A Game of Tax Evasion: evidences from an agent-based model
AA Game of Tax Evasion:evidences from an agent-based model
L. S. Di Mauro ∗ , A. Pluchino † , A. E. Biondo ‡ Abstract
This paper presents a simple agent-based model of an economic system, populated byagents playing different games according to their different view about social cohesion and taxpayment. After a first set of simulations, correctly replicating results of existing literature,a wider analysis is presented in order to study the effects of a dynamic-adaptation rule, inwhich citizens may possibly decide to modify their individual tax compliance according toindividual criteria, such as, the strength of their ethical commitment, the satisfaction gainedby consumption of the public good and the perceived opinion of neighbors. Results show thepresence of thresholds levels in the composition of society - between taxpayers and evaders- which explain the extent of damages deriving from tax evasion.
Tax evasion is quite an age-old phenomenon that has been studied for decades, both theoreticallyand empirically. It can be described as the “illegal and intentional actions taken by individualsto reduce their legally due tax obligations” (Alm 2012, p.55). Tax evasion is a damage to thesocio-economical environment that deprives governments of their fiscal resources and plays animportant role in reducing well-being of societies. The well-known free rider problem rises whena selfish citizen consumes public goods and services without properly contributing to relatedcosts, causing inefficiency and bad allocations of governments expenditures for healthcare, edu-cation, defence, social security, transportation, infrastructure, science and technology, as widelydocumented in a vast literature, among which Andreoni et al. (1998), Slemrod and Yitzhaki(2002), Torgler (2002), Kirchler (2007), Slemrod (2007). Tax evasion is, also, related to socialinequality, as underlined by part of the literature, among which, Alstadsaeter et al. (2017),Bertotti and Modanese (2014, 2016), dealing with the differentiation of the propensity to evadewith respect to income and with redistributive aspects. Finally, it matters in terms of socialjustice, since it specially afflicts poorer people, who do not have the possibility to substitutepublic services with private ones available in the market at higher prices.In his
An Essay on the Nature and Significance of Economic Science , Lionel Robbins (1932,p.15) wrote that “economics is the science which studies human behaviour as a relationship ∗ Department of Physics and Astronomy, University of Catania and INFN Sezione di Catania, Italy;[email protected] † Department of Physics and Astronomy, University of Catania and INFN Sezione di Catania, Italy; [email protected] ‡ Dept. of Economics and Business, Univ. of Catania, Italy; [email protected] a r X i v : . [ q -f i n . GN ] S e p etween ends and scarce means which have alternative uses”. Such a very well-known defini-tion, focuses the point that among all routes between ends and means, each person chooses theone that maximally conforms to her personality. A very intriguing corollary would discuss alldecisions in terms of their ethical acceptability. Unfortunately, the moral evaluation of chosenobjectives is not pertinent of the economic analysis and remains within the attributions of per-sonal free will. More precisely, the individual propensity between cooperation and competitionis determinant in setting the deliberated conduct. While the epistemological analysis of suchaspects would go far beyond the goal of this paper, the model here presented aims to show thecollective relevance of such behavioural elements in driving the decision of each citizen, whichreflects also the perceived quality of the public good and the relational feedback received by hersurrounding social environment.The initial stages of the formal analysis of tax evasion can be dated back to the Seventies, withcontributions by Allingham and Sandmo (1972) and Srinivasan (1973). Despite many similar-ities, such contributions, which are a propagation of an earlier approach advanced by Becker(1968), differ from each other with respect to optimization procedures, taxpayers risk attitudes(which affect second order conditions of chosen objective functions), decision variables, auditprobabilities, tax tariffs, and penalty functions. In particular, Allingham and Sandmo (1972),find that income understatement is decreasing in audit probabilities or in the fine, whereasthe dependence on tax rate is more controversial, reflecting income and substitution effects.Yitzhaki (1974) obtained a conter-intuitive result by modelling fines computed on the basis ofevaded taxes (instead of the understated income): differently from the empirical evidence shownin Clotfelter (1983), Crane and Nourzad (1987), Poterba (1987), as the tax rate increases, theevasion decreases. Many other studies have been done in the attempt to find a positive rela-tioship