IEEE Computational Intelligence Magazine - May 2021 - 67

B. Evolutionary Update Rule

The players can change their strategies
s(i ), 6i ! Z during the whole discreteevent simulation. These changes in strategies come from two evolutionary

mechanisms. First, a player i can imitate
others in the population (generally, their
direct contacts in the social network). Second, players can also change their strategies
by adopting a strategy at random, following a random mutation mechanism with
probability n. The mutation operator does
not take into account if the new strategy
was beneficial in the past in terms of the
fitness values of the players. However, the
social imitation update rule is a social
learning process of the players in the game
[52]. Social imitation update rules consider
the fitness of direct neighbors on a network in the previous steps to make their

decision, either in a deterministic or a
probabilistic way. In our model, we use the
Fermi function as the social imitation
update rule.
Fermi's rule is one of the most wellknown imitation processes [53], [54] and
is applied synchronously: for each step t, a
focal agent i compares its fitness value in
the previous step t - 1, f ti - 1, with one
of its direct neighbors in the social network, j, also in t - 1: f tj- 1. Therefore, the
Fermi's rule is a stochastic pairwise comparison rule, where players can also make
mistakes during the imitation process
(i.e., a player can imitate a neighbor with

αd

DG1
R < P; D1; D2; D3
SH
R > P; D2; D3

R
1 - θ .φ

HG
R>P
0

0

Γ

FIGURE 1 Different games for the payoff matrix in Table I according to parameter values C
and ad. The subjective audit probability is a constant function H($) = H. Here, SH is the Stag
Hunt game; HG is a Harmony Game, and DG1 represents a defection game with conditions
D 1, D 2 and D 3 .

R + Θ ′Γ

PD
R > P ; D1; D2; D3 SH

1 - Θ ′φ

SD
R > P; D1; D3

CG
R > P; D2

(1)
HG
R>P

R
0

DG3
R < P; D2

DG2

DG1
R < P; D1; D2; D3

αd

payoff matrix in Table I satisfies R 2 S
and T 2 P, facilitating the level of cooperation in the game. According to Allen
and Nowak [51], a social dilemma occurs
when R 2 P (mutual cooperation
-benefits both players) and at least one of
the following conditions is met to favor
the adoption of defection: (D 1): T 2 R,
(D 2): P 2 S, or (D 3): T 2 S. Values of
the parameters in Table II determine
which of these conditions are satisfied.
Figure 1 shows possible games according to ad and inspection cost C values; and
assuming a constant audit probability
H($) = H. We see three regions in the figure
depending on the parameters' values: there
is no social dilemma in two of them and
the third is associated with a classical game.
Cooperation prevails if the non-declared
amount ad is below R/(1 - Hz), whereas
defection is the preferred option if ad is
high enough. A Stag Hunt game appears
for intermediate values of ad, where conditions for cooperation R 2 P and temptation to defect (D 2 and D 3) coexist.
In case of a non-constant audit
probability H($), the outcome is more
complicated. The payoff matrix includes
two possible values, H(ad ) = H and
H(2ad ) = Hl. Figure 2 shows the set of
games for different values of H and ad,
assuming a fixed Hl. For large values of
ad (above the horizontal line defined by
(r + Hl C ) /(1 - Hl z)), there is no social
dilemma as mutual defection is always
preferred over mutual cooperation
(P 2 R ). Below the horizontal line, multiple games arise. In general, cooperation
is expected when ad is low and H is
high. More specifically, when the audit
probability is a decreasing function
(H 2 Hl ), the most expected games are
those favoring cooperation (coordination
and harmony games). However, when
the audit probability is increasing
(H 1 Hl ), there is a significant region
where games such as the prisoner's
dilemma or snowdrift prevail and defection is the expected outcome.

0

Θ′

1/φ

1

Θ
FIGURE 2 Different games for the payoff matrix in Table I according to parameter values H
and ad. PD is the Prisoner's Dilemma game; SD is the Snowdrift game; CG is a coordination
game; DG2 represents a defection game with condition D 2 and D 3; DG3 is a defection game
with condition D 2; and (1) is a game with R 2 P and D 2 .

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IEEE Computational Intelligence Magazine - May 2021

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