Diffusion by imitation: The importance of targeting agents

Highlights

• Planner maximizes diffusion of action where agents imitate successful past behavior.

• Optimal targeting strategy depends on probability of success and planner's patience.

• Patient planner concentrates targeted agents when action is likely to be successful.

• And spreads the targeted agents around when action is likely to be unsuccessful.

• For an impatient planner the optimal strategy is exactly the opposite.

Abstract

We study the optimal targeting strategy of a planner who seeks to maximize the diffusion of an action in a society where agents imitate successful past behavior of others. The agents face individual decision problems under uncertainty, make reversible adoption choices and interact locally, so that each agent affects only her neighbors. We find that the optimal targeting strategy depends on two parameters: (i) the likelihood of the action being more successful than its alternative and (ii) the planner's patience. More specifically, for an infinitely patient planner, the optimal strategy is to cluster all the targeted agents in one connected group when her preferred action has higher probability of being more successful than its alternative; whereas it is optimal spreading them across the population when this probability is lower. Interestingly, for an impatient planner the optimal targeting strategy is exactly the opposite.

Introduction

Social interactions play a crucial role in the adoption of products, technologies and ideas (see Jackson, 2008, Rogers, 1995). Recent technological advances have made the collection and analysis of data related to the structure of interactions within societies possible, as well as the rules guiding their members’ behavior. The appropriate use of this information can provide helpful tools for the effective propagation of certain objectives through targeted campaigns.

In this paper, we describe the optimal intervention of an interested party (from now on called planner) who seeks to maximize the diffusion of a given action in a society where agents imitate their neighbors’ successful past behavior. This, for instance, could be a firm that produces a new product and wants to establish it in a new market. Optimal design of social influence campaigns is crucial, in particular when the planner has limited resources available.

The focus of this paper is twofold. First, we highlight the importance of the distribution of targeted agents in the society. That is we examine whether and when the planner should cluster or spread the targeted agents. As it will become apparent, this turns out to be a crucial feature that has been overlooked until now. Second, we compare the optimal strategy of a shortsighted planner versus that of a farsighted one, which turns out to have very distinct characteristics. To the best of our knowledge this is the first paper to discuss thoroughly these two aspects.

More specifically, most of the existing literature on targeting has focused on the importance of central agents (see for instance Ballester et al., 2006, Banerjee et al., 2013). Having a high or a low number of connections (Galeotti and Goyal, 2009, Chatterjee and Dutta, 2011), or diffusing information to others who are poorly connected (Galeotti et al., 2011) are some usual characteristics of influential agents. The importance of these characteristics is beyond doubt. Nevertheless, we show that another factor with significant impact is whether the targeted agents are clustered together or they are spread across the society.

Furthermore, throughout our analysis we highlight the differences between optimal targeting strategies of a patient planner versus an impatient one, i.e. who cares about diffusion in the long run and short run respectively. It turns out that these two cases differ sharply and these differences persist independently of the parameters’ values. This comparison is important in several scenarios, since different targeting strategies may be appropriate depending on the time horizon (Young, 2011).

Our model is quite general, however it would perhaps be more descriptive of a process related to the diffusion of agricultural innovations. Social learning, and imitation in particular, has a prevalent role in the diffusion of agricultural technologies.¹ The introduction of new technologies often occurs through formal private or public intervention and social interactions are vital for their subsequent diffusion (Rogers, 1995). Farmers face great uncertainty regarding their returns to the adoption of a new product or technology. The relative productivity of a new type of crops or the efficacy of a particular fertilizer may vary depending on the composition of soil or the area's climate. In addition to this, uncertainty of returns is further enhanced by unpredictable variations of weather conditions. These characteristics are often not transparent to the farmers beforehand, leading them to base their decisions on past experience, both their own and others’. As a result they often switch back and forth between adopting and abandoning a certain technology.²

Note that, the incentives of the planner may be different than those of the society, as for instance when the planner is a firm that wants to spread its own product. This means that the action of which diffusion is attempted to be maximized might be less effective than its alternative, with the relative efficacy often being known to the planner, but not to the agents. Propagation of suboptimal innovations is commonly observed in both agricultural technologies and other sectors.³ However, it is hard to identify whether the propagators are aware of the lower relative efficacy of their product. A prevalent example, in which that is hard to argue, is the case of counterfeit drugs.⁴ Their substance is similar to that of some original drug, but their relative efficacy is usually lower. This information is available to the producer, without necessarily being known to the consumers. This informational asymmetry has led in some cases to their widespread use, mainly in developing countries.⁵

In fact, in our model actions are differentiated based on their likelihood of being successful, rather than on their expected payoffs. Therefore it might be also the case that a planner wants to spread an action that is more risky than its alternative, but yields much higher payoffs when it is successful. In what follows, we analyze these cases separately, as they lead to different optimal targeting strategies.