between tax rate and evasion (see for example, among others, Yitzhaki (1987), Panades(2004), Dalamagas (2011), Yaniv (2013)).Such a standard theoretical framework inspired several contributions in related literature, con-cerning tax evasion and related issues, such as the shadow economy, as in Buehn and Schneider(2012), psychological perception and society (social norms and moral sentiments like guilt orshame), as in Myles and Naylor (1996), Traxler (2006), Fortin et al . (2007), Kirchler (2007)and many others. However, tax evasion has been discussed also in contributions based on aneconophysics approach, since papers by Lima and Zacklan (2008) and Zaklan et al . (2008)where tax declaration/evasion correspond to the two states of spins in the Ising model (1925) offerromagnetism. More generally, a growing stream of literature presenting agent-based modelsdealing with tax evasion exists. A survey of such papers could be gained by the joint readingof Bloomquist (2006), Alm (2012), Hokamp (2013), Pickhardt and Prinz (2014), Oates (2015),Bazart et al . (2016). The advantage of agent-based models is that they are prone to describethe complexity of aggregate contexts, as documented in previous studies of socio-economic anal-ysis, Pluchino et al . (2010, 2011, 2018), Biondo et al . (2013a, 2013b, 2013c, 2014, 2015, 2017).Simulative models can help investigating relevant questions, as the correspondence between theprovision of the public good and tax evasion, as in Hokamp (2013), the importance of socialnorms and auditing, as in Hokamp and Pickhardt (2010), and the effect of social networks onthe tax compliance, as in Vale (2015). Such aspects, like many others, can be explained in termsof behavioral attributes, seeking for the roots of decisions in the evolution of personal traits,influenced by the surrounding environment.As reported by the IRS (2016), given the extent of the tax evasion, the expenditures paid bygovernmental authorities to induce virtuous behaviours are significant. Nonetheless, in many2ases, free riders remain unpunished. Honest citizens considering the participation to socialcosts as a moral imperative are the sole fully compliant taxpayers. We rephrase the provokingquestion asked by Alm et al. (1992): why should people pay taxes?In order to answer the question, the main motivation of this paper is to combine the agent-basedapproach to the problem of tax evasion with a flavour of game theory. The free rider problem isone of the most classical example of failures of coordination mechanisms. When people try toobtain the best outcome for themselves, the result might be that, considering the final collectiveeffect, everyone gets the worst instead. In other words, we assist to a conflict between individual and collective rationality (Rapoport 1974). This creates the paradoxical outcome, know in thegame-theory literature as the prisoner’s dilemma where, despite two persons apply their domi-nant strategies, they reach a sub-optimal equilibrium. Such a social dilemma motivated a vastamount of literature, regarding the production of public goods, as in Heckathorn (1996), theemergence of social norms and social interaction, as in Hardin (1995), and Voss (2001), amongmany others.The game-theoretic approach re-defines the meaning of the above-asked question: why shouldindividuals ever decide to cooperate if they have incentives to pursue, firstly, their own self-interest? Different reasons can be reported: first of all, because of altruism, as recalled, for justsome examples, in Stevens (2018), Epstein (1993), and Zappal`a et al. (2014); secondly, becauseof imitation, as in Callen and Shapero (1974), in Elsenbroich and Gilbert (2014) and McDonaldand Crandall (2015); finally, because of an assessment of the quality of the public good (i.e.quality of Institutions), as in Nicolaides (2014), La Porta et al. (1999), Feld and Frey (2007)and Torgler and Schneider (2009).The present model will try to analyze the aggregate dynamics of a community in which agentsdecide how to behave, in terms of tax compliance, according to highlighted factors, such asthe ethical orientation, the appreciation of the public good and the imitation. The first, verysimple, setting of the model will be shown to be able to replicate same results obtained byElster (1989), who addresses the simplest case of the many-person prisoner’s dilemma. Severalsuccessive modifications of the same model will be used to validate policy hypotheses and socialimplications. The paper is organized as follows: in Section 2 we present our Tax Evasion Model;in Section 3 we analyze the impact of a fraction of taxpayers in the evolution of the system,by emphasizing the role of cooperative altruism of tax-payers; in Section 4 we introduce thepossibility for the agents to change their behaviors because of two mechanisms (i.e., imitationand assessment of the satisfaction from the public good); in Section 5 we study the effect ofthree policy parameters regulating the tax rate, the