Theoretical Economics Letters, 2012, 2, 8793 http://dx.doi.org/10.4236/tel.2012.21017 Published Online February 2012 (http://www.SciRP.org/journal/tel) Controlling the Tax Evasion Dynamics via MajorityVote Model on Various Topologies* Francisco W. S. Lima Dietrich Stauffer Computational Physics Lab, Departamento de Física, Universidade Federal do Piauí, Teresina, Brazil Email: fwslima@gmail.com Received October 27, 2011; revised November 18, 2011; accepted November 25, 2011 ABSTRACT Within the context of agentbased MonteCarlo simulations, we study the wellknown majorityvote model (MVM) with noise applied to tax evasion on simple square lattices (LS), HonischStauffer (SH), directed and undirected Bara basiAlbert (BAD, BAU) networks. In to control the fluctuations for tax evasion in the economics model proposed by Zaklan, MVM is applied in the neighborhood of the noise critical qc to evolve the Zaklan model. The Zaklan model had been studied recently using the equilibrium Ising model. Here we show that the Zaklan model is robust because this can be studied using equilibrium dynamics of Ising model also through the nonequilibrium MVM and on various topologies cited above giving the same behavior regardless of dynamic or topology used here. Keywords: Opinion Dynamics; Sociophysics; Majority Vote; Nonequilibrium 1. Introduction The Ising model [1,2] has become a excellent tool for to study other models of social application. The Ising model was already applied decades ago to explain how a school of fish aligns into one direction for swimming [3] or how workers decide whether or not to go on strike [4]. In the Latané model of Social Impact [5] the Ising model has been used to give a consensus, a fragmentation into many different opinions, or a leadership effect when a few people change the opinion of lots of others. To some extent the voter model of Liggett [6] is an Isingtype model: opinions follow the majority of the neighbour hood, similar to Schelling [7], all these cited models and others can be found out in [8]. Already Föllmer (1974) [9] applied the Ising model to economics. Realistic models of tax evasion appear to be necessary because tax evasion remain to be a major predicament facing governments [1013]. Experimental evidence provided by Gächter [14] indeed suggests that tax payers tend to condition their decision regarding whether to pay taxes or not on the tax evasion decision of the members of their group. Frey and Torgler [15] also provide empirical evidence on the rele vance of conditional cooperation for tax morale. Follow ing the same context, recently, Zaklan et al. [16] devel oped an economics model to study the problem of tax evasion dynamics using the Ising model through Monte Carlo simulations with the Glauber and heatbath algo rithms (that obey detailedbalance equilibrium) to study the proposed model. I have introduced for the first time the use of local majority rules in social systems. I also include a review paper on all my contributions to the field of sociophysics. Another one shows that a unifying paper on all discrete opinion models. I hope you will find these papers of interest. Grinstein et al. [17] have argued that nonequilibrium stochastic spin systems on regular square lattices with updown symmetry fall into the universality class of the equilibrium Ising model [18]. This conjecture was con firmed for various Archimedean lattices and in several models that do not obey detailed balance [1922]. The majorityvote model (MVM) is a nonequilibrium model proposed by M. J. Oliveira in 1992 [20] and defined by stochastic dynamics with local rules and with updown symmetry on a regular lattice shows a secondorder phase transition with critical exponents β, γ, and ν which characterize the system in the vicinity of the phase tran sition identical with those of the equilibrim Ising model [1] for regular lattices. Lima et al. [23] studied MVM on VD random lattices with periodic boundary conditions. These lattices posses natural quenched disorder in their connections. They showed that presence of quenched connectivity disorder is enough to alter the exponents and from the pure model and therefore that is a relevant term to such nonequilibrium phasetransition with dis agree with the arguments of Grinstein et al. [17]. Recently, simulations on both undirected and directed scalefree networks [2430], random graphs [31] and social networks [3235], have attracted interest of re *This paper is dedicated to Dietrich Stauffer. C opyright © 2012 SciRes. TEL
F. W. S. LIMA 88 searchers from various areas. These complex networks have been studied extensively by Lima et al. in the con text of magnetism (MVM, Ising, and Potts model) [35 39], econophysics models [16,40] and sociophysics model [41]. In the present work, we study the behavior of the tax evasion on twodimensional LS, BAD and BAU networks, and SH networks using the dynamics of MVM, furthermore add a policy makers’s tax enforcement mechanism consisting of two components: a probability of an audit each person is subject to in everyperiod and a length of time detected tax evaders remain honest. We aim here is to extend the study of Zaklan et al. [16], which illustrates how different levels of enforcement affect the tax evasion over time, to dynamics of MVM as an alternative model of nonequilibrium to the Ising model that is capable of reproduce the same results for analysis and control of the tax evasion fluctuations. Then, we show that the Zaklan model is very robust for equili bruim and nonequilibrium models and also for various