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Complex systems research is a growing field in economics as well as other social and natural sciences. Complex systems research aims to understand underlying phenomena that regularly occur across various complex systems, whether those systems occur in physics, chemistry, biology, social sciences, or any other discipline. Thus, throughout the field, one finds models and methodologies being shared across disciplines. For instance, one finds statistical mechanics methods from physics applied to the study of infectious diseases in epidemiology. The study of economics from a complex systems or complexity perspective is closely tied to the study of complex systems in general.
Defining what is meant by a complex system or complexity can be a difficult task. When one looks up the root word complex in a dictionary, one will read something similar to “made up of complicated interrelated parts” or “involved and intricate.” These definitions give one a start in defining how the word complexity is used in the sciences, but one needs a bit more. A complex system is made of interacting parts (usually many), but those parts do not always need to be complicated. Sometimes, seemingly simple systems produce complex behavior. Further, an understanding of each of the constituent parts individually does not lead to an understanding of the entire system. Thus, complex systems research in economics and elsewhere often constitutes a holistic approach to understanding economic systems. It encompasses understanding not only how individual constituent parts (such as individuals or firms) operate or behave but also how those operations or behaviors aggregate to create a system (such as a market outcome or dynamic time series).
What Makes a System Complex?
There are several common features of systems that are associated with complex behavior. These are diversity, structured interactions, bounded rationality, adaption and learning, dynamic systems, and lack of centralized authority. These features are inherently associated with many of the systems that economists often study. It is not the case that all complex systems contain all of the elements listed above, but most contain several of these features. In the next section, this research paper describes these common features of a complex system and gives relevant examples from economics.
Diversity or Heterogeneity
Complex systems frequently are composed of diverse or heterogeneous elements. Elements of a system may be diverse simply because they perform different functions in systems, individuals, firms, and governments, for example. But even agents or objects within a given group or class in a complex system tend to behave, learn, or organize in a multitude of ways. In economics, at the lowest level, the constituent parts are individuals. Individuals in economic systems differ in so many ways it is difficult to count or list them all. Sometimes, this variety is based on characteristics that an agent is born with, such as race, ethnicity, gender, age, or religion. Other characteristics are primarily choices of the agent. Some of these include training and education, residential location, or specialization in a profession, to name a few. Of course, some of these items are interlinked. For instance, one may be born into a Catholic family but convert to another religion. Further, some of the choice characteristics are constrained choices, where the constraints may vary across individuals. For instance, one’s opportunity to acquire education and human capital are constrained by one’s ability to pay college tuition or by the educational opportunities provided by one’s caregivers in the home, and one’s choice of residential location is constrained by the income and wealth that one attains or inherits. Finally, in many economic models, agents are assumed to have homogeneous preferences or tastes. Very few economic models consider that individuals may have different preferences or objectives, yet if one asked each individual in a group of 1,000 to name a favorite restaurant, flavor of ice cream, time to awaken in the morning, and number of hours to relax in a day, one would probably get 1,000 different answers. In summary, there is a great number of ways in which individual economic agents differ, and complex systems models in economics often embrace this diversity.
Structurally Interacting Agents or Parts
In many complex systems models, there exists a specified structure on which interactions occur. Sometimes, this is based on geography; other times, the interactions are a lunc-tion of some other structural constraint such as the neurological connections in the human body. In sum, there is some network architecture central to the interactions of agents.
Although this is currently changing, most traditional economic models assume that agents interact without attention to the details of the interaction. In some cases, agents interact only through a market mechanism like a traditional Walrasian auctioneer. In other instances, agents are supposed to interact through random meetings, as though in a so-called Walrasian soup. For instance, most labor market search models have random meetings between potential employers and employees. Agents then optimize their choice to accept employment if offered a job, based on their expected waiting time for other (preferably better) job opportunities to randomly arrive at or above a reservation wage. However, it is common for individuals to learn of job opportunities through family and friends. In fact, around 50% of jobs are found in this manner (Granovetter, 1995). Further, individuals base many of their decisions on information gained from friends, such as recommendations on which products or brands of products to buy, what restaurants to try when visiting a new city, or what theater performances or movies to attend.