penalty and the audit probability.Time t = 0 , , , . . . , T is a discrete variable indicating the number of played turns. Consider a community where N players { P i } i =1 , ,...,N , all endowed with the same initial amountof capital C i , can play one of two very simple games: • Game A : at each turn, the player gives away d units of capital, which go to reinforce thecapital of other randomly chosen players (1 unit for each); • Game B : at each turn, the player loses h units of capital (which will not be redistributedto other players) with probability p , otherwise she does not pay anything.3igure 1: Tax Model . Rules of a minimal Tax evasion model, for a community of N players.Tax evaders play only game B, while taxpayers play only game A. At each turn their individualcapital increases through an external gain. Parameters d and h are integer constants, with h > d . From an individually point of view,game A is of course a losing game, since the player loses some units of capital donating themto other players. On the other hand, from a collective point of view, it may also be regardedas an altruistic behavior, since the player sacrifices her personal wealth to favor other people.From this perspective, playing game B can be considered as a selfish behavior, since the playerprefers to risk paying a penalty just to have a chance of preserving her capital. Furthermore,if she loses, the penalty is greater of the altruistic donation of Game A and the loss does notincrease the capital of other players.For our purposes, if one looks at the individual capital as the comprehensive monetary valueof goods and services (both private and public) enjoyed by each player at the current turn, itis possible to consider the altruistic conduct of the above-sketched metaphoric game as the taxpayment, i.e., the choice of a person to reduce her own capital in order to increase someone’selse endowment, whereas the selfish behavior stands for tax evasion. In other words, dependingon the chosen game, agents are partitioned in two categories, as depicted in Figure 1: • Taxpayers : altruistic players, indicated as { A i } i =1 , ,...,N a , who always pay taxes playinggame A; • Tax evaders : selfish players, indicated as { S i } i =1 , ,...,N s , who always evade taxes playinggame B.where N = N a + N s . The initial amount of social capital C i (0) = 0 ( i = 1 , , . . . , N ) evolves intime according to the chosen game.Tax’s payment does occur when altruists play game A, in the sense that we interpreted the4igure 2: Tax Model . Average final capital ¯ C ( T ) (red line) over T = 100 turns as a functionof the percentage f of taxpayers. The average final capitals ¯ C a ( T ) (green line) and ¯ C s ( T ) (blueline) for the two categories of players, taxpayers and tax evaders, are also reported and comparedthe one with the other. The percentage f varies from 0 to 100 % with steps of 1 %. All thecapital values have been rescaled in order to have ¯ C ( T ) = 0 for f = 0% . donation of d units of capital towards other d randomly chosen players as a taxation. Thus,being the profits gained from taxes equally distributed among the population, they can beinterpreted as public services. On the other hand, if people who do not pay taxes (the ones whoplay game B) win the game, they get off scot-free and their capital remains the same; losers,instead, are forced to pay a penalty h higher than the tax. The probability p of losing the gameis the probability to fall into an audit. Finally, at the beginning of each turn, people’s capitalis incremented by a certain amount g < d of units, which represents the only external source ofgain for the whole community (see again Figure 1). In terms of benefits for any agent, this latterrule allows to consider the social capital C i as composed by both the accumulated external gainand the value of the enjoyed public services.The null settings for the first part of the paper is: p = 0 . d = 2 units (taxpayment), h = 3 units (evasion penalty) and g = 1 units (external gain). Such values will belater changed to show their incidence on results. In this section we want to investigate the asymptotic behavior of the average social capital¯ C ( T ) = N (cid:80) Ni =1 C i ( T ), calculated at the end of single run simulations with T = 100 turns perplayers, in correspondence of an increasing percentage f of taxpayers present in the community(with f varying from 0 to 100 % at steps of 1 %).In Figure 2 the average final capital ¯ C ( T ) is reported (red line) as function of f , along with itstwo components ¯ C a ( T ) = N a (cid:80) N a i =1 C i ( T ) (green line) and ¯ C s ( T ) = N s (cid:80) N s i =1 C i ( T ) (blue line)5igure 3: Many-person prisoner’s dilemma . Expected benefits as a function of the numberof cooperators for the collective group, for the cooperators and for the free riders. Elster (1989). calculated separately for the altruistic taxpayers and selfish evaders (a horizontal black line at¯ C = 0 is also drawn