Theory development with agent-based models

Agent-based models

In the psychological sciences, relationships of interest are often too complex for straightforward mathematical analysis, and may require a computational specification and analysis through simulation. This is an advantage of ABMs, as their analysis concerns the data generated from iterated simulations of agent interactions, which can encompass quite complex relationships. Some well-known examples include artificial neural network models of cognition (in which the neurons are the agents; e.g., Hopfield, 1982; McClelland, Rumelhart, & the PDP Research Group, 1986); Kalick and Hamilton’s (1986) model of mate preferences; and Nowak, Szamrej, and Latané’s (1990) model of dynamic social impact.

In contrast to most traditional research methods, ABMs are expressly designed to accommodate the three challenges to studying group dynamics outlined before. ABMs are explicitly multidimensional and can efficiently and simultaneously model dynamics at multiple levels of organization while incorporating feedback processes and system memory. ABMs are therefore especially well suited to study complex systems whose dynamics are inherently dependent on heterogeneous actors and organizational structure, which are difficult to model using other formal techniques such as differential equations (Smith & Conrey, 2007). Users can specify rules that describe individual agent behavior, agent–agent interaction, and agent–environment interaction. Various programs and toolkits have been developed to facilitate constructing ABMs. Prominent examples include NetLogo (Wilensky, 1999) and the MASON simulation library (Luke, Cioffi-Revilla, Sullivan, & Balan, 2005).

An ABM is typically comprised of a population of agents in an environment. Each agent is discrete and autonomous, pursuing goals based on local, limited information. Agents can learn from, have memory of, or otherwise be affected by interactions with other agents and the environment. Behavior rules are postulated theoretically and implemented symbolically as a series of computational algorithms. Once initial parameters are specified, the model can run for a given amount of time or until some desired state (such as an equilibrium) has been reached. The end result is the pattern of outcomes that emerge in the population over time. Of primary interest to the study of group dynamics, outcomes can arise from interdependent processes, such that each agent is affected not only by its own decisions, but also by environmental conditions, group-level processes, and the behaviors and responses of other agents in the environment. This is a true strength of ABMs: few other methodologies are as well suited to represent and capture multilevel, interdependent, recursive processes. Additionally, ABMs can generate numerical data like other quantitative research methods and, if suitable, they can allow users to visualize the movement of individual agents and the patterns that these movements produce. In such cases, agents literally move and change on-screen in real time. Put another way, if other research methods take static snapshots of group dynamics, ABMs create motion pictures. (For more detailed discussions on the nature of ABMs in other disciplines, see Epstein, 1999; Gilbert & Terna, 2000; Macy & Willer, 2002; Schank, 2001; Smith & Conrey, 2007.)

Some people may be concerned that explicitly articulating every detail of a model amounts to “setting yourself up for success.” That is, if everything is set up according to the specifications of the theory, the only possible outcome is the one predicted by the theory. However, it is important to remember that individual-level behavioral rules do not necessarily translate directly into simple predictions of outcomes at the group level, because structural organization, feedback, and system memory all affect group-level processes. One of the strengths of ABMs is the ability to directly incorporate those properties of complex social systems, creating the potential for unexpected outcomes to arise despite complete knowledge of initial conditions (Epstein, 1999).

ABMs are especially well suited to the study of group dynamics for a variety of reasons. They can simultaneously represent multiple levels of organization, and can easily accommodate recursion and system memory. We will now introduce an ABM from our own research on social identity dynamics to demonstrate how ABMs can be used to develop theory. We will give careful attention to the processes of developing and analyzing the model, in order to provide a potential guide for future model development.

Optimal distinctiveness theory

Within psychology, there has been a long-standing interest in understanding processes of group formation and change. Group living has been a hallmark feature of human societies throughout our evolutionary history, and the formation of groups is essential to human survival in most ecological contexts. In addition to providing “safety in numbers,” groups also provide a powerful organizational system that can confer great survival advantages. Social categorization and group boundaries help define the limits of cooperative interdependence. Group members can provide aid to mutually acknowledged ingroup members and trust that these ingroup members will reciprocate. In other words, groups can be thought of as bounded communities of mutual trust and obligation (Brewer, 2007; Brewer & Caporael, 2006).