There are several other examples that fit the general idea of the paper. For instance, a government that wishes to reduce criminal activity and is willing to sponsor a number of ex-criminals to change their lifestyle. Or else, a political or religious organization that wishes to propagate its ideology and locates a number of seeds in the society in order to spread the word to their neighborhood. As one can see, the problem of optimal social influence is directly applicable to a bunch of different environments and seemingly unrelated areas.

Undoubtedly, in order to obtain tractable and intuitive results we need to make a set of simplifying assumptions, which might reduce the applicability of our analysis in certain problems. Nevertheless, we provide a framework that can help us understand better which are the parameters that affect social influence crucially and we illustrate how beneficial the knowledge about society's structure may be for the efficient design of marketing and general social influence campaigns.

Formally, we consider a finite population of behaviorally homogeneous agents located around a circle. In each period, all agents choose between two actions simultaneously. The stage payoff each action yields is uncertain and depends on a random shock, which is common for all the agents who have chosen the same action in that period. There are no strategic interactions between agents. After making their decisions, all agents observe the chosen actions and the realized payoffs of their two immediate neighbors. Subsequently, they update their choice myopically, imitating the action that yielded the highest payoff within their neighborhood in the preceding period.⁶

There are several reasons why economic agents adopt simple behavioral rules, such as the imitation of successful past behavior. For example, they often need to make decisions without knowing the potential gains or losses of their possible choices. Such situations may arise with high frequency, or the agents’ computational capabilities may be limited, in which cases they tend to rely on information received from others’ past experience, rather than to experiment themselves.⁷ These arguments are also supported by a recent, but growing, empirical and experimental literature which provides strong evidence in favor of the fact that in several decision problems the agents tend to imitate those who have been particularly successful (Apesteguia et al., 2007, Conley and Udry, 2010, Bigoni and Fort, 2013).

The planner is interested in maximizing the diffusion of her preferred action in the population. She can be either infinitely patient, thus interested in the diffusion of the action in the long run; or impatient, thus interested in the diffusion of the action after just one period. Intermediate levels of patience are also explored via computer simulations. She is assumed to know the structure of the society, as well as how agents behave and can intervene by forcing a change at the initial choice of a subset of the population. Ideally, she would like to target the whole population, but doing so in reality would be extremely costly. Hence, our goal is to identify the planner's optimal strategy given the number of agents that can be targeted.

Observe that, all the agents are identical with respect to any measure of centrality. In fact, none of them has any positional advantage or disadvantage compared to the rest of the population. This is an important feature, given that we want to focus on the distribution of targeted agents in the society. Despite that, we find that expected diffusion changes substantially depending on the subset of the population that has been targeted by the planner.

Looking back at the example of agricultural technologies, it is reasonable to consider that different kinds of crops provide different harvests on average, depending on the fertility of the land and the weather conditions. Firms and governments have better knowledge over this relative efficacy, however the same may not be true for the farmers. This leads them to condition their decision on successful past experience of themselves and their neighbors (see Conley and Udry, 2010), which in turn may lead a whole community to end up planting the same crop even if this is not the most effective one. Moreover, as it is pointed out by Ellison and Fudenberg (1993), the farmers’ technology decisions are guided mainly by short-term considerations, especially when capital markets are poorly developed or malfunctioning.

We show that the optimal targeting strategy depends on two parameters: (i) the likelihood of the planner's preferred action being more successful than its alternative, i.e. yielding higher stage payoff and (ii) the planner's patience. In fact, we observe a sharp contrast between the optimal strategies of a patient planner versus that of an impatient one. More specifically, when the planner's preferred action has higher probability of being more successful, the optimal targeting strategy for a patient planner is to cluster all the targeted agents in one connected group⁸; whereas when this probability is lower, it is optimal to spread them uniformly across the population. For an impatient planner the optimal strategy is exactly the opposite.

The intuition for an impatient planner is quite simple. If the preferred action is likely to be successful, the planner wants to make it directly visible to as many agents as possible, since by doing so she will attract the maximum number of additional adopters in the likely case of a successful realization. Therefore, she should spread the targeted agents across the population. On the other hand, if the action is more likely to be unsuccessful, the planner wants to prevent as many of the targeted agents as possible from observing the alternative action, so that they will not change their choice, even upon the likely event of an unsuccessful realization. Hence, she should cluster them all together.

