IEEE Computational Intelligence Magazine - May 2021 - 69

(1 + a j " i) d ij - (1 - a i " j) d ij
(1 + a j " i) d ij + (1 - a i " j) d ij
ai " j + a j " i
=
, 6Gi, jH ! E
2 + a j " i - a i " j
=

This ratio is between 0 (when both
counterparts declare exactly the same
amount, a i " j = a j " i = 0) and 1 (when
the seller does not declare any amount).
Note that when the fraction of the
amount that is incorrectly declared is
constant and identical for any firm and
transaction, we have a i " j = a j " i = a,
with a being the ratio of unpaid quantity for defectors (Table II). We also
have a i ) j = a, 6G i, j H ! E. Then, the
assigned values for parameter a in the
model would be calibrated by the a i ) j
values in the real data.
Figure 5 represents the cumulative
distribution of the ratio a i ) j in the VAT
declaration network. As can be observed,
the ratio of undeclared transactions is
almost zero for around 75% of operations and below 0.5 for 99.06% of
them. Given this, in our simulations,

0.25

0.5
αi <->j

0.75

1

Cumulative Probability

FIGURE 5 Cumulative distribution of the ratio of undeclared transactions in the VAT declaration network.

0.9
0.8
Final Frequency of Cooperators

ai ) j

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.7
0.6
0.5
0.4

α = 0.1
α = 0.2
α = 0.3
α = 0.4
α = 0.5
α = 0.6

0.3
0.2
0.1

0

100

200

300
Time-Steps

400

500

0.9
Final Frequency of Cooperators

small and big firms (those transactions
below and above quantile 2% in the
edge-weight distribution, respectively).
We assume that the amount to be paid is
a constant fraction of the transaction value.
Therefore, we set the value d ij = " d L, d H ,,
6Gi, j H ! E, to a fixed value of d L = 10
and d H = rd L = 457.59. Edges' weights
d ij are randomly initialized, characterized
by prob dH = 0.02. Note that weights d ij
do not change over time. They are static
and therefore, the same payoff matrix of
the mixed game is used for every pair of
players i and j in this study.
We define the ratio of divergence in
the tax declarations between seller i and
buyer j (a i ) j) as the percentage of VAT
declaration mismatch between the tax
declared by seller i and buyer j. We only
take those mismatches benefiting the
sellers (this is the case where the seller
declares less amount than the buyer). The
ratio is calculated as follows: Given d ij as
the amount that firms i and j need to
declare, let a i " j be the percentage of this
amount that seller i does not declare and
a j " i the percentage that buyer j declares
in excess. Then, the ratio of undeclared
accrued tax between i and j is:

0.8
0.7

α = 0.1
α = 0.2
α = 0.3
α = 0.4
α = 0.5
α = 0.6

0.6
0.5
0.4
0.3
0.2
0

0.2

0.4
0.6
Initial Frequency of Cooperators

0.8

1

FIGURE 6 The upper plot shows the cooperators' evolution for different a values, from 0.1 to
0.6, with an initial cooperators frequency of 0.5. The lower plot shows that the initial frequency of strategies in the population is not relevant for the final state of the model.

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IEEE Computational Intelligence Magazine - May 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - May 2021