Because group membership involves reciprocal obligation, it has been argued that individuals have developed a preference for groups that are moderately sized, such that the number of individuals to which one is obligated is manageable (Brewer & Caporael, 2006): a group that is too small is limited in its ability to generate synergistic benefits, but a group that is too big is vulnerable to freeloading members. As such, there are both lower and upper limits on ideal group size. At the psychological level, optimal distinctiveness theory (ODT; Brewer, 1991) proposes that these ecological pressures—to belong to groups and at the same time limit their size—have resulted in two oppositional psychological motives: the need for inclusion and the need for differentiation. According to ODT, when these two motives operate together, they are hypothesized to give rise to a preference for “optimally distinct” identities, that is, membership in groups that are moderately sized in relation to the population in which the group resides.

Support for the existence of these motives and a preference for optimal distinctiveness has been found across numerous empirical studies (see Leonardelli, Pickett, & Brewer, 2010, for a review). These studies have documented a general preference for groups that are moderately sized. All things being equal, individuals (at least in Western, industrialized settings) identify more strongly with groups that represent a numerical minority of the total observable population than with groups that represent a majority of the population (Abrams, 1994; Blanz, Mummendey, & Otten, 1995; Brewer & Weber, 1994; Ellemers & van Rijswijk, 1997; Simon & Brown, 1987; Simon & Hamilton, 1994).

When individuals update their social identities in response to changes in their social environment (e.g., switch political parties; join the Shriners; go vegetarian), the relative size of every group in the environment also changes, which in turn alters the landscape for every other individual who shares that environment. As individuals leave and join groups, the level of the distinctiveness of each group changes relative to all the other groups in the social environment, which consequently changes the distinctiveness of every individual in every group. Thus, individual-level changes in social identity create opportunities for dynamic cascades and feedback throughout the environment. Though the ODT literature acknowledges such a fluid nature of the social environment (e.g., Turner, Oakes, Haslam, & McGarty, 1994), most research has focused on one individual responding to a static environment. Because there are potentially limitless causal pathways and recursive loops in such a dynamic social landscape, it can be challenging to form—let alone test—anything more than the most simple hypotheses about such a complex social system. Thus, the impact of individual-level preferences for optimal distinctiveness has been studied almost exclusively at the individual level, and group-level outcomes had remained unexplored until recently. This lack of exploration of group-level optimal distinctiveness was due, in large part, to the tools that are traditionally available to researchers interested in the psychological study of groups. As noted previously, lab-based experimentation is an excellent method for testing causal relationships at the level of the individual. However, the lab is not amenable to examining the interactions among multiple individuals over extended periods of time.

When conceptualized from a group dynamics perspective, it is easy to see why laboratory studies on isolated individuals are insufficient for fully capturing the dynamic processes that are logical outcomes of the predictions of ODT. Moreover, because ODT research has thus far only examined individuals, it is largely unknown how these preferences shape group formation and change. An explicit assumption of the theory has been that individuals generally maintain an equilibrium between their opposing needs for differentiation and for assimilation by switching to more “optimal” social identities when the need arises (Leonardelli et al., 2010). However, the group-level consequences of such an assumption had not been tested. As such, it was not clear whether individuals following this type of heuristic for social identity choice would actually sustain either a social equilibrium, in which individuals are relatively stable in their social identities, or even an individual equilibrium in which individuals are satisfied in an “optimally distinct” social identity.

Building an ABM of social identity: The SID model

To address this gap, Smaldino, Pickett, Sherman, and Schank (2012) developed an agent-based model in which agents make social identity decisions (i.e., whether to select or abandon a particular group membership) based on the social identities of their neighbors (hereafter referred to as the SID model). The model was built upon the assumption that agents make these decisions in an attempt to satisfy their preference for groups that are optimally distinct. The overarching goal was to observe the resultant dynamics of group formation as a function of variations in group size preferences and the size of social neighborhoods.

