5.2.1 Purpose

SIMULFIS aims to test the consistency and acceptability of different theoretical hypotheses proposed in the literature on tax compliance [I.i.a]. The purpose of the research line is to fill a number of gaps in previous contemporary research on tax evasion in order to increase the understanding of the causes for the variations of tax evasion levels across time and space. Contrary to most of the studies on tax compliance that test separately different factors that hypothetically affect tax behavior, our model seeks to study the integration and interaction of different mechanisms that have been tested in isolation by previous research (SIMULFIS theoretically oriented aim). The use of an ABM simulation allows to overcome one of the most important shortcomings of standard economic models on tax evasion: to build more realistic models with heterogeneous individuals and social interaction mechanisms properly modeled and to simulate complex social dynamics in a realistic way (SIMULFIS methodologically oriented aim). Once the model is properly calibrated and validated against the Spanish empirical case, it is likely to become a useful tool to assess existing tax policies as well as their possible reforms (SIMULFIS politically oriented aim). The following are the questions that drive the research and development of the ABM tax model:

• To what extent is rational choice theory enough to explain estimated levels of tax compliance?
• What is the effect of normative commitments on tax compliance?
• What is the effect of social influence on tax compliance?
• Which social scenario optimizes tax compliance?
• Which combination of tax policies (deterrence scenario) is able to reduce tax evasion?
• How is an agent-based model calibrated empirically for the study of tax compliance?

[I.ii.b] While SIMULFIS has been mainly developed as a tool for theory development, it also aims to allow quantitative predictions to be contrasted with different empirical cases as external validation strategy. SIMULFIS has been designed to be so open and flexible as to benefit both tax policies researchers and decision makers.

5.2.2 Entities, State Variables, and Scales

[I.ii.a] In SIMULFIS there are two kinds of interacting behavioral entities: humans and institutions. Humans are modeled in different subtypes, while the tax central authority is modeled as an individual institution. Each human agent represents an individual taxpayer with some economic income facing a decision-making process about how much of that income she will declare to be taxed in a compulsory tax system with progressive rate brackets, and some audit and fine procedures with redistributive aims. Humans are embedded in a social network with a small-world-like topology and a homophily structure corresponding to six subtypes as a result of their occupational status and income level.

The space dimension is abstract and meaningless, without spatial units, although the social network topology is relevant for defining the agents' information scope. There is no spacial environment but endowment, institutional, and social environment: the agent constrictions (the decision context) could be considered as composed by (i) a set of income original distribution (ii) a set of normative taxing rules – the tax rates by income brackets, the audit procedure, and the institutional fines; and (iii) a set of neighbors' behaviors.

[I.ii.b] The main attributes or state variables that characterize the tax agents in SIMULFIS are² identity code, gross income (), income level category (high, medium, low), declared income (), class (wage earners, self-employers), homophily group (income level class), fraud opportunity use rate (FOUR; see [II.ii.c] for details), behavioral type (unconditional evader, unconditional taxpayer, conditional taxpayer using the decision filters), perceived sanction risk, perceived personal tax balance, perceived neighborhood tax balance, amount of tax due, amount of tax paid, amount received from social benefits (), social influence coefficient (), support to progressivity, perception of system progressivity, normative satisfaction, period tax behavior (tax abider, tax evader), memory of previous fraud behavior, memory of previous audits, and memory of previous sanctions. Agents' neighborhood is defined as a list of linked agents. Other auxiliary attributes are used for intermediate computations and procedures. The TCA institutional agent (Tax Central Authority), acting as the tax system, has no attributes and no memory other than aggregated period by period records about gross tax revenue – as line chart. All economic attributes use currency as unit (Euro, in the case of Spain).

Notation	Description
	Total number of agents
	Audition probability
	Amount of fine for tax evasion
	Total (gross) income of agent
	Income declared by agent
	Tax rates (over real income)
	Tax rates (over declared income)
	Number of previous tax periods
	Social influence coefficient
	Expected utility of declaring for agent
	Minimum income threshold for receiving the social benefit
	Social benefit received by agent when
	Social benefit received by agent when

Notation	Description
	Expected utility
	Tax balance (personal and in the neighborhood)
	Perceived progressivity of the tax benefit system
	Support for the progressivity principle
	Perception that the system is progressive
	Perceived probability for agent of being caught if she evades
	Social benefit received by agent
	Audits to agent in previous periods
	Audits in agent's neighborhood (including ) in the preceding period
	Agent's number of neighbors
	Fraud opportunity use rate (FOUR) for agent in period
	Median of FOUR in agents' neighborhood in the preceding periods

[I.ii.c] The exogenous parameters of the model include the following, with no change along the simulation run. In the up-to-date publications it is set for the Spanish case at year 2011, but SIMULFIS model can be set to other sociohistorical contexts with ease.