For an infinitely patient planner the arguments are reversed and are driven by the fact that only one of the two actions survives in the long run. In particular, when the action is more likely to be successful, the planner prefers to protect it by concentrating the targeted agents together, thus increasing the number of excess failures that would lead to its disappearance. Given that a high number of excess failures is rather improbable in that case, the action will most likely survive and hence get diffused to the whole population. To the contrary, the planner prefers to spread the targeted agents of an action that is less likely to be successful as much as possible, so as to take advantage of a relatively short sequence of positive shocks that would allow her to capture the whole population. When the likelihood of success is low she knows that by concentrating all the targeted agents together a lot of positive shocks will be needed in order to capture the whole population, which is rather improbable for an action that is expected to be usually unsuccessful.

Note that, an infinitely patient planner disregards completely the speed of the process, whereas an impatient planner fully discounts future adoption. These are two extreme cases that generate the natural question of what happens for intermediate levels of patience, and how this contrast between optimal strategies arises. As a matter of fact, for intermediate levels of patience the previously described conflicting effects of protection versus quick total diffusion remain decisive, although they are combined in complex ways. The major difficulty arises from the fact that the analysis for each time horizon is sensitive to the initial conditions and the exact history of realizations. For this reason, we tackle this problem via computer simulations.

We find evidence that when the action is likely to be successful, there is a smooth transition in optimal strategies from targeting many small groups towards targeting fewer and larger ones, until it becomes optimal to cluster all targets in one group. Intuitively, this means that the planner cares increasingly more about protecting the action from disappearing as a result of more negative realizations, compared to capturing larger parts of the population as a result of few positive realizations. When the action is unlikely to be successful there is a sharper transition between the optima of the two extreme cases, which often occurs directly. Essentially, as long as the planner is sufficiently impatient, she prefers to protect the targeted agents from observing the alternative, by clustering them together. This holds until a threshold of patience is reached, above which protection is no more effective and the planner prefers to try and capture the whole population with as few positive shocks as possible.

In addition to the previous predictions, we intend to understand the extent to which our results have any validity for general network structures. For an impatient planner, we show analytically that the optimal targeting strategy depends on the relation between the number of targeted agents who will also observe the alternative action in the first period and the number of non-targeted agents who will observe the preferred action in the same period. It is apparent that the planner wants to minimize the former quantity and maximize the latter one. The relative weight of the two is determined by the likelihood of success of the planner's preferred action.

For the infinitely patient planner, we use numerical simulations again, in randomly generated networks and we obtain strong evidence towards an intuitive result. Namely, when the action is likely to be successful, the planner prefers to maximize the maximum distance between the set of non-targeted agents and any targeted agent. Intuitively, this means that the planner wants to maximize the number of excess failures that would lead to the disappearance of the action. To the contrary, when the action is likely to be unsuccessful, the planner prefers to minimize the maximum distance between the set of targeted agents and any non–targeted one. Essentially, this means that the planner wants to minimize the number of excess successes that would lead to the full diffusion of her preferred action. The results are also supported by a theoretical analysis on the linear and the star network.

At this stage, one could question how important the role of the particular behavioral rule for obtaining these results is and how robust they would be under either Bayesian (see Gale and Kariv, 2003), or repeated averaging (see DeGroot, 1974, Golub and Jackson, 2010) learning rules, or whether agents were simply allowed to have longer memory. A crucial aspect for the current results is that the agents do not accumulate information over time, a fact that has two negative and one partially positive effect. On the one hand, the society is vulnerable to misguidance by certain unexpected events even at later stages, which for example should not be the case if the agents perform Bayesian updating. Moreover, under any initial conditions there is no guarantee that the society would converge to the planner's desired action (even if this is the socially optimal). On the other hand, the process is less path dependent than De Groot learning, where initial opinions may drive a society towards an inefficient state, sometimes even with certainty. Once again, this would not be a problem under Bayesian updating, which however has been acknowledged by large part of the literature on learning in networks to require quite complex calculations. Having said that, an interesting question that could complement the current paper would be how the results would be altered in the presence of probabilistic imitation, as described by Schlag (1998).

Allowing the agents to have longer memory and assuming they would still imitate the best action within the capacity of their memory would be beneficial for the action that is more likely to be successful, unless the support of realizations of the actions included rare, but extremely positive shocks. On the one hand, longer memory would reduce the impact of an unlikely mildly positive shock for the usually unsuccessful action. On the other hand, for this modified process, the explicit definition of each action's absolute payoffs would be crucial, since the impact of extremely positive shocks would now last longer, thus having a more significant impact on the process.

Finally, we extend our analysis to some additional directions.⁹ We discuss the optimal targeting strategies for a planner, when the number of initial seeds is endogenous and targeting is costly. Moreover, we quantify the practical meaning of infinite patience by characterizing the expected waiting time before convergence occurs. We observe that, for those cases in which the planner's optimal strategy is to cluster all the targeted agents in one group, the process is slowed down substantially. In addition to this, we discuss what happens if we allow for inertia and we show that the results remain unchanged. This extension captures many realistic features, such as the existence of switching costs and some forms of conformity.