There are multiple ways that one can think of this idea of network architecture and the influence of social contacts playing a role in economic outcomes. The first is probably the least controversial: Network structures can act as constraints on the decisions of agents. If information travels through social contacts, then the contacts of an agent help to determine what information that agent holds, whether it is about jobs, products, or another item of economic interest. Second, one may also view the social contacts of an agent in helping to determine one’s behavior through a traditional externality perspective. For instance, if all of one’s friends own Apple computer products, it is more beneficial to own an Apple than if all of one’s friends own PC-based products because of direct reasons such as the (legal or illegal) sharing of software or because of less direct reasons such as troubleshooting when operational problems arise. Third, one can be influenced just by the actions of friends through conformity effects. Rationally, it may be advantageous to attend movies that one’s peers attend just so that one can fit in and engage in conversations about these movies. Perhaps less rationally, one can imagine a lemminglike scenario where one attends movies just because his or her friends attended these movies. In any case, the interactions with one’s social contacts help determine behavior and decision making either through a traditional constraint-based approach or through less traditional conformity effects.
These social contacts may be exogenously given (e.g., family) or endogenous; they may be a constrained choice of the agent, such as the friendships formed at school. One gets to choose one’s friends from the set of other agents that one meets. But this choice is constrained by the opportunities that one has (geography) and, in some cases, by the willingness of other agents to reciprocate the interaction (friendships). In other cases, reciprocation is not needed. For instance, one can pass on some infectious diseases, such as influenza, without asking permission of the recipient agent.
Finally, since each individual has a unique set of friends and family, each person has a unique set of social contacts. Thus, social contacts are yet another way in which agents are diverse.
Again, as in the discussion of structured interaction, traditional economic models often make simplifying assumptions that are unrealistic. This occurs again when one considers the rationality of agents. Many economic models assume that the agents that populate them can compute the answers to very complicated problems almost instantly. Even before the interest by economists in complex systems-style economics, this uberrationality was already being challenged by numerous economists, resulting in the development of a boundedly rational economics model (Rubinstein, 1998). This literature took several directions, and a few of them are discussed here.
Most individuals do not often perform complicated calculations like inverting a large matrix in their heads in a matter of seconds. Thus, some boundedly rational economics models simply assume that economic agents do not have the capacity to easily perform some calculations. There have been a variety of ways in which to implement nonrational agents. Perhaps one of the most notable is the “satisficing” approach of Herbert Simon (1996). Here, agents accept a solution to a problem or an alternative that yields an acceptable but perhaps not optimal solution or alternative.
Rational to Be Irrational
In some cases, the primary constraints on optimization and rationality concern the ability to gather information as opposed to the ability to process this information. For instance, if one wants to find the best price on a common consumer good, say a specific make and model of shoes, in a large city, one could potentially check every retail establishment that sells shoes. But the time cost required to do so may make it irrational to actually complete this comprehensive search for the optimal price. Thus, consumers may find it optimal to simply accept a reasonable price once found. Similarly, in labor economics, many models assume that agents search for jobs and calculate the optimal search behavior, given knowledge about the distribution of offers that exist; they ask themselves if they should accept the jobs offered or wait for better ones, given that they know the distribution of jobs that are available. This line of research still assumes that agents act rationally and also that agents solve a rather complicated optimization problem.
Limited Information Due to Network Constraints
Recall the previous discussion about the transfer of information across social networks. If the majority of information needed to solve a problem or to maximize utility or profits must be obtained from an information source that depends on some contact structure, then agents may optimize, but they do so with potential information constraints. The cognitive abilities of agents may be very powerful, but the agents can act only on the information that is available to them through the given interaction structure. In this scenario, agents may act rationally, given their information, but the actions may appear irrational to outsiders because of the limited information held by an agent. A simple example of this process is contained in the information cascades literature. Here, agents must choose between two similar goods, where one good is superior to the other. Agents receive a private noisy but informative signal about the quality of two goods and observe the choices of other agents choosing prior to them. Through the other agents’ actions and the agent’s own private signal, the agent rationally calculates the likelihood of each good being the superior one and chooses that good. Even though agents act rationally, it is possible that the agents may coordinate on a bad equilibrium where the agents all choose the inferior good. (See Holt, 2007, for simple examples.)