1. 1. Income () brackets and tax rates (), thus offering the possibility of empirical calibration for specific existing tax systems or even counterfactual tax policies. In case of Spain (2008) the corresponding tax rates for annual income brackets, in Euro, are as follows: 0% for less than 5050; 24% for 5051–17,360; 37% for 17,361–32,360; 43% for 32,361–120,000; 44% for 120,001–175,000, and 45% for more than 175,000 after the Personal Income Tax (IRPF) normative.
2. 2. Occupational status, determined by the percentage of wage earners in the population – the rest being self-employed. In case of Spain the corresponding ratio is 81.9% after Social Security affiliation (March 2012).
3. 3. Support for progressivity in the tax system, set as the percentage of agents that support the principle of progressivity in the tax system, so that it may be empirically calibrated with data from attitudinal survey studies. In case of Spain, SIMULFIS use a ratio of 80.0% after a 2010 study (Noguera et al., 2011).
4. 4. Income threshold for receiving social benefits (), which determines eligibility for a means-tested public cash benefit, so that if an agent's net income after tax is below the threshold, it is topped up to reach that level. The marginal withdrawal rate of the benefit is 100%, so that each eligible agent only receives the difference between his declared income and the minimum income determined by the threshold. In case of Spain, this parameter is set to 8700 Euro, 50% of the median income (poverty threshold), after the Spanish Household Budget Survey, 2008 (INE, 2016).

   5 Audit rate, modeled as the probability of any agent being randomly audited. In our Spanish SIMULFIS experiments the rate has been set at different arbitrary values of 0.03, after data from the Spanish Tax Agency for 2008 (ATE, 2008), and 0.25, 0.50, and 0.75. SIMULFIS is aimed, at the present stage, at simulating taxpayer behavior and not tax inspector or tax administration behavior. Since the cost of audits is to be taken into account by the tax administration, it is not included as a parameter of the model. The different values of audit probability are tested hypothetically in order to explore how they affect taxpayers' behavior.

   As in the traditional model of Allinghman and Sandmo, audits are random, that is, there is a random selection of the audited taxpayers. However, alternative auditing strategies (Bloomquist, 2012; Hashimzade, 2015, 2016) should be considered in future developments of SIMULFIS. Particularly, we could include audits that target specific groups of taxpayers: for instance, the most likely or largest tax evaders. Such strategy would improve the realism of the model since real-world tax agencies mostly perform risk-based preselected audits, which have proved to increase tax compliance and tax yield, and decrease audit costs.

   6 Amount of fine (), modeled as a percentage over the evaded taxes charged if an audit succeeded in finding an agent with tax-evading behavior. In the present version, there is no chance for tax evaders to be audited but not sanctioned: when tax evaders are randomly audited, then they pay the fine. In our Spanish SIMULFIS experiments the amount has been set at different rates from 1.5 (after the Spanish 2008 regulation (BOE, 2003)), 3.0, 4.5, and 6.0. In SIMULFIS, taxpayers can end a round with negative income, but the distribution of income starts again from the initial one in each round. However, agents have memory of their past audits. The model aims to test in this sense, how taxpayers adapt and change behavior as their past audit record evolves.

   The combination of audit rates and fines charged will result in a two-dimensional ordered scale of deterrence taxing scenarios.

5. 7. Behavioral or decision filters, which can be activated and deactivated as initial parameters in any SIMULFIS simulation run. Since they are necessary for the agents to make a decision, by default the opportunities to evade (O) and rational choice (RC) are activated (as the base model or zero model for testing). On the contrary, the other two filters, the normative commitments (N) and the social influence (SI), can be activated or deactivated for experimental purposes. Further explanation on decision filters will be done at Section 5.2.5.
6. 8. Social influence coefficient (), as the strength of social influence from her close network neighbors over an agent, modeled as a numerical value from 0 (no social influence) to 1 (full social influence). In our Spanish SIMULFIS experiments three social influence scenarios are considered, with , , and .

[I.ii.d] Space is not included in SIMULFIS model. [I.ii.e] One time step represents one tax period and the simulations were typically run for 100 years, without changing tax system parameters. That was unrealistic, but allows us to explore if outcomes show a stable or convergent tendency.

5.2.3 Process Overview and Scheduling

[I.iii.a] The general model dynamics is as follows: when SIMULFIS is initialized, agents randomly receive a salary and a number of neighbors in the way described at the initialization section [III.ii]. Then they go through the decision algorithm and end up making a decision about how much of their income they declare, as a result of the activated behavioral filters. Then incomes are taxed and random audits and fines are executed. The benefits are paid to those who are entitled, and endogenous parameters are updated for the next time period.