for comparison). The special case f = 0% (i.e. only selfish players, N s = N )is equivalent to playing only game B, whereas the opposite case f = 100% (i.e. only altruisticplayers, N a = N ) corresponds to the execution of game A only. The three values of social capitalhave been rescaled in order to have ¯ C ( T ) = 0 when f = 0%, since – by definition – if no onepays taxes the collectivity must have zero benefits.Looking to the graph, one can see that the average capital ¯ C s ( T ) of the evaders rapidly growswith f . This happens because, for increasing values of f , there are less and less selfish playerssurrounded by a growing number of altruistic ones: these players, besides the external gain,enjoy the public services ensured by taxpayers without giving any personal contribution, sotheir average social capital tends to remain always positive. On the other hand, even the capital¯ C a ( T ) of taxpayers grows with f since, when there are many altruists who randomly donateunits of capital, a large fraction of the donations statistically goes to other altruists, so thealtruistic component has, collectively, a small loss of capital.When the fraction f of taxpayers goes below a certain threshold f th , which in Figure 2 (for theadopted values of the free parameters of the model) is slightly less than 40%, even being thecollective capital ¯ C ( T ) still positive, the average capital of taxpayers becomes negative. Thismeans that – on average – they pay more than what they receive. In any case, the averagesocial capital of taxtaxpayerspayers is always lower than the average social capital of evaders,i.e. ¯ C a ( T ) < ¯ C s ( T ) ∀ f . Therefore, one should conclude that it is always more convenient, froman individual point of view, to choose a selfish strategy, i.e. to evade taxes, regardless of thefraction f of taxpayers. But we know that the average collective capital ¯ C ( T ), for a communityof tax evaders only ( f = 0%), is zero, while the same capital, for a community of taxpayersonly ( f = 100%), reaches its maximum positive value. So, the apparently optimal choice at theindividual level leads, at the collective level, towards a sub-optimal results.This finding is already interesting in itself, since it capture the paradoxical outcome typical ofthe prisoner’s dilemma in the context of game theory. But it is also interesting to notice that,even adopting a minimal number of assumptions, our model gets the same result described by6igure 4: Improved Tax Model . Not rescaled average final capitals ¯ C ( t ) , ¯ C a ( t ) and ¯ C s ( t ) over T = 100 turns as a function of the percentage f of taxpayers. All the curves result shifteddownward with respect to those in Figure 2 and two new thresholds do appear. Elster (1989) and shown in Figure 3. The two heavy lines in the Elster’s diagram indicatehow the expected benefits, for the cooperators and for the free riders, vary with the numberof cooperators (altruists). As in our Figure 2, the line representing the reward to free riders isconstantly above the other one, meaning that noncooperation is individually optimal in terms ofselfish benefits. At the same time, it is better for all if all agents cooperate than if nobody does,indeed,
B >
0. The free riders get the largest benefit C, whereas the worst outcome A is reservedfor the cooperators. If there are at least D cooperators their benefits become positive. The thinline shows how the average benefits for the collectivity varies with the number of cooperators(also for Elster, by definition, it must begin at 0). The distance between the two heavy linesrepresents the cost (per altruist) of cooperation. In the figure the cost doesn’t vary with thenumber of cooperators, but in general it may increase or decrease as more people cooperate.Indeed this topic is a collective action dilemma and tax evasion is a problem of the free riders. Inindividual terms, noncooperation is the most advantageous choice since the capital of tax evadersis always greater than the capital of taxpayers. But the capital of the collectivity increases onlythanks to the contribution of altruistic players. Thus, for the collective point of view, groupswith more cooperators are favored compared to groups with few cooperators. In fact, altruistspay a cost to obtain benefits for the collectivity, and the more they are, the smaller such a costis.However, we could more realistically expect that, if the number of taxpayers is not big enough,selfish tax evaders will get a negative final capital value, because they will need to buy on themarket those goods and services which have not been produced due to their evasion. Actually,this is exactly what happens if we do not rescale the curves shown in Figure 2. The new resultsare reported in Figure 4. The most evident change with respect to the graph analyzed beforeis that all the curves have been now shifted downward. As a consequence, we can recognize no7igure 5:
