8.1 Introduction

With the ongoing increase in computing power, agent-based models have become a preferred tool of choice for the study of complex adaptive systems, especially those systems in which humans are a nontrivial part. Agent-based models are an appropriate choice for these types of systems as they allow the modeler to express the system more naturally, using a logical rule-based approach, rather than with closed form equations that require strong assumptions to be made about the said system (Axtell, 2000b; Epstein, 2006). This is particularly the case when the system is made up of a large (but not infinite) number of discrete, adaptive agents (i.e., humans) that may change, adapt, or coordinate their behaviors over time, recognized as organized complexity (Weaver, 1948). These types of systems currently stymie closed form analysis as well as statistical approximation (Weaver, 1948). Under these circumstances the most efficient way to understand the temporal dynamics of the system is to simulate it (Buss et al., 1990).

With respect to the analysis of a tax system, agent-based modeling (ABM) may be particularly useful. By representing the system as a collection of interacting individuals, displaying bounded rationality, each embedded within a space and social network, one can get around many of the strong assumptions necessary for classical microeconomic analysis such as infinite computing power, a fixed preference ranking, and perfect rationality (Simon, 1972; Axtell, 2000b; Epstein, 2006). There have been many agent-based models of tax compliance that have been created, perhaps most notably by Korobow et al. (2007), Hokamp and Pickhardt (2010), and Bloomquist (2011). Of particular interest here is Korobow et al. (2007), who explicitly introduced the use of network dynamics. This work was extended by Andrei et al. (2014) and combined with the more parsimonious underlying dynamics of the Hokamp–Pickhardt model (Hokamp and Pickhardt, 2010). We use that model as our base here. The model we created here, based upon previous work that was carried out in NetLogo (Wilensky, 1999), was re-implemented in a modeling framework specifically designed to allow researchers to create very large-scale agent-based models.

8.2 Networks and Scale

A great deal of work has been done demonstrating the impact of network structure on system dynamics. We will not attempt to summarize this vast literature here but rather refer the interested reader to Newman (2010) for a general discussion of network dynamics, or Jackson (2008), and Easley and Kleinberg (2010) for more social and economic treatments of network dynamics. However, relatively little work has been specifically applied to the dynamics of tax compliance (see Korobow et al., 2007, and Andrei et al., 2014, for notable exceptions). Andrei et al. (2014) demonstrated that network structure can have a significant effect on tax paying assuming, of course, that there is a social component to tax compliance. This impact was due to how information flowed around networks of different structures. The social networks used by Andrei et al. (2014) were, however, quite small comprising approximately 450 agents. While very few tax systems are populated by only 450 tax paying individuals, or even 1800, insights can be made by a study of such small-scale systems. This is due, first, because no part of our universe is so simple that we can understand it in its entirety. As such, careful study necessitates the creation of models that are simplifications of the real system. Secondly, the value of studying such smaller-scale systems is also a function of whether the system is one that is social or asocial. If the system is asocial, meaning that the humans in a said system make decisions individually without consulting with other individuals, then studying smaller-scale systems can be quite insightful as one is studying the individual decisions of humans. The only need for the creation of a system of more than one human is when humans interact to create the environment in which they make decisions, meaning they directly or indirectly influence each other's behavior and decisions. Moreover, if the system is populated with adaptive agents that interact, the system is now complex, and system behavior may change with scale (Anderson, 1972). Therefore, before one can claim to represent the dynamics of the system in question one must be able to test the dynamics at the appropriate scale. With modern computing hardware, it is now possible to represent taxation systems at a 1:1 scale.

To stress the importance, if the system is social, then the humans within it actively engage with one another to make decisions and have direct influence on decision-making. In this case, it may be necessary to more closely replicate the scale and structure of the system in question. This is due to the fact that the social structure may have a significant impact on how the humans interact with each other and, therefore, how information relevant to their respective decision-making flows within the society. Here, following Korobow et al. (2007) and Andrei et al. (2014), we assume that tax paying is a social process and, therefore, the impact of network structure and the scale of the system should be investigated.