This paper stands within the literature that studies the role of influential agents in diffusion processes on networks, from the perspective of a planner who wants to design optimal targeting strategies.¹⁰ Intuitively, a crucial feature is the centrality of an agent, which quantifies its importance for the connectivity of the society.¹¹

The contribution of our paper is that it addresses the importance of two novel factors in the design of optimal targeting strategies. First, the time horizon of interest to the planner, because optimal strategies might be different when focusing in the short run versus the long run. Second, the importance of clustering versus spreading out the targeted agents.

The role of influential agents in environments with local interactions has been studied in different disciplines, such as computer science (Kempe et al., 2003, Richardson and Domingos, 2002, Kempe et al., 2005, Domingos and Richardson, 2001), marketing (Kirby and Marsden, 2006), physics (Bagnoli et al., 2001) and economics (Galeotti and Goyal, 2009, Goyal et al., 2014).

In the current context, connected agents do not interact strategically, thus network links represent the flow of (truthful) information within the society.¹² Within this context, Galeotti and Goyal (2009) consider a similar question, where an external agent can increase the diffusion of a product by targeting strategically a subset of a population of connected agents. However, their analysis does not focus on either one of the main features of the current paper, as the authors focus exclusively on short run strategies and assume that agents meet randomly. The latter means that the optimal strategies depend only on the degree distribution of the network, thus clustering of targets cannot be discussed. In another recent paper, Tsakas (2015) studies a similar problem of targeting in a setting where agents are able to learn the quality of a product when observing it within their neighborhood. The author finds decay centrality to be an important measure for the characterization of optimal targeting strategies and follows the present paper in that he conditions the optimal targeting strategies upon the relative patience of the planner.

There is also a recent, but growing, literature on competitive influence, where multiple rational agents compete to influence the actions of a network of boundedly rational agents via targeting (Goyal et al., 2014, Lim et al., 2015, Grabisch et al., 2015, Alós-Ferrer et al., 2010, Shi, 2015, Chasparis and Shamma, 2010). In particular, Goyal et al. (2014) study competition between two firms that distribute their resources trying to maximize the long run diffusion of a product in a network. The paper focuses on how efficiently the resources are allocated, as well as on the effect of budget asymmetries on the equilibrium allocations and not on the distribution of seeds or on the time horizon of the players.

The studies on cellular automata and voter models (Durrett, 1988, Liggett, 1985), used mainly in physics, but also in economics (Yildiz et al., 2011), are also related. On cellular automata, the diffusion process studied in Bagnoli et al. (2001) has very similar characteristics to ours, yet the focus is on the long run behavior and phase transition of a system, without focusing on the initial conditions. On voter models, Ortuño (1993) studies optimal targeting in a standard voter model and points out the potential impact of the distribution of seeds on expected diffusion.

Regarding agents’ behavior, we focus on imitation of successful past behavior. Similar rules have been studied in several theoretical settings,¹³ with the focus being mostly on the characterization of stochastically stable configurations. Moreover, the assumption that agents imitate without ever making a mistake may be in contrast to the usual assumptions employed in the literature in evolutionary games, where the presence of rare mutations is crucial (see for instance Robson and Vega-Redondo, 1996). But there is a reason for this. In the current setting, the process can be described by a Markov chain that would be ergodic under the existence of mutations. This would make the problem of initial targeting irrelevant, as the long run distribution would be independent of the initial conditions.¹⁴ Second, large part of the literature on evolutionary dynamics focuses on finding the minimal number of agents who can trigger a transit from one steady state to another, by changing their actions. This cannot be the focus here, because the nature of the process ensures that any action chosen even by a single player can be spread to the whole population with positive probability.

Notice also that in the literature on evolution, the studies that deal with the speed of the process look from a perspective different from ours. Essentially, they look mostly at the long run behavior of the system and intend to understand which kind of dynamics or initial conditions lead to this stable behavior faster (see Arieli and Young, 2016). To the contrary, our analysis of short run behavior, focuses on periods in which the process has not been stabilized yet.¹⁵ The speed of diffusion is also studied in Young (2011), in the context of a coordination game. The environment may be significantly different than ours, yet in both papers an innovation's rate of diffusion is affected by its relative payoff compared to the status quo.

There is also a recent but yet growing empirical literature¹⁶ that provides empirical evidence on the adoption of imitative in real environments. In particular, Beaman et al. (2015) test several network theoretical models of targeting in a field experiment regarding adoption of agricultural technologies in Malawi, considering also the idea of clustering or not the seeds. In a more general framework, the current analysis builds upon the work on learning from neighbors,¹⁷ where most of the papers focus mainly on conditions under which efficient actions are spread to the whole population and not on optimal influence strategies.