topologies used here. We show that the choice of using the Ising (equilibrium dynamics) or MVM (nonequilib rium dynamics) used to evolve the Zaklan model is ir relevant, because the results obtained in this work are about the same for both Ising and MVM. The Zaklan model also is robust, because it works on LS, SH net work, BAD and BAU networks. We show that for dif ferent topologies the Zaklan model reaches our objective, that is, to control the tax evasion of a country (Germany and others). This does not occur with other models as AxelrodRoss model for evolution of ethnocentrism [41], because the results are different depending of the topol ogy of the network. The Ising model also is not robust, because on directed BA network occur with other models as Axelrodthis no phase transition present as also on directed LS, 3D, 4D and directed hypercubics lattices [42]. As described above, the MVM was proposed by M.J. Oliveira in 1992 [22] in order to improve the crite rion of Grinstein et al. [17], initially described above. In the order to achieve his goal he used 44 (LS) Archi medean lattice. However, also with the aim of improve this criterion other researchers studied MVM on several other topologies that are not Archimedeans [39,4348]. To their surprise all results obtained for the critical ex ponents are different from results obtained by M. J. Oliveira, and are also different for each topology used. Pereira et al. [49] then concluded that MVM has differ ent universality classes which depend only on the topol ogy used, and that all have one thing in common that is their effective dimension, obtained by critical exponents for each topology used, equals Deff = 1. Here, we show that the Zaklan model behavior is identical for all to pologies or dynamics studied here. Therefore, we believe that this model is very robust, different the other models cited above. Galam [5053] introduced for the first time local majority rules in social systems to the field of so ciophysics using discrete opinion models. Here, we also hope to introduce for the first time the use of MVM to the field of sociophysics or econophysics using discrete opinions as in the Zaklan. Therefore, we do not live in a social equilibrium, any rumor or gossip can lead to a government or market chaos and we believe that nothing is better than a nonequilibrium model (MVM) to explain events of nonequilibrium. Stock market generalized to market, in order to include currency exchange. The re mainder of our paper is organised as follows. In Section 2, we present the Zaklan model evolving with dynamics of MVM. In Section 3 we make an analysis of tax eva sion dynamics with the Zaklan model on twodimen sional square lattices using MVM for their temporal evolution under different enforcement regimes; we dis cuss the results obtained. In Section 4 we show that MVM also is capable to control the different levels of the tax evasion analysed in Section 3, as it was made by Zaklan et al. [16] using Ising models. We use the en forcement mechanism cited above on various structures: SL, SH network, BAD and BAU network; we discuss the resulting tax evasion dynamics. Finally in Section 5 we present our conclusions about the study of the Zaklan model using MVM. 2. Zaklan Model On a square lattice each site of the lattice is inhabited, at each time step, by an agent with “voters” or spin vari ables σ taking the values +1 representing an honest tax payer, or −1 trying to at least partially escape her tax duty. Here is assumed that initially everybody is honest. Each period individuals can rethink their behavior and have the opportunity to become the opposite type of agent they were in previous period. In each time period the system evolves by a single spinflip dynamics with a probability given by 1 1112 2 i k ii wqS i (1) where Sx is the sign 1 of x if . 0x 0Sx if 0x , and the summation runs over all i nearest neighbour sites k i of i . In this model an agent as sumes the value 1 depending on the opinion of the majority of its neighbors. The control noise parameter q plays the role of the temperature in equilibrium systems and measures the probability of aligning antiparallel to the majority of neighbors. Then various degrees of ho mogeneity regarding either position are possible. An ex tremely homogenous group is entirely made of honest people or tax evaders, depending the sign Sx of the majority of neighbhors. If of the neighbors is zero the agent Sx i will be honest or evader in the next Copyright © 2012 SciRes. TEL
F. W. S. LIMA Copyright © 2012 SciRes. TEL 89 time period with probability 12. We further introduce a probability of an efficient audit . Therefore, if tax evasion is detected, the agent must remain honest for a number k of time steps. Here, one time step is one sweep through the entire lattice. p q k are triggered in order of to control the tax evasion level. The individual remain honests for a certain number of periods, as explained before in Sections 2 and 3. We also extend our study to other networks as the SH net work, BAD and and BAU networks with N = 400 sites. As before the initial configurations is with all honest agents (i ) at fixed “Social Temperature” q. Here, we have been performed simulations of 25,000 time steps. 