Adaption or Learning
Central to the idea of bounded rationality is the fact that agents must face some constraint on their ability to optimize. The constraint may be limited cognitive abilities, limited information, or limited time. All of these items force an agent to act in a way that does not guarantee optimization. In some of the cases described, agents simply act rationally given the constraint. But since many if not most economic models include a dimension of time, an alternative and increasingly popular approach is to allow agents to learn over time. The learning often takes on two different dimensions that this research paper calls experiential and imitative learning. With experiential learning, the agent uses his or her past experiences to try to improve on economic outcomes. This approach may include some type of trial-and-error learning where agents apply a heuristic to a given problem and view the results. Then when faced with the same problem, or a similar problem, the agent adjusts his or her behavior to attempt to reach a better outcome. Or it may proceed according to a more traditional economics approach where agents use a rational cognitive model where an optimal decision is made according to the available information at a given time. Then as more information is revealed, the agent reoptimizes according to the new information. With imitative learning, agents use the experiences of others around them to try to improve their outcomes. For instance, one might copy the strategy of a neighbor who has fared well in a labor market in one’s own search for a job. Imitative learning often takes on many dimensions. For instance, one can copy the actions of neighbors or the strategies of neighbors. This distinction must be considered carefully especially in light of the fact that actions are often more observable than strategies.
There are several common methods for incorporating either form of learning. Most involve some type of error or mutation process that allows agents to improve. For instance, suppose that one observes the outcomes of a selection of agents in a population (an agent’s neighbors). One method of learning would be for the agent to simply replicate the agent he or she observes who has the best outcome. Another would be for the agent to replicate the best agent in most cases but sometimes make a mistake and, when doing so, replicate another randomly chosen agent. Or in another method yet, an agent could choose another agent to replicate as a function of the other agent’s performance; better performing agents are more likely to be replicated than poorly performing agents, but all agents have some positive likelihood of being replicated. The ability of agents to make mistakes often allows the population and the individual agent to improve on their outcomes in the long run. For instance, suppose that every agent can observe the outcome of every other agent in a population. If every agent copies the best agent, then all agents will have the same strategy in the next time period, and no additional learning can result. This is fine if the best agent was acting optimally, but if the agent was not acting optimally, then the population will never act optimally. Thus, one can get stuck with a suboptimal strategy. Replication of some subperforming agents can maintain diversity, which may allow the population to reach a better long-term outcome. Another component common in many learning models is the ability of agents to simply make mistakes or errors. Similar to not always replicating the best agents, errors can allow for continued diversity in a population and the ability to more fully explore the set of possible solutions to a problem or game. It is common for learning models to incorporate both of these elements. For instance, a genetic algorithm mimics the reproduction found in nature (Holland, 1995). Better performing agents are more likely to be reproduced. But when an agent does reproduce, it does not produce an exact copy of himself or herself; mutations to strategies occur, and sometimes, strategies of agents are combined (as in genes of an offspring being a combination of parents’ genes). Finally, there is also a literature that discusses things like optimal rates of learning and optimal rates of errors. Further, it is sometimes best for rates of error to change across time. One may want a high rate of error early on in order to explore and cover a large range of the possible solution space, but once so-called good solution regions are identified, it may be best to begin to limit errors so that these good regions can be better and more fully explored. (See the discussion of simulated annealing in Miller & Page, 2007.) Thus, there can be a balance between the amount of exploration and exploitation in problem solving (March, 1991).
Dynamic, Complex Adaptive System
Models in complex systems are almost always dynamic. The preceding paragraph listed various ways in which agents learn. Learning is inherently a dynamic phenomenon. But dynamic properties of complex systems are not limited to this area. It is common that agents in a complex system model are changing in some way. Sometimes this change includes learning. But it may also include change in the form of new interactions for the agents, revelation of new strategies, or the creation of new types of agents, firms, or institutions in an economic or social system. Thus, not only are individual agents often evolving but the systems that guide the agents also are changing or evolving. Of course, this produces feedbacks between the system and the agents that make up the system. A strategy or action that does well today may not be the best strategy tomorrow or next year.
As an example, consider Brian Arthur’s (1994) “El Farol Bar Problem.” In the problem, individual agents in a population of fixed size must decide whether to attend an event (an Irish music night at a local bar, in the original example). If more than x% of agents attend the event, each agent that attends receives less utility than if he or she stayed home; the event is too crowded. But if less than or equal to x% of agents attend, then each agent attending receives more utility than if he or she had stayed home. The dilemma here is that if all agents use the same strategy, then everyone attends or no one does. More importantly for this discussion, the best choice for each agent depends on the choices of all other agents. Thus, an agent’s best strategy today may not be a successful strategy tomorrow if other agents learn and adapt to the system. For instance, suppose that agents rely on their friends to report attendance at the event to them in order to project attendance in the following week. If x = 75 and 50% of agents attend this week and honestly tell their friends that they had a great time, then one might expect to get more than 75% attending next week. This may lead agents to develop more sophisticated strategies that may include misinforming other agents. Further, the organizers of the event may also have an incentive to report attendance figures that may or may not be accurate in order to maximize attendance according to a profit function. The important thing to note is that even some simple scenarios or games can easily lead to complex behavior.