Designing an ABM of a complex social system requires the researcher to articulate the specific components of that system and to specify the properties of those components and how they relate to one another. There are myriad ways in which any particular system might be decomposed into parts, and there is not necessarily a uniquely best decomposition (Kauffman, 1971). The questions being asked about the system will drive how the system is modeled. The model designer should therefore strive to create a model system that captures the aspects of the real-world system in which she is interested, with enough complexity to create a credible analogy between the model and real-world systems, but otherwise as simple as possible so as to maximize generalizability and minimize obscuring artifacts. The complexity of a model may depend on how much is already known about the system in question, as well as the complexity of the theoretical mechanisms under investigation. A complicated theory may demand a complicated model. Yet simple models can often be extremely revealing. The overarching lesson in the history of complexity science is that complex or unexpected behavior often emerges from an instantiation of relatively simple relations and behaviors.

Because ODT had not been previously studied using an ABM, the SID model is by necessity as simple as possible in order to test the basic assumptions of the theory. Each agent has only two attributes: membership in a particular group (i.e., its social identity) and a preference for that identity to have a particular degree of distinctiveness (i.e., its optimal distinctiveness). Later, we will also discuss the introduction of a third attribute: spatial location. Each agent follows two simple behavior rules. First, the agent observes the social identities of all of the other agents in its neighborhood, noting how many individuals (including itself) possess each social identity. Second, if the agent observes a social identity that is closer to its optimal degree of distinctiveness than its current social identity, it switches to that social identity. Thus, an agent will stay with its current social identity only when it observes no other social identities that are closer to its optimal degree of distinctiveness.

The formal description is necessarily more precise, if perhaps less rhetorically pleasing. A population of N agents is initialized so that each agent is randomly assigned to one of k social identities. Each agent has a preference for optimal distinctiveness, d*, which is the desired proportion of the agents in its neighborhood with whom the agent shares a social identity (e.g., if d* = 0.2, then each agent would be maximally satisfied if exactly 20% of the agents in its neighborhood shared its social identity). For simplicity, we specified that d* is the same for all agents and does not change over time. The model dynamics proceed in discrete time steps. At each time step, each agent, in random order, assesses the distinctiveness, d_g , of each social identity g in its neighborhood, which is equal to the proportion of agents who have that social identity (i.e., d_g = n_g /N, where n_g is the number of agents with social identity g). The agent then switches to the social identity with a distinctiveness closest to d*, so that it minimizes |d_g—d*|, the difference between its optimal and actual degrees of distinctiveness. If two or more social identities are equally close to optimal, one is selected at random unless one of these is the agent’s current social identity, in which case it retains its current social identity. Each agent acts in turn, so that one agent’s choice may directly influence the choices of agents that “move” after it. This process continued either until a stable equilibrium was reached, in which no agent had a better option than its current social identity, or for 5,000 time steps, whichever came first. For the initial runs, all agents in the population were defined to share the same neighborhood, and a stable equilibrium was always reached. Next, we explored the role of the structure of social networks. Agents were situated on a lattice and only assessed the social identities of those agents within a limited radius (their local neighborhood; see following lines). In these cases, an equilibrium was not always reached, as agent choices sometimes led to perpetual feedback and identity switching, and the results reflect the average situation after 5,000 time steps.

Model results

Simulations demonstrated that, in a well-mixed population in which each agent knew and responded to the social identities of every other agent (i.e., all agents shared the same big neighborhood), all agents ended up in groups that were overly large and therefore suboptimally distinct (Figure 1A). This outcome happened at all levels of desired optimal distinctiveness, d* (see Smaldino et al., 2012, for a detailed explanation of the causal dynamics). At first glance, it therefore might appear that the assumptions of ODT do not lead to individuals who are satisfied with their degree of distinctiveness. However, in considering this result, it became apparent that individuals in real populations rarely have full knowledge of everyone in their environment, but rather are organized in social networks with limited knowledge beyond their local connections.

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Figure 1.

Results of the SID model from the well-mixed and spatial versions. (A) Agents’ average distinctiveness at the end of a simulation run, as a function of their optimal distinctiveness preferences (d*) and neighborhood size (r). Data is averaged over 30 runs for each parameter condition. The well-mixed model represents the case in which each agent’s neighborhood is the entire population. The dotted grey line indicates an ideal case in which the average distinctiveness is equal to agents’ optimal distinctiveness. (B–E) Spatial organization of agents at equilibrium for example runs with different optimal distinctiveness preferences (d*) and local neighborhood sizes (r). Each agent’s color represents a particular social identity. <d> refers to the average distinctiveness across the population. All of these runs settled to highly stable spatial patterns except (C), in which around 25% of agents switched identities each time step.