Pseudo-code of SIMULFIS (v.11, February 2013):

• User sets the initial parameters – using interface sliders and monitors,
• Procedure Setup,
• System generate agents, set individual attributes, set network links between agents.
• For each tax period (sequential order, asynchronous update):
• Procedure Decision about FOUR:
• For each agent (random order, asynchronous update)
  • Agent run Opportunity filter (O). Update FOUR,
  • Agent run Normative filter (N). Update FOUR,
  • Agent run Rational Choice filter (RC). Update FOUR,
  • Agent run Social influence filter (SI). Update FOUR,
  • Next agent.
• Central Authority Collect Taxes,
• Central Authority Run random Audits and Execute Fines,
• Central Authority Pay Social Benefits,
• System Compute the Outcome Variables,
• System Update Plotting display and Print outcomes,
• If period = 100 stop running,
• Otherwise Next tax period.

There is no endogenous stop condition for the simulation. The experimental plan has been performed with a limitation of 100 tax periods, after checking that it is enough to assure that a stable state has been achieved by the system.

5.2.4 Theoretical and Empirical Background

[II.i.a] Specifically, the model includes the possibility of complex interactions between the three main types of mechanisms that have been considered by the literature as likely determinants of tax evasion decisions: fairness concerns, rational choice or utility maximization, and social influence. SIMULFIS design relies on the assumption that independently none of the single mechanisms mentioned allows for an explanation of the aggregate level pattern on tax evasion. By means of (i) the ABM agents' interaction and (ii) the quasi-experimental manipulation of the mechanisms triggered in the decision process, SIMULFIS aims to study the complexity of a socioeconomic system with tax evasion decisions.

[II.i.b] The agents' decision model is based on a set of assumptions supported by both (i) established theories about microeconomic decision models, as bounded rationality with cognitive and with social network informational constrictions, and (ii) observational mechanistic explanations, as social influence. In line with an analytical, sequential, and procedural decision-making approach, after Jon Elster's view of decision process (Elster, 1984, p. 76) (Elster, 1989, p. 13), agents decide how much tax they evade after going through four successive “filters” that affect their decision: (i) opportunity, (ii) normative commitments, (iii) rational choice, and (iv) social influence.

[II.i.c] This approach aims to capture recent developments in behavioral social science and cognitive decision theory, which disfavor the usual option of balancing all determinants of decision in a single individual utility function (Bicchieri, 2006; Elster, 2007; Gigerenzer et al., 2011).

[II.i.d] Some submodels used in SIMULFIS are based on sets of empirical data. The reference value for some exogenous parameters uses household surveys, statistical census, and field or laboratory experiments. Specifically, the income tax rates and brackets and the income threshold for receiving social benefits come from the Spanish regulations, and the occupational status distribution data are taken from Social Security Affiliation official records, while support for progressivity in the tax system data are from a specific study (Noguera et al., 2011).

Some laboratory experiments provide data about the ambivalent effect of social influence () (Muchnik et al., 2013), but they were considered inconclusive at the time SIMULFIS experiments were developed, so instead of using a reference value, we explore a discrete and reduced space of the parameter. There might be specific reasons and tests designed to choose optimal values for the social influence coefficient (Lorscheid et al., 2016). However, since our aim here is just to give a description of the model, we do not intend to choose an optimal parameter setting but just to explore sufficiently different value combinations along the parametric space.

[II.i.e] The data mentioned were available at subpopulation aggregation level, so they were useful in calibrating SIMULFIS for the case of Spain.

5.2.5 Individual Decision Making

[II.ii.a] The subjects of decision making are the agents and the tax central authority. The tax central authority makes system-level top-down decisions about whom to audit and then about applying the corresponding fines if necessary and redistributing taxes by means of social benefits. Each taxable agent makes her individual-level decision about the amount of income economic units she declares as subject to taxation, that is, the percentage of her income she will conceal. The use of sequential filters approach for modeling the final decision making could be understood as an aggregation of multiple levels of decision, powered by different situational logics.

[II.ii.b] In SIMUFIS, the conditional agents are modeled so that they pursue an explicit set of objectives: to conceal as much amount of their income as possible given their occupational social context, and keeping a general normative fairness perception, while avoiding a sucker feeling given their close social network. Individual agents use a number of success criteria to generate a final decision along a sequence of four decisional filters. Each decision outcome from a filter is taken as an input for the next, so that different basic rationalities are behind each of the decision-making stages. The first filter (opportunity) is modeled after a routine-based rationality, without considering any objective, as a maximum fraud opportunity correlated with the level-class category. The second filter (normative) is modeled following a satisficing approach, as bounded rationality based on personal normative perception and comparison with the social environment. The next filter (rational choice) is modeled as a cognitive procedure where agents maximize a utility function to decide. The last filter (social influence) is modeled as a reference group convergence function where the agent adjusts and moderates her previous decision about what percentage of her income she will conceal.

