IEEE Computational Intelligence Magazine - May 2021 - 70

we consider the range of realistic values
for parameter a between 0 and 0.5.
D. Model Setup

We set the model up for 50 Monte-Carlo
runs, with 1,000 time-steps in each run,
thereby ensuring that all the realizations
reach a stationary stable state (as we can
see in Section V-A). The simulation results

were obtained by averaging the last 25%
of the simulation time-steps in the independent Monte-Carlo runs. Source code
and data files are available at https://
bitbucket.org/mchserrano/evolutionary
-game-tax-fraud.
We set the remaining parameters-
when not explicitly specified-as follows:
inspection cost C = 1 and reputational

0.8

V. Analysis of the Results

0.7

A. General Dynamics of the Model

0.6

α Values

0.5
0.4
0.3
0.2
0.1
0
0

0.4

0

0.8
1.2
Inspection Cost (Γ )

1.6

0.2
0.4
0.6
0.8
Final Frequency of Cooperators

2

1

FIGURE 7 Sensitivity analysis on a and C. a controls the difficulty of the game, directly affecting the cooperation level. C is the inspection cost when a player is inspected due to mismatch
declaration. We see how inspection cost C is only significant for a values approximately
between 0.2 and 0.6. If a is either lower or higher, C has no impact on the cooperation level.

TABLE III Main features of the social network topologies.
NETWORK

70

reward for cooperators R = 1. Fine value
z was set to 1.5 (50% fine plus the undeclared quantity) as per previous models
[43]. The values used to generate the linear audit probability function were set to
HlH = HlL = 0.5, when the analysis was
not focused on these subjective
-probabilities (see Section V-C).The mutation probability of the evolutionary
dynamics was always set to n = 0.01.

AVERAGE DEGREE

CC

D

REAL NETWORK FROM DATA

1.9952

0

111

SF (BA WITH m = 2)

3.0090

0.0035

12

SF (BA WITH m = 4)

5.0070

0.0044

9

SF (BA WITH m = 6)

7.0100

0.0065

9

SF (BA WITH m = 8)

9.0220

0.0082

8

ASS. SF ( p = 0.5, m = 2)

2.9970

0.0029

20

ASS. SF ( p = 1, m = 2)

2.9850

0.0120

283

DISS. SF ( p = 0.5, M = 2)

2.9935

0.0016

18

DISS. SF ( p = 1, m = 2)

2.9860

0.0002

15

ASS. SF ( p = 0.5, m = 8)

9.0040

0.0069

11

ASS. SF ( p = 1, m = 8)

9.0460

0.0325

53

DISS. SF ( p = 0.5, m = 8)

9.0120

0.0097

8

DISS. SF ( p = 1, m = 8)

8.9820

0.0005

19

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2021

We first show the dynamics of the
model for the base parameters. The
upper plot in Figure 6 shows the evolution of the model over 1,000 timesteps for different values of a. The plot
also shows the max-min range of the
simulations for the 50 Monte-Carlo
realizations. The stationary state is
quickly reached and even 500 timesteps are s-ufficient in this case. Additionally, the max-min ranges in the plot
highlight that the deviation of the
model is low. The lower plot in Figure  6 shows how the model dynamics
is independent of the initial strategy settings of the population. By enabling a
player to randomly change their strategy using the mutation operator, we also
eliminate differences in the outputs in
case of extreme conditions (e.g., the
initial frequency of cooperators is either
0 or 1). Therefore, we fixed the initial
frequency of cooperators to 0.5 for the
rest of the analysis.
We also analyze the impact of
changing the a values and inspection
cost C by running a sensitivity analysis.
Results can be observed in the heatmaps of Figure 7. This graph is in
agreement with the set of possible
games analyzed in Figure 1. These
results show how the model is sensitive
to the values of a, which smoothly regulates how the population converges to
cooperation or defection and therefore,
the difficulty of the game. The impact
of inspection cost C is less significant in
the model dynamics when a has either
high or low values. In fact, when a values are high (or low) and therefore,
cooperation (or defection) is restricted,
https://bitbucket.org/mchserrano/evolutionary-game-tax-fraud https://bitbucket.org/mchserrano/evolutionary-game-tax-fraud https://bitbucket.org/mchserrano/evolutionary-game-tax-fraud
IEEE Computational Intelligence Magazine - May 2021

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