Adaptation of Cipolla Diagram . Players’ behavior sketched as function of both thepublic services S available for the collectivity and the individual contribution C i of each playerto this services. The celebrate four Cipolla’s categories can be recognized, strictly depending onthe fraction f of taxpayers present in the community. longer one but three thresholds, namely a , b , c . • If f < a , social capital is negative for all of the three categories. As expected, taxes paidby few altruists aren’t enough to guarantee the public services for everybody, so evadersdamage the community and also themselves. Cooperators suffer twice because they paytaxes and may also need to buy substitutes of public goods. • If f = a , tax evaders begin to gain, since there is a sufficient fraction of taxpayers. Taxespaid by altruists suffice to ensure that evaders spend, for substitutes of public goods, lessthan what they gain (thus making their social capital, on average, positive). Altruistsremain in the negative, as the collective social capital. • If f > b , the average social capital of the collectivity approaches positive values, but stillat the expenses of taxpayers whose capital remains, on average, negative. • If f > c , average capital for taxpayers becomes positive. Above such a threshold, publicservices are sufficient for everyone (even if the collected resources are less than they shouldbe). Costs are distributed among many altruists and they can finally benefit from theiraltruistic action. The few evaders still present in the community do continue, of course,to be better off than altruists, since they benefit from the public services without pay-ing taxes and all community members (consciously or unconsciously) settle for a lowerquality/quantity of the public service.The scenario just outlined can be effectively summarized by means of an adaptation of thewell known Cipolla diagram (Cipolla (1976)), as showed in Figure 5. Let’s call S the totality of8igure 6: Tax Model with mixed players . Rules of the taxpayersTax evasion model, for acommunity of N players. Tax evaders play only game B, taxpayers play only game A and mixedplayers randomly alternate between game A and game B. public services available for the collectivity and C i the individual contribution of the each playerto this services: S is negative when services are scarce and positive when they are sufficient forguarantee the social wealth, while C i is negative when the agent only benefits from this serviceswithout any effort and positive when he contributes to create them. Following these definitions,four quadrants can be clearly recognized in the Cipolla diagram, which identify four types ofplayers depending on the fraction f of taxpayers present in the community:1. Smart players: are the taxpayers for f > c , since they contribute with their individualcapital to ensure a positive level of public services for everybody;2. Naive players: are the taxpayers for f < c , since they maintain positive the collectivesocial capital at their expenses;3. Bandits: are the tax evaders for f > a since, with their selfish behavior, they get anadvantage (free riders) at expenses of the rest of the community;4. Stupid players: are the tax evaders for f < a since, with their selfish behavior, not onlydo penalize the rest of the community, but also themselves. In this section we introduce the possibility, for both the altruistic and selfish players, to assumean intermediate behavior. For this purpose, we insert a third category between the two onesdescribed in previous sections. Agents belonging to this new category randomly alternate (withprobability 0 .
5) between game A and game B. They will be named ”mixed players” because oftheir dynamic strategy setting, alternatively aimed to preserve their capital (thus evading taxes9igure 7:
Single run simulation . An example of the small-world lattice (left panel). Taxpayersare represented with green nodes, evaders with blue nodes and mixed players with yellow ones.The showed configuration of the network has been captured at the end of a single run simulation.The time behavior of the fraction of players belonging to the three considered categories (topright panel) is shown together with that one of the corresponding average social capital (bottomright panel). The initial percentage of taxpayers has been fixed to (above the critical valuefor both IF and CF equal to 1). by game B), and to cooperate (by game A), as explained in Figure 6.In order to select the transition rules to change category, each player has been given a newvariable, named, ”believeness” 0 < B i <
1, which specifies the level of commitment in the mem-bership to categories. For both taxpayers and evaders, B i = 1 means that the agent is a zealotof her group, whereas the more B i approaches the zero value, the more it means that the agentis easily influenced and so more inclined to change strategy, becoming a mixed player. For anyof the mixed players, the meaning of parameter B i is slightly different: if B i = 0 .