[II.ii.c] In SIMULFIS, agents decide what portion of their income they will conceal, beyond binary or ternary choice, which is typical in previous models. As an example, if two taxpayers – say, A and B – both have a gross income of 100 units and they both decide to hide 10 units, but taxpayer A had the objective chance to hide 50 while taxpayer B could only hide 20, although in absolute terms they comply the same, in relative terms taxpayer B is making use of 50% of his opportunity to evade tax while taxpayer A only makes use of 20%. Agents make their decisions about tax compliance as produced by a specific sequence or combination of different mechanisms or “filters,” most of which may be activated or deactivated in order to run controlled experiments. The decision process sequence includes the opportunity filter (O), the normative filter (N), the rational choice filter (RC), and finally the social influence filter (SI). If a filter is triggered then the agent modifies the concealed part of her income that results from the previous filter, so that the decision process is modeled as a complex interaction between this kind of cognitive mechanisms and the perceived socioeconomic environment. It is worth noting that filters have different logics or rules to generate the corresponding outcome: the O filter is structural deterministic, the N filter is relative and satisfaction based, the RC filter is utilitarian and optimization based, while the SI filter is ruled by convergence to the social environment. See detailed information about each filter in Section 5.2.17.

As an example of the FOUR decision process, Figure 5.1 displays a numerical case of how agents' decision algorithm operates. The encircled numbers express percentages of a hypothetical agent's concealed income, and how they change when the agent goes through the consecutive behavioral filters; as explained above, the opportunity filter results in a maximum percentage of concealable income, while the normative, the rational choice, and the social influence filters lead the agent to a decision as to what percentage of her income she will conceal. The bold numbers at the middle level express the equivalents to these two latter percentages in terms of FOUR once the agent has gone through the rational choice filter, the resulting percentage is transformed in terms of FOUR (doted ascending arrow) this FOUR is compared with the agent's neighborhood's FOUR; then, the application of the social influence formula results in a different FOUR (bidirectional horizontal arrow), which is again transformed into a percentage of income to be concealed by the agent (doted ascending arrow).

Specifically, the example works as follows:

1. 1. Let us assume that in a given period of a simulation, agent after the O filter may conceal 60% of her income.
2. 2. Let us assume that she is satisfied only with respect to her tax balance but not with respect to the progressivity of the system; then, she reduces by one third her opportunity to conceal so that the maximum she can conceal after applying the N filter is 40% of her income.
3. 3. Let us assume that after applying the RC filter, this percentage is further reduced to 20% of her income. Therefore, her FOUR () will be 33.3%, resulting from 20/60.
4. 4. Let us also assume that in the previous period her neighborhood's median FOUR () was 10%. If the social influence coefficient () is set to 0.5, the agent's FOUR after the SI filter will be: ) = 33.3 + 0.5 (1033.3) = 21.7%.

   5. The resulting FOUR of 21.7% is equivalent to concealing 13% of the agent's income, since she is making use of 21.7% of her initial opportunities, which were 60%, so: 21.7% 60% = 13%.

   In this example, the SI filter affects the agent's compliance by reducing the percentage of income she decides to conceal from 20% to 13%.

[II.ii.d] The agents adapt their behavior to changing endogenous and exogenous state variables. For instance, the perceived audit probability () of agents at each period of the simulation depend on agents' decisions in the previous periods and on information gathered from their close social network. Other parameters that co-evolve as a function of previous decisions of the agent and other agents are eligibility for benefits, tax balance (personal and in the neighborhood), perceived progressivity of the tax benefit system, or neighborhood's compliance rate.

[II.ii.e] Social norms or cultural values play a role in the decision-making process, because agents' normative commitments and fairness concerns toward taxation are explicitly modeled. The decision model takes into account both factual and normative beliefs, and also relative deprivation feelings (Manzo, 2009, 2011). The agents' support for the progressivity principle in the tax system models their normative beliefs on progressivity, and is used in behavioral filter N to moderate agents' decision about the maximum amount of concealable income. In that way, sociocultural differences could be introduced in specific SIMULFIS settings.

Spatial aspects do not play any role in the decision process [II.ii.f], because the space in SIMULFIS is modeled as an abstraction. On the contrary, temporal aspects play a relevant role in the decision about the concealed income [II.ii.g]. Specifically, in the expected utility function () that rules the rational choice filter, (i) individuals use a discounting function for taking into account the risk of being audited and sanctioned – the cube root included in the aforementioned function – and also (ii) individuals keep a temporal memory to update their perceived probability of being sanctioned if they evade (), as a function of their own audit record in all previous periods and the audit rate in their neighborhood in the immediately previous period. In that way, uncertainty is included in the agents' decision rules [II.ii.h] by means of their explicit consideration of uncertain situations or risk while applying the rational choice decision filter.

5.2.6 Learning

[II.iii.a] In the strong sense of the concept, individual learning is not included in the decision process modeled in SIMULFIS, because individuals never change their decision rules over time as consequence of their experience. In a weaker sense, agents can change aspirational levels depending on a combination of past experiences and present perceptions – in normative and social influence filters – and also can change reference values for risk aversion – in rational choice filter – so that their behavior evolves in time as they update the memory records. No collective learning is implemented in the model [II.iii.b], beyond the convergence decisional mechanism implied in the social influence filter.