3. Controlling the Tax Evasion Dynamics Here, we first will present the baseline case, i.e., no use of enforcement, for different network structure. We use for LS, BAD and BAU network, and SH network. All simulation are performed over 25,000 time steps, as shown in Figure 1. For very low noises the part of autonomous decisions almost completely disappears. The individuals then base their decision solely on what most of their neighbours do. A rising noise has the opposite effect. Individuals then decide more autonomously. For MVM it is known that for c, half of the people are honest and other half cheat, while for c states dominated by cheating or by correlated changed into dominated; you always have correlations compliance prevail for most of the time. Because this behavior we set some values close to qc, where the case that agents dis tribute in equal proportions onto the two alternatives is excluded. Then having set the noise parameter, , close to (qc = 0.075) on the square lattice, as suggested in Sec tion 3, we vary the degrees of punishment (k = 1, 10 and 50) and audit probability rate (p = 0.5%, 10% and 90%). Therefore, if tax evasion is detected, the enforcement mechanism and the period time of punishment q qq q p In Figure 1 we plot the baseline case k = 0, i.e. , no use of enforcement, for the LS (a), SH (b), BAU (c), and BAD (d) for dynamics of the tax evasion over 25,000 time steps. Although everybody is honest initially, it is impossible to predict which level of tax compliance will be reached at some time step in the future. Figure 2 illustrates different simulation settings for square lattice, for each considered combination of degree of punishment (k = 1, 10 and 50) and audit probability rate (p = 0.5%, 10% and 90%), where the tax evasion is plotted over 20,000 time steps. Here we show that even a very small level the enforcement (p = 0.5% and k = 1) suffices to reduce fluctuations in tax evasion and to es tablish mainly compliance. Both a rise in audit probabil ity (greater p) and higher penalty (greater k) work to flat ten the time series of tax evasion and to shift the band of possible noncompliance values towards more comp li ance. However, the simulations show that even extreme enforcement measures (p = 90% and k = 50) cannot fully solve the problem of tax evasion. Figure 1. Baseline case for different network structure. Where we use q = 0.95qc on different networks. All simulation are performed over 25,000 time steps.
F. W. S. LIMA 90 In Figure 3 we display tax evasion for BAD and BAU networks, SH networks for different enforcement for k = 1, 10, and 50 with the same audit probability p = 1%. We observe for BAD ou BAU network that the tax evasion level decreases with increasing time periods k of punish ment, similar behavior also occurs for SH network. Figure 2. The square lattice model of tax evasion with various degrees of enforcement q = 0.95qc and 20,000 time steps. Figure 3. Display tax evasion for different enforcement regimes for BA and SH Network and for degrees of punishment k = 1, 10, 50 and audit probability rate pa = 4.5%. Copyright © 2012 SciRes. TEL
F. W. S. LIMA 91 Figure 4. Display of the tax evasion for different enforcement regimes for BA and SH network. Again, we use 25,000 time steps. In Figure 4 we plot tax evasion for BAD and BAU networks, and SH network, again for different enforce ment k = 1, 10, and 50, but now with audit probability . For BAD and BAU, and SH networks the tax evasion level decreases with increasing audit probability showing that an increase of the audit probability fa vors the control of tax evasion. In all case studied here, we observed that the time period of punishment is important to control tax evasion. 4.5%p p k 4. Conclusion In summary, tax evasion can vary widely across nations, reaching extremely high values in some developing countries. Wintrobe and Gёrxhani [54] explains the ob served higher level of tax evasion in generally less de veloped countries with a lower amount of trust that peo ple have in governmen tal institutions. To study this problem Zaklan et al. [16] proposed a model, called here call the Zaklan model, using Monte Carlo simula tions and a equilibrium dynamics (Ising model) on square lat tices. Their results are good agreement with analytical and experimental results obtained by [915,54]. In this work we show that the Zaklan model is very robust for analysis and control of tax evasion, because we use a nonequilibrium dynamics (MVM) to simulate the Zaklan model, that is the opposite of the study done by [16] equilibrium dynamics (Ising model), and also on various topologies used here. Our results are qualitatively and quantitatively identical the results obtained by Zaklan et al. [16] giving the same behavior regardless of dynamic or topology. Here, we also hope to have introduced for the first time the use of MVM to the field of sociophysics and econophysics using discrete opinion model as Zaklan model. As we do not live in a social equilibrium and any rumor or gossip can lead to a government or market chaos, we believe that nothing is better than a nonequi librium model (MVM) to explain events of nonequilib rium. Therefore, as the Zaklan model is a sociophysics and econophysics model, we also believe that the best topology used for simulations of this model are social networks of BAD and SH type. 5. Acknowledgements The author thanks D. Stauffer for many suggestion and fruitful discussions during the development this work and also for the reading this paper. We also acknowledge the Brazilian agency FAPEPI (TeresinaPiauíBrasil) for its financial support. This work also was supported the sys tem SGI Altix 1350 the computational park CENAPAD. UNICAMPUSP, SPBRAZIL. REFERENCES [1] L. Onsager, “Crystal Statistics. I. A TwoDimensional Copyright © 2012 SciRes. TEL
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