5, the agent isand remains undecided, whereas values lower that 0 . . B i is based on both the imitation of their acquaintances and the individual eco-nomic situation.In order to account the first point, we need to introduce an opportune topological structurefor our community. In the previous section, the simulations have been performed by assuminga fully connected topology, where each player was able to interact with each other. Now weassume a more realistic social structure, in particular a small-world lattice (Watts and Strogatz1998), where each player is a node connected with short-range ties (that mimic strong socialrelationships) to her four neighbors, but with a small rewiring probability ( r = 0 .
02) of sub-stituting one of those ties with a long-range one (representing a weak social relationship). An10igure 8:
Single run simulation . Another example of the small-world lattice at the end ofanother single run simulation, with the same plots of the previous figure but in the case of aninitial percentage of taxpayers equal to (below the critical value for both IF and CF equalto 1). example of such a kind of network is reported in the left panel of Figure 7, where taxpayers,evaders, and mixed players are represented, respectively, by green, blue, and yellow nodes.For taxpayers and evaders, if at a given time step the number of nearest neighbors belonging totheir same category is lower than the sum of the players of the other categories (included themixed one), the believeness value B i decreases of a quantity IF × δB , where IF is the ”ImitationFactor” and δB = 0 .
01; otherwise, this value increases of the same quantity (of course neverexceeding 1). If, after this, for a certain player of one of these two categories it happens that B i ≤
0, that player becomes a mixed player and her new value of B i is randomly chosen in theinterval [0 , B i decreases of the quantity IF × δB ifthe evaders are more than taxpayers, otherwise it increases of the same quantity. If, after this,for a given mixed player it happens that B i ≤
0, that player becomes an evader; instead, if forthe same player B i ≥
1, she becomes a taxpayer. On the other hand, if the number of mixedplayers is greater than (or equal to) the sum of the players belonging to the other two categories, B i moves towards 0 . IF × δB and the agent maintains her mixed behavior.The second mechanism which influences the change of category concerns the economic situationof the players. If the social capital of a given player is negative, the agent will be unsatisfied ofher own economic situation and so more prone to change her strategy. So, when their capital isnegative, for both evaders and taxpayers the believeness value decreases of a quantity CF × δB ,where CF is the ”Capital Factor”; on the other hand, for mixed players, B i increases or de-creases of the quantity CF × δB depending on their actual state: if B i ≥ . Thresholds diagram . Threshold values for the initial percentage of taxpayers asfunction of the capital factor CF and for different values of the imitation factor IF . B i < . IF and CF ),below which the global situation gets always worse and above which gets always better. InFigure 7 and in Figure 8 are reported the results of two typical single run simulations with IF = CF = 1 and in correspondence of an initial percentage of taxpayers which is, respectively,above (60%) and below (50%) the critical value (that, for these values of IF and CF turns outto be around 55%). In the first simulation, the final economic condition appears to be good forall the categories and in the community there is a majority of players who pay taxes; instead,in the second one, the final economic situation is good only for the evaders and these playersrepresent the majority of the agents.The threshold values for the initial percentage of taxpayers as a function of CF are showed inFigure 9 for different values of IF . As we can see, for a given value of IF , generally the criticalvalue of the initial percentage of pay taxes rapidly increases with CF , then it tends to oscillatearound a stationary asymptotic value which decreases with IF . For IF = 0, i.e. without imita-tion, a change in strategy is due to CF only: when CF is low, i.e. when the dissatisfaction for anegative economic situation is not significant, a small initial percentage of taxpayers is enoughto induce a final positive trend for the whole community; on the contrary, when CF is high,i.e. when a negative capital heavily acts on the personal dissatisfaction, the initial percentageof taxpayers has to be more consistent in order to counterbalance the tendency of the altruisticcomponent to be penalized with respect to the selfish one. Such an effect is reduced by the12resence of imitation, which in average helps the initial fraction of taxpayers – that should be,in any case, always greater than 50% – to spread around the community.Following such results, a reasonable policy implication could be to induce a sense of satisfactionin taxpayers, in order to reduce the temptation to evade, even when the personal economicsituation is bad. Thus, it should be better, for the Government, to care of taxpayers than ofthe evaders: for instance, an educational policy spreading a tax morale is expected to be moreeffective than a tax amnesty, because it operates in such a way that individuals feel themselvesrewarded by institutions. This can also impact on the number of taxpayers, which has beendescribed as a key factor in determining the average social capital. In this last section we want to analyze the evolution of the system as function of the three mainfree parameters of our model, namely, the tax, the penalty and the probability of an audit. Itis, indeed, worth to notice that changing the value of the external gain g would produce only asymmetric rescaling effect on the final capital of all the social components. Therefore, we neglectits variations and let g = 1, without loss of generality. In previous sections, parameters valueswere: d = 2, h = 3, and p = 0 .