5.2.7 Individual Sensing

[II.iv.a] Agents are assumed to sense and consider in their decisions both endogenous and exogenous state variables. These perception processes are not subject to noise or errors, but are constrained by the structural position of agents in their close social network.

[II.iv.b] In the opportunity filter, agents are aware of their own occupation status and income level. In the normative filter, agents perceive (i) their own normative beliefs on progressivity (exogenous); (ii) the income after taxes and benefits for all the population – to compute if the income ratio between the richest 10% and the poorest 10% is reduced by more than 30% after each tax period; (iii) their own personal previous tax burden () and their own previous received benefits () – to compute whether they are net contributors or net recipients; and (iv) their neighbors' personal previous tax burden () and previous received benefits () – to compute whether the majority in the neighborhood are net contributors or net recipients.

In the rational choice filter, agents perceive (i) their own audits in all previous periods (), the audits in their neighborhood in the preceding period (), the number of neighbors () – to update their own audit risk (); (ii) their own gross income; and (iii) some relevant system parameters, such as the applicable tax rates, the fines for evading taxes, and the expected social benefit.

In the social influence filter, agents are aware of (i) the exogenous system parameter of social strength (), (ii) their own FOUR after the RC filter, and (iii) the median FOUR in their neighborhood () – to compute their final FOUR as a convergence to the group median pondered by ().

There is diversity in the spatial scale of sensing in SIMULFIS [II.iv.c]. Although some global or aggregate variables are perceived from the whole model space, most of the information used by individuals is available from their local network. On the contrary, the central tax authority is assumed to perceive any internal attribute for individuals subject to audit processes. No mechanism by which agents obtain information is modeled explicitly [II.iv.d], and no costs for cognition and gathering information are included [II.iv.e].

5.2.8 Individual Prediction

[II.v.a] Individual agents use a number of sources to predict future conditions and to make inference about tendencies. Extrapolation from experience is explicitly modeled by implementing a memory of past events – audits – that have affected both the agents and their neighborhood. While the space is an abstraction in SIMULFIS, these predictions are in fact based on close network observations.

[II.v.b] Rational choice filter (RC) is the only procedure that models how agents are assumed to estimate the consequences of their decisions: By using a temporal discount modeling function, individuals compute their expected utility under uncertainty about the probability of being audited, and make a decision about how much to conceal by optimizing this function.

The strictly local nature of memory data in SIMULFIS model includes the possibility for agents to be erroneous in the prediction process [II.v.c].

5.2.9 Interaction

In SIMULFIS, interactions among agents and entities are assumed as direct [II.vi.a] and depending on social distance, or network proximity, configured by the type of agents – occupational status and income level [II.vi.b]. No explicit representation of the communication involved in social interactions is modeled [II.vi.c]. Since agents take into account the behavior of their direct ties, there is a direct network effect on agents' decisions [II.vi.d]. Agents are randomly linked with a number of neighbors under some constraints: each agent has a minimum of neighbors, and a subset of each agent's neighbors are similar to him or her in terms of occupational status and income level. The generated social network type is similar to the so called “small world,” and does not change over time.

5.2.10 Collectives

[II.vii.a] In SIMULFIS, individuals form aggregations that affect, and are affected by, other individuals. These aggregations are imposed by the modeler: A social structure in terms of local constrictions to interaction are imposed at each initialization, without any change or relinking during the simulation run.

[II.vii.b] These collectives are represented by generating link-agents between pairs of individuals, not as a separate kind of entity with its own state variables and traits but just a set of related agents.

To generate the homophily network, agents are randomly linked with a number of neighbors under some constraints: each agent has a minimum of neighbors, and a percentage of each agent's neighbors that are similar to him or her in terms of occupational status and income level. Both parameters – minimum and percentage – can be adjusted in the graphic user interface. Agents are also randomly assigned an occupational status (wage earners or self-employed) and an income level (high, the top decile of the income distribution; low, the three lowest deciles; and middle, the six deciles left in between). Both classes are calibrated using the Spanish case (Llacer et al., 2013). Each agent is assigned one of six subtypes of occupational status crossed by income level following Table 5.3.

5.2.11 Heterogeneity

[II.viii.a] Agents are heterogeneous in terms of market position. Different degrees of opportunity to conceal part of their income are assigned to different categories of agents, as a function of their income level and occupational status. This is theoretically justified because the FOUR is a much better indicator of the intensity of agents' tax fraud efforts than the amount of money evaded or the percentage of their income they conceal.

In SIMULFIS there is no heterogeneity in the decision making for conditional agents [II.viii.b]. Once a simulation is set in the initializing procedure – a subset of decisional filters are activated by the user – all individuals with conditional strategy will follow the selected decisional stages. The possibility of including subpopulations of unconditional agents is open as an option in the initial setting: these agents do not decide using the activated filters, but behave as unconditional tax evaders or unconditional tax compliants.