The principle contribution of this chapter is the demonstration that, with modern computing hardware and software, researchers in the tax evasion field (and other economic sectors) are no longer forced to study small-scale systems but may now study systems that approach a 1:1 ratio of artificial agents to “real” taxpayers. This is important as scale may impact one's results. As shown by Gotts and Polhill (2010), system scale in land use with imitative agents can show different results as the scale increases. However, as reported by Nardin et al. (2014) scale may not impact system level results. While more systematic investigation of the importance of scale are not yet available, given the theoretic argument by Anderson (1972) and recent conflicting results by Gotts and Polhill (2010) and Nardin et al. (2014), researchers should be cautious in drawing conclusions about system level results if they have not investigated the impact of scale. Having said that, one should also keep in mind that these sorts of analytic systems should be used for the purpose of gaining insight and for understanding the potential dynamics of a system rather than point prediction. Therefore, how much effort should be spent on investigating the impact of scale needs to be determined on a case-by-case basis at the discretion of the researchers involved. For example, one can gain insight into the dynamics of a Schelling (2006) segregation model at relatively small scales, while one may need much larger scales to understand epidemics in a major metropolitan area (Parker and Epstein, 2011).

While scale is an important issue, there are surprisingly few systematic analyses of how system dynamics may change with scale (notable exceptions, mentioned include Gotts and Polhill, 2010, and Nardin et al., 2014). Since Anderson (1972) published his seminal article, the potential issue of scale has been highlighted as a feature of complex systems that should not be ignored. Unfortunately, we have been unable to locate published work that addresses the problem of determining what the appropriate scale is for modeling the dynamics of a particular system. For example, a single water molecule does not possess the property of wetness. It takes a number of water molecules to create the emergent phenomenon of wetness. But how many water molecules must be included in a model to create the characteristic of wetness to claim we have captured the relevant features? What if we decide to model wave action instead of wetness? Similarly, a single ant will have a great deal of trouble surviving on its own. An ant colony, however, can be quite adaptive for farming, going to war, creating structures, and calculating optimal foraging paths (see generally, Hölldobler and Wilson, 2009). How many ants must be modeled to capture these emergent properties?

Just like water molecules and ants, there is growing literature suggesting that as human systems change in scale the dynamics associated with them may also change. This work grew out of findings in biology that showed very specific allometric scaling relationships among various organisms. For example, organisms enjoy economies of scale, for example, an elephant is more metabolically efficient than a mouse (West et al., 1997; West and Brown, 2005). Human social systems appear to function in a similar way. Human societies become more resource efficient as they grow in size (Bettencourt et al., 2007). The efficiency is for such things as miles of road, miles of electric lines, and so on. Unlike nonhuman organisms, the “social production” of human societies may be subject to scaling dynamics. Starting with West and continuing with Bettencourt and others, many have reported that signals have been found that indicate that the dynamics of human populations may change in systematic ways as the scale of human society changes. Specifically, for our purposes here, there appears to be specific changes to income, gross adjusted income, and tax revenue as human society changes in scale (this study looked specifically at cities within the United States). As described in Gulden et al. (2015), while incomes tend to grow superlinearly as cities increase in size because the United States has a progressive tax, tax revenue increases at an even faster rate than does income as the size of a city increases. However, there is more to this story, if one categorizes income into a number of income bands then one can see that lower incomes increase linearly while the highest incomes grow superlinearly. It is this growth in the highest income categories that causes overall income to appear to grow superlinearly. This, in turn, will obviously have impacts on the overall growth in tax revenue as cities increase in size. This being the case, it is our contention that researchers should be cognizant of the effects of scale on their analyses and their conclusions and test their models at various scales.