Lack of Top-Down Administration
In most examples of complex systems, there does not exist a central authority that is responsible for overseeing and coordinating activities of the various agents in the system. Thus, outcomes in the system occur as a result of ex ante uncoordinated actions. Coordination may occur in the system, but coordination does not occur as a result of an exogenously specified central authority. (It is possible, though, that such an authority may emerge from the activities of the system.) Thus, complex systems modeling is sometimes referred to as social science from the bottom up (Epstein & Axtell, 1996).
Economic Outcomes From a Complex System Perspective
One of the hallmark features of most complex systems is that one cannot understand the whole by independently understanding the sum of the parts. For instance, one can understand the incentives of buyers and sellers participating in a market as well as the rules or laws that define a market but still not fully understand the aggregate behavior of the market. In traditional economics, one might focus on something like a price as a market outcome and take this observation as an indicator of market performance or behavior. The study of complex systems embraces a larger goal of also understanding how the market emerges, how relationships between participants may form and fail, and how institutions, laws, consumer strategies, and firm organization change over time.
One of the reasons that complex systems are difficult to understand is because the interactions between system components tend to create complicated feedbacks and nonlinear relationships between various parts of the system. Another hallmark feature of many complex systems is the existence of multiple feedback relationships in the system. Feedback can be positive or negative. Positive feedback exists in a system if a change in variable x causes the system to respond to the change by creating further change in x in the same direction. Negative feedback exists in a system if a change in variable x causes the system to respond to the change by reversing the direction of the change in x. As an example of positive feedback, consider the juvenile crime rate in a neighborhood. Suppose that this crime rate is affected by the number of businesses in the neighborhood. More business activity leads to more jobs for young people, which lowers the crime rate. But a lack of businesses leads to fewer job opportunities and more crime. Further suppose that businesses are adversely affected by high crime rates. Now suppose that an exogenous change occurs in the system and several new businesses open in a neighborhood. By the relationships described, crime rates would decrease; this would lead to even more businesses entering the neighborhood and a further reduction in crime rates. As an example of negative feedback in the same example, suppose that the decrease in the crime rate is met by a decrease in enforcement of laws. This lax enforcement might then lead to an increase in the crime rate.
Of particular importance, positive feedbacks can lead to there being multiple equilibria in a system. As another example of positive feedback, suppose that there are two competing operating systems for a computer, X and Y. Further, suppose that the value to an agent of using a given operating system increases in the number of other agents who use the same system. Thus, if a large percentage of consumers use System X, the value of an agent’s using System X is higher than if a small percentage were using System X. One can think of the same scenario with Operating System Y coming to dominate the market. Thus, an equilibrium could be reached where there are a large percentage of Operating System X users and a small percentage of Operating System Y users. Or one could have the opposite scenario, with a large percentage of System Y users and a small percentage of System X users.
Understanding Complex Systems: Theory and Policy
The possibility of multiple equilibria in a complex system makes the issue of equilibrium selection even more important. For instance, in the simple supply and demand model taught in introductory economics courses, there is only one equilibrium and thus one prediction for the outcome of that model. But if a system has a multitude of equilibria, how does one make predictions? Further, how does one understand the process of selecting and attaining a specific equilibrium?
To begin answering these questions, one needs to consider that not all equilibria are created equal. An equilibrium that is associated with positive feedback is inherently unstable, while an equilibrium that has negative feedback surrounding it is stable. As an example, consider the following equilibrium: a pencil perfectly balanced on its end. This is an equilibrium for the pencil: As long as no one changes the conditions around the pencil by perhaps blowing on it or shaking the table on which it is balanced, the pencil will stay balanced as it is. But if the pencil tips just a bit, the positive feedback introduced by gravity will lead the pencil to tip a little further and eventually fall over. Positive feedback in any one direction away from an equilibrium can result in forces that drive a system away from the equilibrium. On the other hand, negative feedback is associated with stability. Consider the simple supply and demand model of an introductory economics course. In this model, prices act as negative feedback. If the price deviates from equilibrium, perhaps by dipping too low, the shortage created in the market acts to push prices upward and back to equilibrium. On the other hand, if prices increase, the surplus created in the market will pull prices back down toward the equilibrium. So if prices deviate in any direction away from equilibrium, the negative feedback in the system acts to restore the system to the equilibrium.