5.2.12 Stochasticity

[II.ix.a] In the initialization process, each individual agent is randomly assigned an occupational class (wage earners or self-employers) – after the ratio established in the initial settings for the Spain case. A random uniform algorithm assigns an income decile to each agent. A mean income by deciles by occupational class is then assigned to agents – using a decile range table for the Spain case – so that the income distribution follows a uniform distribution for each range of income decile (Llacer et al., 2013, Table 1). The final income distribution is modified so that 10% of the highest deciles agents get a top level income – in the range (109,116–2,000,000) for wage earners, and (144,000–2,000,000) for self-employers – with a uniform distribution.

In the initialization process, a random uniform distribution in the range (0.0–1.0) is used to initialize the individual's perceived risk of audits. Agents are also assigned some binary attribute values with a probability of 0.5: perception of the progressivity of the system, personal tax balance, neighborhood tax balance, and behavior in a previous period.

In the auditing process, the central tax authority agent samples the population using a simple random sampling without replacement with the probability set at initialization ().

5.2.13 Observation

[II.x.a] Since our main focus is behavioral, SIMULFIS outcome data go beyond traditional indicators for compliance – such as the amount of their personal income agents conceal – toward determining how much relative advantage agents take of their opportunities to conceal income, or (FOUR). The outcome indicators collected from the ABM are as follows:

1. 1. The mean FOUR of all agents – typically 1000 – computed at each tax period of a simulation
2. 2. The mean FOUR of all agents in all periods of a single simulation run – typically computed after 100 periods.

When agents' FOUR is converted into the equivalent amounts of evaded tax (in Euro), we may have also outcomes such as the following:

1. 1. Mean percentage of gross income concealed by each agent in all periods of a simulation
2. 2. Mean percentage of gross income concealed by all agents in each period of a simulation
3. 3. For each agent, the absolute amount of tax evaded, which is the result of . Note that progressive tax rates applied to different income brackets are not necessarily the same for total gross income as for declared income, so this calculation is necessary.

From these individual outcomes it is possible to compute aggregated outcomes for the system, such as (i) aggregated concealed income as a percentage of total income in the system, (ii) aggregated fiscal pressure as a percentage of total income in the system, (iii) aggregated absolute amount of tax evaded, and (4) aggregated tax evaded as a percentage of total tax due, or “aggregated tax gap,” which is the result of

5.1

Finally, all these results may be cross-tabulated by different initial settings, categories of agents, values of a parameter, and so on, thus allowing SIMULFIS users to run controlled virtual experiments on tax compliance behavior.

The aforementioned aggregated results are an outcome from decisions at the individual level [II.x.b]. The behavioral filters make each decision locally dependent on the agent position, her previous events history, and her neighborhood behavior, and while a random initial state is generated by each simulation run, the system aggregated results could be understood as emergent properties of the ABM.

5.2.14 Implementation Details

[III.i.a] The model was originally implemented using the Netlogo modeling environment version 4.1.3. Several versions and enhancements of SIMULFIS were developed from October 2011 (v0.1) until November 2013 (v12.0.1). Most of the simulations executed for the published experimental outcomes (Llacer et al., 2013; Noguera et al., 2012, 2013) were run in the stable version v11, using Netlogo v5.0.4, on a quad-core PC with Windows 7 OS. A typical simulation runtime – with 1000 agents, 100 tax periods, and all filters activated – takes around 10 minutes (626.57 seconds).

Any simulation is run in 10 different “worlds” – different random initial assignations of neighbors and income – and the outcome means are used to contrast the corresponding hypothesis.

In the present state SIMULFIS model is not public [III.i.b], but authors can distribute versions of the code to interested researchers on demand.

5.2.15 Initialization

SIMULFIS can be easily set for a diversity of initial states [III.ii.a] of the virtual society at , providing a tool for experimenting in different socioeconomic contexts and tax systems. In the Spain case, the following initial parameters were used: 1000 tax agents, 81.9% of wage earners in the population, income is distributed following a data-based decile range table (Llacer et al., 2013, Table 1), each agent has a minimum of 10 neighbors with 80% neighbors of the same category, all individual agents use conditional strategies, the estimate risk of audition is randomly distributed between 0 and 1, and the tax rates are set empirically after the rate brackets for the case.

Some initial parameters vary among simulations [III.ii.b], so that SIMULFIS allows to assign different values to initial parameters (e.g., the weight to social influence in agents' decision algorithm). This variability is useful in the design of systematic controlled experiments, where the space of values of a parameter – or combination of state variables – is explored in discrete steps with the corresponding simulations.

Our published simulation outcomes (Llacer et al., 2013; Noguera et al., 2012, 2013) explore, for the case of Spain, a set of scenarios that involve the following initial settings:

• Activation of normative filter,
• Activation of social influence filter,³
• Audit probability rates of 15%, 30%, 45%, and 60%,
• Fines of 1.5, 3, 4.5, and 6 as multipliers of the amount of evaded tax,⁴
• Social influence weight of 0.25, 0.50, 0.75.