Given the bounded rationality of human actors, scaling up a social system may have other effects, such as changes to network diameter (a measure of how far apart components of the network are). This is one way to think about how long it may take information from one node to move to another node. This is a function of network structure. In a fully connected network, one in which every node is connected directly to every other node, moving information from one node to another is trivial. However, if we make the rather mild assumption that humans have finite ability to process social information, which in this context is the number of other individuals they can know (Dunbar, 1992), then as the size of the whole network grows beyond the number of individuals a human can know that we can no longer have a fully connected network and that information must flow around the network via intermediaries. Clearly, this will impact how information will move from one individual to another. How much this impacts the flow will be a function of the structure of the network. For example, the Small Worlds (Watts and Strogatz, 1998) network structure can be particularly efficient in this case; however, it cannot fully mitigate the impact of network size growing much larger than the cognitive ability of the humans. This reinforces the importance of knowing the network structure underlying the system of interest. For example, if we assume the number of relevant social connections is held fixed and constant at 50 (again this can be any finite number much less than the overall population), as can be seen in Figure 8.1, when network size is less than or equal to the social connections number the graph may be fully connected and information can flow directly from any individual to any other individual. However, as the network increases in size beyond that of the cognitive capabilities of the humans that make up the network the closeness of the network decreases rather abruptly, which is another type of phase transition. As the network size overtakes the social connection number, it is more difficult for information to flow from one individual to all others. These dynamics may be exacerbated by other network formation characteristics such as homopholitic or assortative dynamics (Newman, 2010). This being the case it is important to explore the dynamics of the system of interest at the appropriate scale, or as close as is possible. This is simply a thought experiment that motivated the work reported within this chapter. Here, we explore these dynamics with a simple income tax system and explore how its dynamics change with the structure of the social network and the size of the society.

While this chapter focuses on the analysis of large-scale social systems via distributed ABM, another approach used to analyze very large-scale economic systems is that of econophysics (Mantegna and Stanley, 1999). Econophysics uses the tools of statistical physics to study economic systems. While many insights can be generated by this approach, as discussed in Chapter 11 of this book, the use of these techniques requires strong assumptions about the human component, such as the exclusion of learning and adaptation, heterogeneity of strategies and behavior, and systematic biases that may limit their ability to generalize. This being the case, we feel that this approach should not be the only way to study large-scale human complex systems. Therefore, we focus our efforts here on the creation of large-scale agent-based models in an attempt to show others in the field that such simulations can be created, run, and analyzed efficiently and thereby be an effective analytical tool. Moreover, keeping track of the dynamics of a social system as a temporal trajectory may be very important and likely cannot be done in a way other than via simulation (Buss et al., 1990), therefore, in this case, we turn to agent-based models. Finally, in addition to tax system dynamics, agent-based models have been used to study other puzzling human economic system dynamics. For example, in the United States when the retirement age was lowered from 65 to 62.5, that did not result in an immediate change in the median retirement age. Rather, there was a multi-decade lag before the median age lowered. Axtell and Epstein (1999) was able to demonstrate conditions sufficient to cause such a lag using a parsimonious agent-based model.

In the sections that follow, the model and our analysis of the impact of scale on model dynamics are discussed in more detail. If scale has an impact here, we expect it to manifest in the population variance. More specifically, we expect to see increased variance in tax payment and audit risk perception. However, it should be noted that, given this particular tax regime model, the network micro structure is fairly constant, and there are no homophilitic biases in the population. Our intuition is that we will see very little change across different scales.

8.3 The Model

We will use the ODD+D Protocol to describe our model (Müller et al., 2013). For details regarding this protocol used to describe agent-based models please see Chapter 1.

8.3.2 Design Concepts

8.3.2.1 Theoretical and Empirical Background

The model was derived from the prior work by Hokamp and Pickhardt (2010) (additional details about this foundational model can be found in Chapter 9), (Korobow et al., 2007; Andrei et al., 2014). Of note, while this study implements the Andrei et al. model, it is not, strictly speaking, a replication of that previous work. As stated above, the contribution from this piece is the demonstration that massive-scale agent-based models can be used productively by researchers engaged in tax evasion analyses, and not that the prior results from the work of Andrei et al. (2014) can just be replicated. As is clear in the following discussions, we, in fact, do not replicate the results of Andrei et al. (2014). While that is interesting, one should not place too much stock in this deviation as we have not attempted a formal replication here. For example, some of our parameter settings and our activation regime differs from the previous work of Andrei et al..