4. In all simulations we also fix IF and CF to 1, and we startwith two different initial percentage of taxpayers, above and below the 55% threshold (whichhas already been shown in Figure 9).In the first setting, taxpayers are 60%, mixed players 0% and evaders 40%. In Figure 10 thefinal percentage of agents (panel a) and the final average capital (panel b) are reported for thethree categories and after 2000 turns, in correspondence of increasing values of the tax d (weverified that 2000 turns are sufficient to stabilize the final composition of the population). Itclearly appears that for d < h ) the finalcomposition of the population is dominated by taxpayers, with percentage above 70 %, andthere is a corresponding good economical situation for all the three categories. For higher valuesof the tax (i.e. for d ≥ h ), the evolution of the system leads to a majority of tax evaders and thissituation fits good only to evaders. This means that, as one could expect, keeping the amount oftaxes below the penalty induces tax payment and results in a decrease of the evasion. Figure 10also shows the final percentage and the final average capital for the three categories and after2000 turns, but for increasing values of the penalty h (panels c and d) and the audit probability p (panels e and f), respectively. Looking at the average capital in panels (d) and (f), we cansee that for values of the penalty h > p > .
8, the economicsituation of the evaders becomes worse than the ones of taxpayers and mixed players. But,curiously, this does not induce necessarily to a reduction of tax evasion: in fact, panels (c) and(e) show that the final percentage of evaders does not result to be lower for h > p > . Final percentage and final average capital after 2000 turns of the three categories forincreasing values of, respectively, tax (a-b), penalty (c-d) and audit probability (e-f ). The initialpercentage of taxpayers is always equal to 60%.
Final percentage and final average capital after 2000 turns of the three categories forincreasing values of, respectively, tax (a-b), penalty (c-d) and audit probability (e-f ). The initialpercentage of taxpayers is always equal to 50%. d = 1. However, differently from theprevious case, an increase of either the penalty or the audit probability, has now the beneficialeffect of reducing the number of evaders, who tend to change strategy becoming mixed players.It is interesting to notice that, in correspondence of the maximum values h = 10 and p = 1 . Conclusions
In this paper we presented a very simple toy model of tax evasion in order to explore, by meansof extended agent-based simulations, under which conditions people behave altruistically, bypaying taxes, or, on the contrary, behave selfishly, by evading taxes.In the first part of the paper, the impact of a varying fraction of altruistic players on the finalcapital of a fully connected community has been shown. Surprisingly, the simulation resultsshowed that, despite its simplicity, the model is able to capture the real trend of the expectedbenefits for a collective group in presence of a variable number of cooperators and free riders,as already described by Elster (1989). A further improvement of the model, which accounts forthe need of substitutes to non-produced public goods (because of high levels of tax evasion), hasallowed to identify a threshold, in the fraction of taxpayers, below which evaders not only createa damage for the collectivity but also for themselves (thus falling in the category of ”stupidpeople” in the famous 1976 Cipolla’s diagram).In the second part of the paper, the model was enriched by the introduction of some extensions:a small-world network topology for the social community (driving the imitation), and a thirdcategory of ”mixed” players (sometimes altruist, sometimes selfish). New interesting resultshave been obtained, showing the presence of a threshold, in the initial percentage of taxpayers,able to ensure an average economic advantage to this category of players at the end of thesimulations. Such a threshold is influenced by the individual propensity of agents to imitateand by their sensitivity with respect to their personal economic situation. Finally, an extendedparametric study has shown how the amount of taxes to pay, the penalty for evaders and theaudit probability, do influence both the final composition of the considered community and thefinal average capital of its three social components (taxpayers, evaders and mixed players).
Acknowledgments
Authors thank Andrea Rapisarda for useful discussions.
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