The initial values are in part chosen arbitrarily and in part based on data [III.ii.c]. A number of parameters are established in terms of the experimentation plan with clear hypothesis to be tested.

Income distribution: Each run agents are assigned an amount of annual income following an exponential-shape distribution. In the Spanish case setting, SIMULFIS is calibrated empirically for an income decile distribution of the Spanish. The top of the distribution has also been adjusted for including a small percentage of big fortunes. After the distribution has been completed, the model assigns each agent to one of three income levels: “high” (the top decile of the distribution), “low” (the three lowest deciles of the distribution), and “middle” (the six deciles left in between). This distribution also may be differentiated for self-employed workers and wage earners in order to calibrate the model in a more realistic way, after data from the Spanish Household Budget Survey (INE, 2016).

Social network: Each run agents are randomly linked with a number of neighbors under some constraints: each agent has a minimum of neighbors (10 by default), and a percentage of each agent's neighbors are similar to him or her in terms of occupational status and income level (80% by default).

5.2.16 Input Data

[III.iii.a] SIMULFIS does not use any input from external sources to represent processes nor exogenous variables that change over time.

5.2.17 Submodels

[III.iv.a] The submodels that represent the processes listed in “Process overview and scheduling” are the following:

1. Opportunity filter (O): The outcome is a function of the agent income level and occupational status. The maximum concealable income as a result of the O filter will be used as the denominator for computing the agents' FOUR ().
2. Normative filter (N): The outcome is a reduction of the agents' concealable income resulting from their previous opportunity filter, using a three-way conditional function: (i) If an agent is satisfied in both “progressivity” () AND “tax balance” () she reduces by two thirds (66%) the proportion of income she may conceal; (ii) if an agent is satisfied only in “progressivity” () OR “tax balance” (), the reduction is by one third (33%); and (iii) if an agent is NOT satisfied in any sense, she may conceal the full percentage of income resulting from the opportunity filter.

   Rational choice filter (RC): Over the percentage of their gross income resulting from previous filters O and N, agents rationally maximize their net income by estimating the expected utility of concealing under uncertainty conditions (). An agent's decision to declare income () is a function of tax rates (), fines (), her real income level (), and the probability of being caught if she evades (). Homogeneous risk aversion is assumed by using cube roots in both addends of the equation, and the expected social benefit, which agents compute by determining their eligibility in the immediately preceding period of the simulation, is also included. The usual square root is avoided to prevent applying it to eventual negative numbers in the second addend – which could be the case, for example, if agents declare a low percentage of their income and fines are very hard.^{5, 6}

   5.2

   where

Note that when but , that is, when the agent is not legally eligible for benefits but she evades enough to be so, then . Similarly, when , that is, when the agent is not eligible whether she evades or not, then and .

Social influence filter (SI): After applying the previous RC filter, agents' FOUR () converge to the median in their neighborhood (), according to . The result of this calculation is the agents' final FOUR, which is expressible in terms of the percentage of the agents' income that is declared or concealed, and in terms of absolute amount of income concealed.

The model parameters, dimensions, and reference values [III.iv.b] are different in a diversity of experimental conditions simulated by running SIMULFIS.

The reference values for opportunities to conceal (O filter) – in percentage of agents' gross income are 80% for high income level self-employed, 60% for the rest of self-employed, 30% for high income level wage earners, 10% or 20% for medium income level wage earners, and 10% or 20% for low income level wage earners.

Normative filter (N) has two dimensions of satisfaction: (i) “progressivity” of the tax system (), and (ii) personal “tax balance” ().

(1) Satisfaction with the progressivity of the tax system (PTi = ) is a function of two parameters: If agents support progressivity (SPi = ) = 1) and perceive that the system is progressive (PSt = ) = 1), they are “satisfied”; otherwise, they are not.

(1a) The agents' support for the progressivity principle in the tax system () is an exogenous parameter, available through survey research. Agents are randomly assigned normative beliefs about progressivity () according to the aggregated percentage (support = 1, do not support = 0).

(1b) The agents' perception of “real” progressivity in the tax system is updated after each tax period. The system is perceived as progressive if the mean income ratio – after taxes and benefits – gap between the richest 10% subpopulation and the poorest 10% subpopulation is reduced by more than 30%. Perfect information is assumed for agents' estimations.

(2) Satisfaction with the tax balance () is a function of two parameters: If agents are net contributors while the majority of their neighbors are net recipients they are “unsatisfied”; otherwise, they are satisfied.

(2a) Agents' personal tax balance is computed by comparing their personal tax burden () with the benefits they eventually receive (): agent is a “net contributor” if , or a “net recipient” if .

(2b) To compute their neighborhood's tax balance, agents observe whether the majority (more than 50%) of agents in their neighborhood (including themselves) are net contributors or net recipients. Perfect information is assumed.