Agents are embedded within a social network. The network is one of seven potential types: Erdös–Rényi (Boccaletti et al., 2006), small worlds (Newman, 2010), power law (Clauset et al., 2009), ring world (Boccaletti et al., 2006), von Neumann (Weisstein, 2002), Moore (Weisstein, 2002), and finally no network may be present (all agents are disconnected from each other). Agents are of three basic types (specific proportions are discussed later): honest (these agents always declare all of their income), dishonest (these agents calculate the lowest amount of income to declare based upon their risk calculations discussed further), and imitating (these agents declare income based upon the behavior they observe among their social network neighbors). Examples of the networks used in the simulation are shown in Figure 8.2. We chose these networks for two reasons: first, these were the networks used by Andrei et al. (2014), and second, they represent a nice spread of network structure and characteristics as shown in Figures 8.3 and 8.4. For example, as can be seen in Figures 8.3 and 8.4, these networks have statistically distinct distributions of degree centrality and closeness centrality. Differences in these measures will impact how information can spread across the network, here meaning information about tax evasion and audits. This, in turn, may have an impact on agent learning and decision-making (Barkoczi and Galesic, 2016).

8.3.2.2 Individual Decision-Making

There is one major decision made by agents in this model: that is, how much of their income should be declared and, thereby, taxed. As discussed earlier, there are three types of tax paying agents in this model and how much income an agent decides to declare will depend on what type a given agent is. An honest agent will simply declare all of its income. Dishonest agents will declare the least income they “feel” is necessary. Here, necessity is a function of their risk-aversion and how likely they feel they are to be audited. Conforming agents look to the ratio of declared to actual income of their network neighbors to decide how much income to declare. Details of the decision-making can be found further in the Section 8.3.3.5.

8.3.2.3 Learning

This model includes mild learning. Over model time taxpayers will adjust how much of their income to declare based upon an increase in information about how likely they are to be audited. It should be noted that this learning is heavily biased to overestimate the likelihood of being subjected to an audit.

8.3.2.4 Individual Sensing

Agent sensing takes place via the social network, if present. Agents only attempt to collect information about the tax paying environment and ignore all other features. Specifically, agents attempt to collect information on their social network neighbors regarding audit experiences and tax paying strategy.

8.3.2.5 Individual Prediction

The only prediction agents undertake in the model is determining how likely they think it is that they will be subjected to an audit. Given agent decision-making it is very likely that this value will be much higher than the actual probability of being subjected to an audit.

8.3.2.6 Interaction

As this model focuses exclusively on tax paying behavior, agents have relatively simple interactions. These interactions include observation of the ratio of reported to actual income of their immediate network neighbors and whether or not a neighbor has been audited.

8.3.2.7 Collectives

There is a single collective in the model made up of all taxpayers. This collective is represented as a social network and is imposed exogenously at instantiation. Furthermore, it is static for the duration of the run.

8.3.2.8 Heterogeneity

Agents are heterogeneous in a number of ways. Each agent is provided with an income level and risk-aversion that is potentially unique. Moreover, each agent is in a unique location within the social network. This position will provide each of them with a unique view into the social dynamics going on within the population and unique set of experiences over the course of the simulation run creating the overall dynamic. These features were varied for each agent for each replicate of the simulation.

8.3.2.9 Stochasticity

Stochasticity is in the model in a number of places. First of all, agents are randomly assigned to “locations” within the social network. Secondly, each agent is assigned a risk-aversion (random uniform distribution from 0 to 1) and an initial value for the perceived probability of being audited. It should be noted that, differing from Hokamp and Pickhardt (2010) and Andrei et al. (2014), we have given all agents a homogeneous income of 100 tokens (tokens are used here as an abstract unit of income). We made this choice to limit sources of randomness. Given that agent decision-making is not a function of income level, we feel this was a reasonable simplification to make.

8.3.2.10 Observation

A number of data are collected as the model runs. Data on the model's network structure between agents as well as individual-level data for each of the agents are collected. In all iterations of the model, complete initialization items for the agent are recorded, including income, personality type (honest, dishonest, imitator), risk aversion, and the individual's initial probability of being audited. At every time step of the model, the agent's declared/actual income is also recorded. The entire model records the mean VMTR over the course of all time steps across the model, where VMTR is the voluntary mean tax rate, specifically: taxes paid divided by total income for each agent averaged over the population.