One way to predict which of the many equilibria in a system will occur is to consider whether an equilibrium is stable or unstable. Like the pencil balanced on its end, any unstable equilibrium requires very specific conditions to occur. If any of these conditions deviate slightly, the system leaves that equilibrium. As an example, throw a pencil in the air and let it land on the desk 100 times; how many times does it land balanced perfectly on its point? Unstable equilibria are almost never observed. Thus, when predicting which equilibrium will occur in a system, one can rule out any unstable equilibria as good predictors for the system.
However, one may still be left with many stable equilibria in a system. Which equilibria will be attained is another focal point for complex systems research, and many angles have been taken in addressing this issue. There are formal theoretical interests such as measuring the size of the basin of attraction of an equilibrium. (The basin of attraction for a given equilibrium is the set of states that lead to the equilibrium.) A larger basin of attraction should imply that the equilibrium will be attained more often.
More interesting from an empirical perspective, different methods of learning or behavior can lead to a different equilibrium. For example, consider the following simple situation. An agent wants to meet a friend for lunch and knows that they are going to meet at one of two restaurants, A or B, in 10 minutes, but the agent’s phone is broken, and he or she cannot contact the friend to coordinate on which restaurant. So the agent chooses one of the restaurants. Suppose that the agent fails to coordinate with the friend, who chose the other restaurant, so the agent eats alone. Now suppose that the next day the same situation occurs. What should the agent do? What strategy does he or she follow? One option is to follow a pure best response to what happened in the previous period. This strategy would lead the agent to choose the opposite restaurant from the last time. If the friend follows the same strategy, they will fail to coordinate again. On the other hand, suppose that the agent plays a best response to the entire history of the friend’s choices. The agent chooses according to the fraction of times the friend chooses each restaurant. Thus, as the agent keeps choosing the wrong restaurant time after time, his or her frequency of visiting each one approaches one half. Thus, choosing Restaurant A 50% of the time and Restaurant B 50% of the time, the agent will be able to meet the friend for lunch in 50% of the cases. (One quarter of the time, both choose A; one quarter of the time, both choose B; one quarter of the time, the agent chooses A and the friend chooses B; and one quarter of the time, the agent chooses B and the friend chooses A.) But one could do even better in this case by weighting the more recent choices more strongly.
Then as chance meetings occur, the probability of another chance meeting increases. But recall that one can’t go too far. If one plays a pure best response to only the last period, one will return to the possibility of never coordinating. As a side note here, suppose that one prefers Restaurant A and the friend prefers restaurant B. One could also analyze this game from the perspective of an altruist and an egoist strategy. An egoist chooses the restaurant that he or she prefers all the time, and an altruist chooses the restaurant that his or her partner prefers all the time. If two altruists or two egoists (again with conflicting restaurant preferences) play the game, they never coordinate. But if one of each plays the game, they always coordinate. Thus, there can be situations where diversity of preferences, tastes, or types can lead to better outcomes than if diversity is lacking.
Equilibrium selection may also be a function of path dependence (Page, 2006). For instance, in an example given previously, once Operating System X is chosen over Y by a majority of the population, it may be very hard to break out of this equilibrium. Further, even if X and Y start out equal in terms of quality, it may be the case that one comes to dominate the market. But if the minority technology makes improvements and becomes the superior choice, it may be hard to get the population to switch to the better equilibrium. Simply put, history can matter. Further, institutions matter too. As a simple example, there may be a role for the government to help push the public toward coordinating on a superior equilibrium if the population is stuck at an inferior equilibrium. Public policy can be used as a lever to push systems toward or away from one equilibrium or another, depending on the best interest of society.