Rational choice filter (RC): Agents' perceived probability of being sanctioned if they evade tax by concealing income () is randomly distributed in the initial period of the simulation and afterwards is updated endogenously as a function of the agents' audit records in all previous periods and the audit rate in the agents' neighborhood in the immediately previous period, according to the following equation.

5.3

Social influence filter (SI): Agents' decision about declared income converge to that of their neighborhood, and SIMULFIS models the strength of social influence by an exogenous parameter equal for all agents, ranging (0, 1) from no social influence to full social influence. Note that, when (full social influence), the individual effect of the RC filter is cancelled; and when (no social influence), the agents keep their original FOUR resulting from the RC filter. The reference value for the social influence coefficient is set according to three social influence scenarios, with , , and .

[III.iv.c] Submodels are designed and parametrized with specific assumptions or justifications. In the opportunity filter (O), in order to determine values for different agents' opportunities, we adopt some simple – and arguably realistic – assumptions: (i) wage earners have fewer opportunities to conceal than self-employed workers, because their income is typically withheld at origin in a more substantial proportion. (ii) Agents with high income – the top decile of income distribution – have more opportunities to conceal than the rest, because they receive income from many different sources or they have the resources, techniques, and abilities needed to successfully conceal a higher proportion of it. (iii) For similar reasons, agents with middle income have more opportunities to conceal than agents with low income – although in other set of experiments low-income workers have higher probability to participate in the shadow economy (Noguera et al., 2013). (iv) For all agents there is some percentage of their income that they cannot conceal, since the government always has some information on at least a minimum proportion of every agent's income.

Normative filter (N) models agents' normative attitudes and satisfaction feelings toward the tax system's design and performance in terms of its fairness. This filter combines two elements in order to determine the outcome level of satisfaction of each agent: (i) agents' satisfaction with the progressivity of the tax system, and (ii) agents' satisfaction with their tax balance when compared with that of their neighborhood. The agents' support for the progressivity principle in the tax system models their normative beliefs on progressivity, while agents' satisfaction with their tax balance is determined by comparing their personal tax balance with that of their neighborhood. This method tries to capture the well-known findings of the literature on relative deprivation, which show that people's feelings of satisfaction with their endowments depend more on the comparison with their reference group than on the amount enjoyed in absolute terms (Manzo, 2011).

Rational choice filter (RC) models the agents' cognitive process of maximizing expected utility according to an adaptation of the classical tax fraud expected utility function (Allingham and Sandmo, 1972). SIMULFIS uses a discrete computational approach to determine a sequence of 11 expected outcomes in terms of agents' net income, which results from reducing concealed income by intervals of 10% – from evading 100–0% – of agents' concealable income. Since the expected utility formula we are using introduces substantive complications in relation to Allingham and Sandmo – such as a progressive tax rate and a social benefit – this way of formalizing expected utility makes the computational working of the model easier, while ensuring the consistency of agents' decisions. Despite other filters and mechanisms affecting agents' compliance, rational choice has always some weight in the final decision – except under full social influence. We take this as a realistic assumption, since an economic decision such as tax compliance is almost always rationally considered by taxpayers, and, in most cases, assessed by experts or professionals.

Social influence filter (SI) models the extent to which agents' decision about declared income (in terms of FOUR) converge to that of their neighborhood as a result of any kind of social influence (Bruch and Mare, 2006; Centola and Macy, 2007; Durlauf, 2001, 2006; Rolfe, 2009; Salganik and Watts, 2009; Watts and Dodds, 2009). In SIMULFIS, the strength of social influence is determined by an exogenous parameter () equal for all agents, ranging (0, 1) from no social influence to full social influence. Note that, when (full social influence), the individual effect of the RC filter is cancelled; and when (no social influence), the agent keeps his original FOUR resulting from the RC filter. The operation of social influence mechanisms affecting tax evasion behavior has been recently questioned by some scholars (Hedström and Ibarra, 2010) on the basis of the “privacy objection”: since tax compliance is taken to be private and unobservable by peers, no social influence could take place. However, as survey studies repeatedly show (IEF, 2012; CIS, 2011), citizens usually have an approximate idea on the tax compliance level in their country, occupational category, or economic sector, its evolution in time, and its main causes. These ideas may be formed from information received through mass media, personal interaction, or indirect inference (e.g., shared social characteristics, when compared with economic lifestyles, may be proxies for inferences about neighbors' and peers' tax compliance). Additionally, in countries where there is low tax morale and high social tolerance toward tax evasion – such as Spain – it is usual to have access to public “street knowledge” about personal tax compliance, and to give and receive advice between neighbors and peers on how to evade. Finally, there is some literature modeling social influence mechanisms triggered by estimated or revealed information on criminal and dishonest behavior (Diekmann et al., 2011; Gino at al., 2009; Groeber and Rauhut, 2010).