8.3.3 Details

8.3.3.1 Implementation Details

While the original Andrei et al. model was created in NetLogo 4.3 (Wilensky, 1999) and is reported in Andrei et al. (2014), this implementation of the model was created in Python using our own ABM framework. The framework and this model are freely available here: https://github.com/ceharvs/mabm. While Python is not as high performance as other compiled languages such as C++, it is highly accessible, easy to learn, and has a very strong user community with many packages and libraries that can be leveraged by users. This being the case, and because one did not already exist, we chose to create this framework in Python.

8.3.3.2 The Agent-Based Modeling Framework

We chose to create our own ABM framework for a number of reasons. The main reason was that distributed computing has the ability to facilitate very large-scale agent-based models by addressing the traditional scaling limitations. However, reliable and efficient communication strategies are needed to synchronize agent states across multiple processors. The protocol used to synchronize agent states across processors has a significant impact on the efficiency of the tool. While there are many management methods to synchronize activities across a processor, including conservative methods that perform a complete synchronization of all agent state information at every time step (Fujimoto, 2000), we chose an alternative approach that uses an event-driven technique to synchronize agents. This procedure only synchronizes changes to pertinent information in the model at each time step. This technique requires less information to be broadcast, which reduces the run time of the simulation while maintaining consistency in the model. This method works in this context due to the relative simplicity of our model and the fact that coordination among agents is “loose,” meaning if agents occasionally receive information from a previous time step the overall system dynamics are not negatively impacted. These features allow us to make the trade-off between performance and perfect synchronicity.

Our massive-scale ABM framework was developed in Python using the Message Passing Interface, or MPI, capabilities to support distributed agents. The framework was designed using both of the aforementioned synchronization techniques: conservative and event driven. A sample rumor model was used in this experiment, where each agent had exactly two assigned neighbors that were distributed according to the probability of having a neighbor on a foreign processor. Trials were run with 20 million and 80 million agents with 4, 8, 16, and 32 processors. The completed timing analysis shows that the event-driven technique was significantly faster for this model than the conservative method. In addition to the overall speed increase, the event-driven technique also scales up in number of processors as well as number of agents per processor more efficiently than the conservative method. As agent activation regimes can have a significant impact on the dynamics of the model (Axtell, 2000a), one should be careful when choosing a synchronization method when moving from a “traditional” computational platform such as a laptop to a cluster computing system. Knowing that conservative synchronization may be needed to create the dynamics needed for a particular model (strong synchronization overriding execution efficiency), we included both in our framework.

8.3.3.3 Initialization

The process of initialization is always the same; however, depending on the input files used the resulting tax system may not be the same as any other. Upon initialization, the simulation creates the taxpayer network. Initially, this is simply a set of links and nodes for each type of network at each scale. Once the “skeleton” network is created, via functions available in python-igraph Python package (igraph.org), the nodes are given individual characteristics that then make the tax paying system. Further, this allows us to ingest the networks once and then run many other design points on this network without needing to recreate the basic network structure. Given the time necessary to generate the network, this is a more efficient method for running parameter sets against the same network structure. By randomly assigning taxpayer attributes to nodes within the network we are able to test the impact of having different types of agents in different network locations without the computational expense of destroying and recreating the network for each run of the model. It should be noted that this is a significant deviation from the initialization procedure used in Andrei et al. (2014) where the networks are recreated each time the simulation is instantiated. As agents are randomly created for each simulation run and randomly assigned to nodes in the network, akin to randomly distributing agents across a fixed topological landscape, we felt this was a reasonable approach to take given the trade-offs in performance we faced with network generation. Moreover, to further control randomness and isolate the impact of network structure, where appropriate, we held the ratio of nodes to edges as close to 1.2471 times the number of nodes, rounded to the nearest integer. This number was used as it allowed us to create the networks while holding some characteristics constant: in this case the ratio of nodes to edges. Clearly, this specific ratio does not work for all topologies, such as a ring or a von Neumann network as each node has a fixed degree of 2 and 4, respectively; rather, this heuristic was used for power law, preferential attachment, and random network topologies. Finally, it should be noted that the impact of holding the network structure fixed was not explicitly tested.

In the original model, the population of agents was made up of 441 total agents, 50 of which were designated as honest and 50 of which were designated as dishonest. As one of the points of the current study is to change the scale of the population this specific number will be replaced by a percentage. To correlate with the original work we will use 11.34% of the population to be designated as honest and an additional 11.34% of the population to be designated as dishonest.

