IEEE Computational Intelligence Magazine - May 2021 - 73

D. Population Diversity in the
Subjective Inspection Probabilities

Next, we analyze how diversity in the
individuals of the population, with
respect to their subjective probabilities,
affects cooperation. In order to run this,
we considered HlH = HlL = 0.5 as the
mean n of the normal distribution
N ( n, v) of the subjective probabilities
of the whole population, and we modified the standard deviation v of the distribution. Figure 10 shows the output
of seven simulations with different standard deviation v values: from 0, corresponding to the default configuration of
the experiments in this work, to 0.4,
where individuals are highly diverse.
Figure 10 shows how population
diversity is beneficial for promoting
cooperation when the game is hard
(high values of a), but cannot promote
cooperation when the game is easy.
Similar trends were observed when
changing the density and other properties of the networks in Section V-B. This
diversity changes the cooperation levels
because of the polarization of the entire
population, as observed by Antonioni

Our model represents players of the tax system as agents
on nodes of a heterogeneous social network... When
being a tax evader, the player does not pay a fraction
of the transaction value, saving this cost as a personal
benefit (free rider).
et al. [59]. Figure 10 also reveals that
diversity always induces a shift of the
population to a 50% polarization (gray
horizontal line).
E. Impact of Rewarding and
Penalty Policies

In this final section of our model analysis, we focus on ascertaining if policies
to increase the reward for cooperators
are more efficient than those to
increase the punishment for defectors
via the fine values. Punishment versus
reward has been studied in different
public goods games and common pool
resources [53], [60], [61]. For our analysis, we increased the values of reward
R from 1 to 2 and fines from 1 (most
liberal-defectors just have to return
the unpaid tax) to 2 (the fine is double
the unpaid quantity). Figure 11 shows the
impact of different reward and fine values on cooperation under a sensitivity

analysis of a and for three different
scenarios of subjective audit probability
HlL and HlH .
We first observe how the impacts of
both reward and fine policies differ
depending on the subjective audit probability. When high transactions have a
higher subjective audit probability (the
third scenario), increasing the reward, R,
is more efficient for promoting cooperation than increasing the fine, z. The
highest cooperation percentage is
obtained with R = 2 while keeping the
base fine of 1.5. However, for all the a
values, increasing the fine up to 2 does
not generally induce as many new
cooperators as rewarding policies do.
When the subjective audit probability for low transactions is higher than for
high transactions, the output of the
model changes dramatically. If we are in
the balanced second scenario (HlH = 0.5
and HlL = 0.5), penalizing defectors

Final Frequency of Cooperators

The analysis reflects important variations when modifying these subjective
probabilities to favor a particular transaction type. We can see how tax fraud is
limited when the subjective probability is
higher for low transactions than high
transactions. In fact, differences are not
relevant when HlL 2 0.5. However,
when HlL decreases, the number of cooperators declines almost independently of
HlH . The number of cooperators declines
and differences are significant when a
values differ (e.g., cooperators are the
dominant strategy in the final population
only when a = 0.05 and the game is
easy). The results change when the probability of high and low transactions is equal
(Figure 9, lower plot). In this case, there
are no major differences in the frequency
of cooperators when HlL and HlH
change.These results show that the significant effect of subjective probability
on tax fraud, for low transactions in the
upper and middle heatmaps, is mainly
due to the larger number of low transactions in the network.

σ=0
σ = 0.01
σ = 0.05
σ = 0.1
σ = 0.2
σ = 0.3
σ = 0.4

0.8

0.6

50% Cooperators

0.4

0.2

0.7

0.6

0.5

0.4
0.3
α Values

0.2

0.1

FIGURE 10 Analysis of the diversity of subjective probabilities HlL and HlH when setting individuals of the population by generating a normal distribution N( n, v). We set n as 0.5 and plot
different values of v.

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IEEE Computational Intelligence Magazine - May 2021

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