Though ODT does not explicitly articulate network constraints in shaping distinctiveness, we nevertheless elaborated upon the well-mixed initial model by introducing a simple structure, such that agents responded only to the social identities of their neighbors. Agents were placed on a square lattice with periodic boundaries (so as to eliminate edge effects, the agents on the far right edge are neighbors with the agents on the far left, and similar for the top/bottom edges). The lattice was a 50 × 50 grid containing 2,500 agents, one to a cell. An agent’s local neighborhood was defined as all the agents within a square centered on the agent and extending r cells in each of the four cardinal directions (up, down, left, right), such that a neighborhood consisted of (2r + 1)² agents. When this change was implemented, stable, spatially contiguous groups emerged in which all agents achieved high levels of satisfaction, that is, their groups were close to optimally distinct (Figure 1). As such, the SID model identified one boundary condition to optimal distinctiveness previously unspecified by ODT: network structure, which is analogous to spatial, social, or organizational distance in the real world.

The inclusion of a network structure proved to be critical to our model of social identity dynamics. We followed up on this finding by probing what the model could tell us about the effects of different types of network structures on agent satisfaction. We systematically varied the size of each agent’s local neighborhood, r. Figure 2 shows agents’ average distinctiveness at equilibrium as a function of local neighborhood size for three different values of d*. These results indicate that agents’ satisfaction with their distinctiveness is best achieved through social neighborhoods of intermediate size, which complements and extends ODT’s predictions regarding preferences for moderately sized groups. When a neighborhood is too small, agents may become trapped either in homogeneous groups or in groups in which they are unique. However, once neighborhoods exceed a certain size, agents again end up in groups that are too large, leaving them with an unsatisfied need for increased distinctiveness.

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Figure 2.

Average distinctiveness in the population as a function of neighborhood size for three values of d* (indicated by the dotted lines). Data are averaged over 30 simulation runs for each parameter condition.

Implications for theory development

The SID model is a step toward a more precise body of theory linking individual psychology with group behavior. At a basic level, the model provides a clear demonstration of how individual decisions can feed back into stimuli for others—a point that previously existed only rhetorically in the optimal distinctiveness literature. The SID model also highlights the previously unarticulated role that network structure and size plays in how individuals assess and modify their distinctiveness. One novel prediction generated by the model is that a moderate neighborhood size is ideal for generating optimally distinct groups. The question of how such moderately sized local neighborhoods are formed—perhaps through a combination of physical space, social structure, and cognitive constraints—also represents a new and potentially illuminating direction for both empirical and theoretical research.

The findings of the SID model represent a theoretical advance in the study of optimal distinctiveness. Empirical work remains to be done to investigate how individuals perceive their relevant networks, and how neighborhood size and population density influence perceptions and decisions related to distinctiveness. As we have stated before, modeling does not and cannot replace carefully designed empirical studies. Ideally, lines of communication between modelers and empiricists should remain open and active in both directions. Models can demonstrate possibility and suggest new directions for research, but a single model cannot solve the problems of an entire theory any more than can a single experiment. There are assumptions inherent in any model—just as there are for any other empirical method—and future work can test the robustness or boundaries of these assumptions. Nevertheless, ABMs are showing themselves to be an essential tool for theory development in the study of group dynamics and related fields. In the next section, we discuss general strategies for researchers hoping to develop theory with ABMs.