Understanding Complex Systems: The Tools
There are a variety of tools used to understand complex systems. Traditional analytical modeling and statistical and econometric techniques are sometimes helpful in understanding complex systems. But their use is often limited by the severe nonlinearity of many of the systems. As mentioned previously, the nonlinearity results from the multiple interdependent relationships and feedback effects common in complex systems models. Because mathematical and statistical methods for dealing with nonlinear systems are limited, computational simulations and particularly agent-based computational experiments are commonly used in the study of complex systems (Miller & Page, 2007).
An agent-based model begins with assumptions about the preferences and behavior of individual agents, firms, and institutions and the interrelationships among them. After specifying the initial conditions (agent endowments of wealth, firm endowments of capital, etc.), a group of artificial agents is set forth and studied. Economists can then vary conditions in the model, changing, say, initial wealth endowments or different learning rules and perform a controlled experiment of the computational system. Such computational models are already common in many natural sciences—physics, for example. Use of these methods in economics is beginning to take hold, and there are a growing number of researchers engaged in agent-based computational economics (ACE). (See Leigh Tesfatsion’s Web site for a great overview of the current literature.)
As ACE grows, there are opportunities for complex systems research to merge and collaborate with the equally exciting and emerging field of experimental economics. Experimental economists use populations (usually small) of human subjects to engage in controlled experiments of economic relevance, perhaps risky gambles where individual risk preferences can be uncovered. One limit to human subject experiments is the scale to which the experiments can be run. Typical experiments may have dozens or sometimes a couple hundred participants. An agent-based computational experiment does not suffer from this limit. In fact, it is the goal of one group of economists using agent-based methods to build a model of the entire U.S. economy with one artificial agent for each person and firm residing in the United States. Of course, one strong limit of agent-based models is the fact that the agents are artificial, not real people. Thus, there is fertile ground that can be covered by using agent-based models to scale up human subject experiments and to use human subject experiments to add realism to artificial agent experiments.
An Example: Schelling’s Residential Segregation Model
As an example of a simple complex systems model in economics, consider Thomas Schelling’s (1978) residential segregation model. The model is so simple that one may be surprised that it generates segregation at all. To begin, assume that there is a town where all houses are located along one street like the one below:
Further assume that some of the homes are inhabited by people from the land of p, some are inhabited by people from the land of k, and some of the homes are vacant. The vacant homes are denoted with a v.
In this example, there are nine homes, seven families (four p families and three k families), and two vacant homes. Now assume that all houses have the same value to all families and that a family can swap its house for a vacant one at any time. Further assume that a family is satisfied with its home as long as one of the neighbors is of the same type as the family; a p is satisfied as long as he or she has one p neighbor, and a k is satisfied as long as he or she has one k neighbor. Thus, in this example, the only two families that are happy are the two p families at the fifth and sixth homes. All the other five families are unhappy about their current living conditions.
Now introduce dynamic movement into the model as follows. Take the citizens in order of location from left to right and ask each if he or she is satisfied at his or her current location. If the resident is not satisfied, ask if he or she would be satisfied at any of the homes currently vacant. If the resident prefers a vacant home to the current home, move him or her there. If there are two homes the resident would prefer to the current home, move him or her to the home nearest to the current location. If the resident is not satisfied but there is no vacant home in which he or she would be satisfied, the resident waits until the next opportunity to move and checks the vacant homes again. Thus, one would start with the family at the first home. This is a k family with one neighbor, who is a p. The resident is not satisfied with the current home. There are two locations vacant (third and fourth homes), but neither of these locations has a k neighbor. Thus, the resident would be unsatisfied at both of these locations as well and remains at the first home. Now move to the family at the second home. This is a p family with one k neighbor and one vacant neighbor. Since this family does not have a p neighbor, it is not satisfied either. Thus, it will move if it can find a vacant home with a p neighbor. The fourth home fits this criterion. Thus, the family at the second home moves to the fourth. One now has as follows:
The next three occupied homes (fourth, fifth, and sixth) are all satisfied. Thus, they remain at their current locations. The family at the seventh home is not satisfied since it has two p neighbors. But it could move to the second home and be satisfied. Thus, one now has as follows:
At the eighth home, there is a p family that is not satisfied. There are two locations this family could move to in order to be satisfied, the third and seventh home, and they move to the closest one, the seventh.
Finally, there is a k family at the ninth home that is not satisfied. It can move to the third home and become satisfied. This move yields the following:
If one checks all of the families again, one sees that they are all satisfied. And what one also may notice is that the neighborhood has become completely segregated with three k’s and four p’s all living next to each other. Note that this high level of segregation occurs even though the families required only that one of their neighbors be similar to them. Even with these modest preferences, the model generated a large amount of segregation.