8.3.3.4 Input Data

The model uses two input files. The first file specifies the network size and structure. This file specifies the tax paying population. For efficiency, we created the network specification separately from the simulation (as discussed above). The input file for the social networks was created with the igraph python package. The second file specifies the agent population and global parameters. These values will be discussed in more detail in our discussion on the design of experiments used here.

8.3.3.5 Submodels

There are three submodels used in this agent-based model: auditing, declaring income, and updating an agent's perceived likelihood of being audited.

Auditing Submodel

This submodel is quite simple. At each time step agents draw a random uniform number from 0 to 1. If this number is less than the objective audit probability, then the agent undergoes an audit. The agent then draws another random number, if this random number is less than the probability of apprehension then the audit is assumed to uncover any income not declared. If this occurs and there is undeclared income, the agent is severely penalized and then updates its perceived probability of audit (see further text). As the number of taxpayers in the simulation increases this becomes an unrealistic design element of the simulation. We are aware of no taxing authority that performs audits as a fixed proportion of the population they serve. We chose to implement this audit submodel to do as little disruption to the original formulations as possible. As discussed in the conclusion, we intend toadd a budget/personnel constraint on the taxing authority in our future work.

Declaring Income Submodel

Honest agents always declare all of their income; therefore

8.1

where is agent 's declared income at time , and is agent 's actual income at time . Dishonest agents calculate a minimum income to declare based on experience and risk. The assumption for the following three equations is: if the agents feel they have a high likelihood of being audited they will declare all of their income even though they would prefer not to, if the agents feel there is very little risk of being audited then they will declare none of their income, finally, if the agents feels there is some risk they will declare some of their income. Following Hokamp and Pickhardt (2010), if

8.2

then will be 0. However, if

8.3

then will be as the agents assume it is too risky to not declare all their income. In the above equations is the subjective probability of apprehension, is the agent's risk aversion, is the penalty rate, and is the tax rate. However, if falls within the interval defined by Eqs (8.2) and (8.3) then an agent's declared income becomes

8.4

Once again, following Hokamp and Pickhardt (2010), imitating agents observe the behavior of their social network neighbors and base their declared income on the product of the ratio of actual to declared income of their neighbors and their own income

8.5

where is the number of link neighbors an agent possesses. Following Andrei et al., if an agent is not apprehended and its (the objective probability of apprehension) then the agent decreases its by 0.2. If an agent is apprehended, it adjusts its declared income and subjective probability of apprehension as follows

8.6

These dynamics were used to incorporate the human decision-making heuristic of availability (Tversky and Kahneman, 1973) and the fact that humans are loss adverse (Kahneman and Tversky, 1984).

Audit Likelihood Submodel

This submodel is also relatively simple. If an agent is audited it sets its subjective probability of audit to 1.0. Again, following Andrei et al., if the agent is not audited in a given time step, then the agent decreases its subjective probability of audit by 0.2. The agent's subjective probability of audit is bounded by 0 and 1.

8.4 The Experiment

The model is run for 40 tax cycles. Each time step is a year and consists of a full “tax cycle.” Meaning, agents may be audited and may be apprehended by the taxing authority, discuss what happened within their social network the previous year, decide what to do in the current tax year, and then file a tax return.

We ran a simple full factorial experimental design for this study. The factors that were varied included the objective probability of being audited, , the structure of the social network, the number of taxpayers, and the number of processors on which the model was run. The experiment is summarized in Table 8.1. All other parameters are held constant, save agent risk aversion and subject audit probability, , which, as discussed earlier, are drawn from uniform distributions.

