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participant in the white (black) market. The following results are not sensitive to the
specific lattice geometry and we implement the model on a two-dimensional square
lattice with dimension 1000
×
1000. Eq. (1) contains also the coupling of the spins to a
local magnetic field
B
i
which can be associated with the morale attitude of the agents
and corresponds to the parameter
i
in the theory of Nordblom and Zamac (2012). In
addition, our model contains a local temperature
T
i
which measures the susceptibility
of agents to external perturbations (either influence of neighbors of magnetic field).
We then use the heat-bath algorithm [cf. Krauth (2006)] in order to evaluate statistical
averages of the model. The probability for a spin at lattice site
i
to take the values
S
i
=
γ
±
1isgivenby
1
p
i
(
S
i
)=
(2)
1 + exp
{−
[
E
(
−
S
i
)
−
E
(
S
i
)]
/
T
i
}
and
E
(
−
S
i
)
−
E
(
S
i
) is the energy change for a spin-flip at site
i
. Upon picking a random
number 0
≤
r
≤
1 the spin takes the value
S
i
= 1when
r
<
p
i
(
S
i
= 1) and
S
i
=
−
1
otherwise. One time step then corresponds to a complete sweep through the lattice.
Following Hokamp and Pickhardt (2010) we consider societies which are composed
of the following four types of agents: (i)
selfish a-type agents
, which take advantage
from black market activities (
S
i
=
−
1) and, thus, are characterized by
B
i
/
T
i
<
0and
|
>
J
; (ii)
copying b-type agents
, which conform to the norm of their social network
and thus copy the behavior with respect to black or white market participation from
their neighborhood. This can be modelled by
B
i
<<
J
and
J
i
/
T
i
B
i
|
1; (iii)
ethical c-type
agents
, which have large moral doubts about participating in black market services and
thus are parametrized by
B
i
/
T
i
>
0and
>
J
;(iv)
random d-type agents
, which act by
chance, within a certain range, due to some confusion about the attribution of services
to the black or white market. We implement this behavior by
B
i
<<
J
and
J
/
T
i
<<
1.
The parameters distinguishing the different agent types are taken from Pickhardt and
Seibold (2013) and Seibold and Pickhardt (2013).
Furtheron we implement different enforcement schemes into our model. Here we
first consider the case where the detection of a black market participating agent enforces
its compliance over the following
h
time steps. This is the procedure which has been
invoked in Zaklan et al. (2008, 2009); Lima (2010); Pickhardt and Seibold (2013) and
also implemented in a randomized variant in Lima and Zaklan (2008). Second, we also
study lapse of time effects, i.e. the situation where a detected agent is also screened
over several years in the past by the (tax) authorities (i.e. backaudit). This variant has
been studied within an econophysics tax compliance model in Seibold and Pickhardt
(2013). If tax evasion is detected in the current time period, the backaudit comprises
also an inspection of the preceding
b
p
time steps. Denote with
n
e
the number of time
steps over which the agent was evading within the backaudit plus current period. Then
the period
k
over which the agent is reinforced to be compliant is set to
k
=
n
e
∗
|
B
i
|
h
.
For example, for a convicted agent in the current time step, inspection of the preceding
b
p
= 5 time steps reveals three periods where he was evading. Setting
h
= 2 yields a
number of (3 + 1)
2 = 8 periods where he is forced to be compliant. Thus the above
limit of fixed compliance period
h
is recovered in the limit of zero backaudit
b
p
= 0
since then
k
=(0 + 1)
∗
∗
h
=
h
.
