Theoretical Economics Letters, 2012, 2, 87-93 Published Online February 2012 (
Controlling the Tax Evasion Dynamics via Majority-Vote
Model on Various Topologies*
Francisco W. S. Lima
Dietrich Stauffer Computational Physics Lab, Departamento de Física, Universidade Federal do Piauí, Teresina, Brazil
Received October 27, 2011; revised November 18, 2011; accepted November 25, 2011
Within the context of agent-based Monte-Carlo simulations, we study the well-known majority-vote model (MVM)
with noise applied to tax evasion on simple square lattices (LS), Honisch-Stauffer (SH), directed and undirected Bara-
basi-Albert (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 Ising-type
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
[10-13]. 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 detailed-balance 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
up-down 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 [19-22]. The
majority-vote model (MVM) is a nonequilibrium model
proposed by M. J. Oliveira in 1992 [20] and defined by
stochastic dynamics with local rules and with up-down
symmetry on a regular lattice shows a second-order
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 non-equilibrium phase-transition with dis-
agree with the arguments of Grinstein et al. [17].
Recently, simulations on both undirected and directed
scale-free networks [24-30], random graphs [31] and
social networks [32-35], have attracted interest of re-
*This paper is dedicated to Dietrich Stauffer.
opyright © 2012 SciRes. TEL
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 two-dimensional 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
Axelrod-Ross 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 Axelrod-this 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 [39,43-48].
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 [50-53] 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 two-dimen-
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 spin-flip dynamics with a
probability given by
 
Sx is the sign 1
of x if . 0x
if 0x
, and the summation runs over all i nearest-
neighbour sites
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
will be honest or evader in the next
Copyright © 2012 SciRes. TEL
Copyright © 2012 SciRes. TEL
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.
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
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
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 non-compliance 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.
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
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. 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 [9-15,54]. 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 (Teresina-Piauí-Brasil) for its
financial support. This work also was supported the sys-
tem SGI Altix 1350 the computational park CENA-PAD.
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