8.5 Results

The simulation results are measured by average VMTR at the end of the 40 tax cycles. In general, while some differences are observed among the networks, the differences are not as dramatic as those reported by Andrei et al. (2014). Likely, this is a function of our implementation. As discussed earlier several implementation decisions were made that could cause differences in results. However, it is worth noting these differences as it may point to fragility of the results reported by Andrei et al. As shown in Figures 8.5 and 8.6, while there are few end state differences between networks of different scales and the basic outcome of the simulation, there are differences between networks of different types, which observation is consistent with the findings reported by Andrei et al. These differences hold as we increase the scale of the population of agents. The dramatic increase in declared income observed in the no network case was caused by the imitator agents defaulting to honest behavior if they had no neighbors. However, as can be seen in the other cases, the mean level on income reported is approximately 40%. Thus, the taxing authority is faced with a very dramatic tax gap. Rather than receiving 25 tokens of tax from each individual the taxing authority is only receiving 10 tokens on average, producing a gap of 15 tokens.

8.5.1 Impact of Scale

In this case we did not see a noticeable change caused by changing the scale of the system, consistent with our intuition about the specifics of this model, as discussed earlier. What was observed, however, is a much stronger signal about mean behavior, which is likely attributable to the central limit theorem. However, what is interesting is that while there was a similar collapse in the variance it was not as dramatic. We have observed this dynamics in other complex systems. As the system scales up there is a clearer and clearer signal regarding mean behavior but the variance in agent behavior continues to remain high, the so-called macro-level stability with micro-level dynamism. This sort of stability/dynamism can be seen in real-world systems such as employment in the United States. Although there is a relatively stable nationwide unemployment rate there is a great deal of dynamism at the employee level. Roughly five million individuals change jobs, two million leave the workforce, and two million join the workforce (Fallick and Fleischman, 2004). Another way to think about this dynamics is comparing it to a standing wave in a river. A standing wave is caused by water flowing downhill over an obstacle such as a rock. The wave remains stationary and collocated with the obstacle. Despite this stability, the water molecules that make up the wave are constantly changing. Just like nationwide employment, at one level we see stability, the wave, while witnessing micro-level dynamism. While being able to represent that both the micro-level dynamism and macro-level stability are not necessary for all analyses, it is important to be able to create analytics that can capture both when necessary.

One issue we do not see at the large scale is the impact of the unrealistic audit regime. If a fixed proportion of audits had a disproportionate impact, we would expect to see higher voluntary tax rates as scales increased. This is, however, not what we see (Figures 8.7 and 8.8), while variance does decrease, we do not see a systematic increase in taxes paid.

8.5.2 Distributing the Model on a Cluster Computer

Here we show that a massive-scale agent-based model of tax evasion can be created based upon previously published model formulations, specifically Hokamp and Pickhardt (2010) and Andrei et al. (2014). The typical reasons for moving to a parallel and distributed simulation architecture involve memory needs and how much computing power each agent will need, which is usually a function of how cognitively sophisticated the agent is. Given the simplicity of the agents involved in the simulation used here, one could argue that parallel execution is likely to be unnecessary. However, as shown further, even relatively simple agents at a US national scale consume more memory than most individual computers possess.

As discussed earlier, simulations are an important tool for social research. Furthermore, simulating social systems at the appropriate scale may be very important for a complete understanding of their likely dynamics. However, designing, building, and executing massive-scale agent-based models are nontrivial. When working with a simulation run across a set of computers that do not share a common memory, maintaining synchronization among the computing processors and designing agent communications so as to allow for social processes while not slowing the simulation down with relatively slow cross-processor messaging can be quite difficult.

Furthermore, agent activation can be tricky. As shown in Axtell (2000a), the specified agent activation regime can impact the results of a simulation significantly. Activation refers to how agents are “woken up” and do things. Although in the real world events happen simultaneously, it is difficult to do this en silico. Executing an agent-based model on a single processor necessitates waking up agents in a given order, one at a time. In order to simulate simultaneity under these conditions one typically activates agents in a new random order at each time step and “buffers” the agent states until after all agents have been activated. Buffering the state means the agents determine their new state but do not “reveal” it while other agents are still being activated. In this way, all agents are dealing with information from the same time step. There are other activation regimes that can be used, of course, and one should choose the method that is most appropriate for the system being simulated (Axtell, 2000a). As one moves to multiple processors, events can happen at the same time. However, now the simulation may become out of sync across the processor if, due to other processes running on the processors, execution time differs from one processor to another. For example, this could mean that one processor may be executing simulation time step 23 while other processors are executing time step 26 because an agent on the first processor had a particularly time-consuming set of calculations. Typically, this requires a central process to track the simulation progress on each processor and ensure that processors advance together. This means the simulation cannot move faster than its slowest process. Of course, the focus of the experiment described here was that of system scale, and so it should be noted that we did not explicitly test agent activation regimes in this study, rather we chose a single, logical one (discussed earlier) for this situation.

