Status Conformity
Like theories of status competition and maintenance, middle-status conformity theory is about one's proper place in society. The main claim is that the relationship between status and conformity to norms--and hence susceptibility to social contagion--is an inverted U (e.g., Homans 1961, Philips and Zuckerman 2001). Because high-status actors feel confident in their social acceptance, they feel comfortable...
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