Note that perfect segregation is not the only outcome in the model. The following configuration would be an equilibrium as well:
And there are other equilibria similar to this that would not be perfectly segregated. However, what one sees in this simple model is that even modest preferences for wanting to have neighbors similar to oneself can lead to large amounts of homogeneity within neighborhoods and large amounts of segregation.
One can make the model slightly more complex by introducing more dimensions to the neighborhood. Assume that there is a population of individuals that live in Squareville. Squareville is a set of 36 residential locations like the one below:
xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx
Again, one can populate the neighborhood with a set of agents of two types and some vacant locations. Again, let the families be satisfied if at least one half of their neighbors are of the same type as them. This time, take each family in a random order and check to see if it is satisfied. If it is, leave the family there. If it is not, look for the nearest location where the family can be happy and move it there. (Break ties with a flip of a coin.)
One can do this in the following way: Take 24 coins and place them on a grid like the one above. Place 12 of the coins on the grid heads up and 12 of the coins on the grid tails up at random locations. Now roll a six-sided die two times. Let the first number be the row and the second number be the column. Thus, if one rolls a two and then a three, look at the coin located at row two and column three. If there is not a coin there, roll again. If there is a coin there, determine if that coin is satisfied or unsatisfied. If it is satisfied, roll again. If it is unsatisfied, find the nearest location where the coin would be satisfied and move it there.
What one will notice is that as one proceeds with this algorithm, patches on the grid begin to develop where there are mostly heads and others where there are mostly tails. And eventually one will reach a point where every coin is satisfied, and in a majority of the cases, there is a very large degree of segregation. Even though each coin would be satisfied if it had only one half of its neighbors like it, many of the coins have only neighbors like them. The grid will have patches of heads only and tails only, with some borders in between.
One can also try this model with different parameters. What happens if families require only one third of their neighbors to be the same as them? What about three quarters? What if there are more vacant spaces? Fewer vacant spaces? What one will find is that the details of the outcomes will vary (for example, the location of the tails and heads neighborhoods on the grid), but the amount of segregation will still be surprisingly high. For instance, if one sets the tolerance parameter to be one third, significantly more than one third of a family’s neighbors will be the same as the family. (In addition to the exercise described, there are several simulation applets available on the Web that can be found in a quick search with such terms as Schelling segregation model simulation. One of the simplest to use is the Net Logo Schelling segregation model.)
This simple example displays many of the previously described characteristics of a complex system. The system is dynamic. There are multiple equilibria in the location choices of residents. The coordination on a given equilibrium may have very little to do with the preferences of agents; it may be a product of historical chance. Positive feedback results as neighborhood composition changes. For instance, a decrease in one neighborhood of type p individuals makes it less likely that other type p individuals will remain in the neighborhood. It may not be obvious from understanding the individual incentives and preferences of the agents that large amounts of segregation are likely to result. Simple agent motives lead to complex behavior and outcomes. Neighbors result from a specified interaction structure, in this example a line or a grid, but more complicated structures can incorporate actual neighborhood structures. Finally, even though the model is simple, diversity exists both in the types of agents and in the neighbor of a specified location.
This research paper has outlined the emerging field of complex systems in economics. All of the hallmark aspects of complex systems are present in almost all economic contexts. Complex systems are generally composed of boundedly rational, diverse agents and institutions who interact within a specified structure in a dynamic environment. Further, these agents and institutions often learn and update their behaviors and strategies and lack a centralized authority that oversees control of the system. These characteristics make the study of economics from a complex systems perspective natural.
Once one embraces a complex systems thinking within economics, many new avenues and tools for research open for one’s consideration. There is a rich history in fields such as physics that customarily deal with the nonlinear nature of complex systems models. As such, tools from disciplines such as nonlinear mathematics, computational simulations, and agent-based computational experiments are becoming more and more common in economics. These tools help economists to work through the complicated feedbacks and existence of multiple equilibria that are common in many complex systems.
In addition, it should be recognized that the complex systems approach to economics considers many tools from a variety of approaches that are already common in economics. For instance, learning models are not unusual in contemporary economics, nor are interactions across social networks or diversity. But as with the idea of complexity, the sum of combining many of these constituent parts often leads to a more rich environment and understanding than the analysis of these parts individually.
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