Another issue with distributed simulations is the communication among processes on different processors. This is typically very inefficient and can even negate any efficiency gain that is achieved by distributing a simulation to multiple processors. As can be seen in Figures 8.9 and 8.10, there is actually very little difference in the runtime of this simulation as it is run across additional processors. This is because the agent level activity is quite simple and, in terms of time consumption, the speedup associated with additional processors is surpassed by that needed to synchronize communication across multiple processors and keeping everything in sync. Additionally, the network structure may heavily impact the execution time of the model, as it may increase or decrease the amount of cross processor messaging that will take place during a run. Certain networks, such as the ring, as well as the absence of any network have less connections between agents that span multiple processors. Simulations that run with these network structures will be the least time consuming as there is little to no communication required between the processors. Another simulation with a high level of interaction between agents on disparate processors, such as a random network with high connectivity, will be much more time consuming. As the model scales with the number of processors used, the amount of communication needed between agents to travel between processors also increases.

As an example of single run performance, for 10,000 agents on the preferential attachment network, the average time to run the simulation is 34.7 seconds on a single processor, which decreases to 10.7 seconds when run on 8 processors. This speedup is very small compared to the potential, since this model, for this size of a population, is already quick to complete. Looking at the same timing for simulations with one million agents, the run time decreases from 2360 seconds on 2 processors to just 827 seconds on 16 processors. The preferential attachment network is highly connected between processors. A better example of the advantages of parallelization is shown when we use the ring world network: the run time decreases from 1728.5 seconds on 2 processors to 226.4 seconds on 32 processors (aggregated performance is shown in Figure 8.10).

Memory usage may also necessitate distribution. In the present case, even with very simple agents the memory footprint of the simulation grows quite large as the agent population approaches national scales. A simulation with only 10 agents consumes 23 MB in memory, but as the size of the simulation increases to one million agents, the memory usage increases to roughly 2686 MB. Continually increasing the number of agents will scale this problem up to a point where it is unreasonable to run the simulation on a single machine and the problem needs to be distributed, especially on the scale of millions to billions of agents. While other languages are more efficient with their use of memory, such as C++, this problem cannot be fully mitigated; regardless of chosen language, scale will eventually necessitate distribution (given current hardware availability). Moreover, it is not unusual for the memory use of an agent-based model to scale super-linearly with the number of agents, especially when modeling a social network among the agents. Meaning, one can very quickly run into memory issues when scaling models up, especially network-focused models. In the case presented here, clearly we do not need to use a cluster computer due to memory demand, rather we could test and demonstrate the use of this agent-based modeling framework with a simple model already existent within the tax evasion analysis literature.

8.6 Conclusion

This chapter introduced a relatively efficient, Python-based, framework for creating massive-scale agent-based model which may be run on cluster computers. The model used as a tax evasion test case was based upon the work previously reported by Hokamp and Pickhardt (2010) and extended by Andrei et al. (2014). Our primary concern, here, is to test the importance of scale within this model of tax evasion. In this particular case, we found little sensitivity to scale, with VMTRs staying reasonably consistent. However, we continue to feel that this is a feature of complex social systems that warrants additional careful study. While our findings do differ from those reported by Andrei et al. important differences exist between our implementation and prior work. These differences exist for computational efficiency when using a cluster computer and to limit some sources of variation. We chose to reduce the sources of variation in this case to better isolate the potential impact of scale. In future work, we intend to extend the model to improve its verisimilitude as scale increases including, for example, a budget/personnel constraint on the taxing authority. Furthermore, we feel that careful analysis of the interrelationships among network structure, size, and simulation memory use is important. It would be useful for the ABM community, in general, to better understand the trade-offs involved in networks, agent population size, and agent verisimilitude. In conclusion, with modern software and hardware, researchers are now able to carefully explore the importance of scale when undertaking a study of tax evasion and other social complex systems.

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