Resources for theory development with ABMs

There are three types of resources, all important, that we recommend for researchers who wish to add ABMs to their methodological toolkit. First, seek out published models, which illustrate how other researchers have applied ABMs to tackle problems in theory development. We have presented one example of an ABM applied to a question from the behavioral sciences, but there are many more. Some modeling topics of interest to group dynamics researchers include opinion dynamics (Axelrod, 1997; Centola & Macy, 2007; Deffuant, Neau, Amblard, & Weisbuch, 2000; Nowak et al., 1990), mate choice (Kalick & Hamilton, 1986; Simão & Todd, 2002; Smaldino & Schank, 2012), cooperative group formation (Gray et al., 2014; Smaldino & Lubell, 2014), and distributed social cognition (Smith, 2014; Smith & Collins, 2009; Stasser, 1988). Second, read philosophical literature on how to think about models. There are many discussions that go far beyond the present discourse on that matter (e.g., Bedau, 1999; Beisbart, 2012; Epstein, 1999, 2008; Kauffman, 1971; Parke, in press; Schank, 2001; Weisberg, 2007; Wimsatt, 1987). Such readings can help the aspiring modeler in both model design and interpretation, as well as providing broad support for the epistemic value of computational models. Finally, take time to learn to model. Of course, becoming an expert at model design and programming is not a requirement for every researcher using modeling. Just as experimental inquiry utilizes experts in statistical analysis, exploiting ABMs may similarly involve specialists. Nevertheless, it will benefit anyone employing ABM to have some experience with building and testing models. Thankfully, it has never been easier to learn the requisite programming to construct an ABM. For example, the NetLogo modeling environment (Wilensky, 1999) was specifically designed to be accessible to nonprogrammers, and has a shallow learning curve. NetLogo is free, open-source, and comes bundled with a large library of demo models and a detailed tutorial. For those requiring more formal instruction, a recent textbook on agent-based modeling (Railsback & Grimm, 2012) teaches modeling techniques along with a practical guide to programming models in NetLogo. Additionally, an introduction to ABM using NetLogo (Janssen, 2010) has recently been published free online (www.openabm.org/book/1928/games-gossip). For more advanced programmers who desire more precise control over the details of their models, MASON (Luke et al., 2005) is an open source simulation library written in Java, is extensively documented with numerous demo models and tutorials, and is maintained by an active community of users and developers. Several other software tools and tutorials are also available depending on a user’s needs and expertise.

Combining agent-based modeling with other methods

In addition to adding rigor to thought experiments, ABMs may also be combined with data as a more systematic alternative to null hypothesis testing common in behavioral research. This allows for fine-grained model comparison in which the models being tested propose explicit mechanisms rather than mere statistical relationships. Here we will discuss two examples of this application of ABM. Because the ABM literature on organizational psychology is still sparse, the examples will be drawn from elsewhere—the first on human crowd dynamics, the second from animal behavior. Though neither study comes directly from the field of organizational psychology, both involve group dynamics and social organization.

Moussaïd, Helbing, and Theraulaz (2011) investigated the dynamics of pedestrians maneuvering toward an exit through a crowd. Though other models of pedestrian dynamics had been previously proposed, their model was based on psychological theories of decision making and social norms rather than on physics-based approaches that lack psychological underpinnings, and involved a formal instantiation of two simple decision heuristics. Their model was tested against empirical data derived from video analyses of pedestrians moving through a corridor in which a confederate was either standing still or moving in the opposite direction, as well as data from crowds of different densities moving through a bottleneck. They were able to fit their model to match all these cases. The wide range of conditions against which the model was tested provides strong support for their theories of perception and decision making among pedestrians. Importantly, validating their model with real-world data allowed them to generate new questions concerning more complex scenarios, and provides predictive power for future industrial design and emergency preparation.

Moussaïd et al. (2011) used empirical data to validate their model, but it is also possible to generate a range of models and then use data-fitting to select the best among models. In this way, the data can be used directly to select which theory, formalized as an instantiation of an ABM, is best supported empirically. Schank (2008) investigated the development of social behavior in rat pups using a combination of experiments and modeling. He filmed the pups moving in an open arena alone and in groups of eight, at both 7 and 10 days of age. The pups’ positions, directional headings, and surroundings were catalogued at regular time intervals. At the same time, he developed an ABM of probabilistic movement based on the individual’s surroundings (e.g., relative location of walls, other pups), and a genetic algorithm (Mitchell, 1997) was implemented to optimize model parameters to fit the data. The best-fit model parameters showed that the group behavior of 7-day-old pups could be explained as the aggregation of many pups, each behaving as if it was alone. In other words, the pups reacted to one another no differently than they reacted to a wall, suggesting that their huddling behavior involved no social cognition. In contrast, the group behavior of 10-day-old pups was not explained by the parameters that best fit their individual behavior. Ten-day-old pups behaved differently alone and in groups, supporting the hypothesis that social cognition comes online in rats around 10 days of age. In this instance, the agent-based model functioned as a tool for data analysis, providing more precise hypotheses than would be possible by verbal theories alone.