IEEE Computational Intelligence Magazine - May 2021 - 66

observed behavior of the relevant agents
must always be an end to pursue. Given
that the data necessary for this type of
analysis is inexorably linked to individual data protection rights, collaboration
between researchers and tax administration is a must. This is especially the case
if we consider increasing our knowledge
of taxpayer compliance behavior, as
indicated by Bloomquist [49].
III. Model Description
A. Game Strategies and Payoff Matrix

Players of the game are a finite set of Z
agents (companies) occupying the nodes
of a social network, with edges denoting
economic transactions between them.
The network is undirected but weighted: a weight d ij means the accrued tax
to be declared and paid in relation with
all cumulative transactions between both
players i and j. Under a correct behavior,
this value d ij is the consumption tax
involved in the transactions between

TABLE I Payoffs for the mixed tax
fraud game.
C

D

C

R

R - HC

D

ad - H 6C + zad@

ad - Hl6C + zad@

both companies i and j. We differentiate
between two quantities to be paid, high
(d H ) and low (d L), based on high and
low transaction values, respectively.
Therefore, d ij = " d L, d H ,, 6Gi, j H being
an edge of the social network. In this
sense, the game can be called a mixed
game [23], [50] where two players play
a different game depending on the type
of transaction, given by parameter d ij .
Each player i chooses a strategy s
from two possibilities at every time step
(s(i ) = " C, D ,): being a cooperator or
tax payer (C), or being a defector or tax
evader (D). When being a tax evader, the
player does not pay a fraction of the
transaction value d ij , saving this cost as a
personal benefit ( free rider). We model
this fraudulent fraction by parameter
a ! 60, 1@, which also measures the difficulty of the social dilemma. Higher
values of a correspond to higher economical benefits for free riders when
declaring transactions.
In order to set the fitness of a player i
for a specific time step t ( f ti ), the player
accumulates all the payoffs w tij from the
pairwise interactions over all its direct
contacts in the social network (i.e.,
neighborhood in the network):
f ti = 1/Gk Hi / Gjk=Hi1 w tij, where Gk Hi is the
degree of player i. The payoff w ij of focal
agent i with respect to neighbor j is

TABLE II List of parameters of the tax fraud evolutionary game. H($) is a linear
function.
NAME
R

REPUTATIONAL AND SOCIAL REWARD FOR CORRECTLY PAYING TAXES

C

INSPECTION COST

d = " d H, d L ,

AMOUNT OF TAX DEBT TO BE PAID FOR THE INVOLVED TRANSACTIONS

z

FINE MULTIPLIER TO BE PAID BY A DEFECTOR UPON INSPECTION

a ! 60, 1@

H = " H H, H L ,

RATIO OF UNPAID TAX DEBT BY DEFECTORS

TAXPAYER'S SUBJECTIVE PROBABILITY OF BEING AUDITED WHEN ONE PLAYER DEFECTS

H H = H (ad H)

SUBJECTIVE PROBABILITY FOR HIGH TRANSACTIONS WHEN ONE PLAYER
DEFECTS

H L = H (ad L)

SUBJECTIVE PROBABILITY FOR LOW TRANSACTIONS WHEN ONE PLAYER
DEFECTS

Hl = " HlH , HlL ,

66

DESCRIPTION

TAXPAYER'S SUBJECTIVE PROBABILITY OF BEING AUDITED WHEN BOTH
PLAYERS DEFECT

HlH = H(2ad H)

SUBJECTIVE PROBABILITY FOR HIGH TRANSACTIONS WHEN BOTH PLAYERS
DEFECT

HlL = H(2ad L)

SUBJECTIVE PROBABILITY FOR LOW TRANSACTIONS WHEN BOTH PLAYERS
DEFECT

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2021

obtained by considering their strategies
in the previous time step and the specific
payoff matrix played depending on the
type of transaction (either low or high).
Table I shows the payoff matrix
defining the mixed game. Parameter R is
the social and reputational reward for a
player when acting in accordance to its
tax duties. C is the inspection cost a
company should pay when a tax agency
examines the company and its documentation regardless of its own behavior
(i.e., playing strategy). z is the fine multiplier a company must pay when the
tax agency audits the company and discovers fraudulent behavior.
H() is a linear probability function to
define a player's perception of how probable a tax audit is. This probability is subjective, even if all the players have the
same perception of this probability (we
can say there is a " shared collective perception " ), and depends on the taxpayer's
subjective probabilities about being audited, following previous studies such as that
of Hashimzade et al. [43]. The function
H() depends on the difference in the
amount declared by the players, and is
only applied when the tax agency discovers a transaction mismatch for a pair of
players. The probability function is built
from two values (2ad L and 2ad H ), which
set the probability of being inspected for
both low and high transactions. For clarity,
we have defined H H , H L, HlH , and HlL
as the values of the probability function
based on its arguments. TableĀ  II shows a
summary of all the model parameters.
The payoff matrix in Table I includes
different two-strategy games depending
on the values of the parameters. These
games determine the decision of every
player to either cooperate or defect, and
therefore influence the final outcome of
the evolutionary game. To characterize
the included games we consider a general payoff matrix:
C

D

C

R

S

D

T

P

where S, T, and P represent the expressions of Table I. The definition of the



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