![Plots of fractions of conformist cooperators (CCs), conformist defectors (CDs), profiteer cooperators (PCs), and profiteer defectors (PDs) against MCS. BA is generated by the growth and preferential attachment, where the parameters are set as N = 900 and m = 3. RE is generated by a square lattice network with 900 nodes. In the game model, b = [1.1, 1.3, 1.5] and δ = [0.7, 0.8, 0.9] are set for cross experiments. Subplots (a) and (b) show the network evolutionary game with conformists and profiteers on the BA and regular network, respectively.](https://www.researchgate.net/profile/Minyu-Feng-2/publication/358640119/figure/fig2/AS:1173227804078081@1656730714628/Plots-of-fractions-of-conformist-cooperators-CCs-conformist-defectors-CDs-profiteer_Q320.jpg)
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Chaos 32, 023117 (2022); https://doi.org/10.1063/5.0081954 32, 023117
© 2022 Author(s).
Evolutionary multigame with conformists
and profiteers based on dynamic complex
networks
Cite as: Chaos 32, 023117 (2022); https://doi.org/10.1063/5.0081954
Submitted: 11 December 2021 • Accepted: 25 January 2022 • Published Online: 16 February 2022
Bin Pi, Ziyan Zeng, Minyu Feng, et al.
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Chaos ARTICLE scitation.org/journal/cha
Evolutionary multigame with conformists and
profiteers based on dynamic complex networks
Cite as: Chaos 32, 023117 (2022); doi: 10.1063/5.0081954
Submitted: 11 December 2021 ·Accepted: 25 January 2022 ·
Published Online: 16 February 2022
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Bin Pi,1Ziyan Zeng,1Minyu Feng,1,a)and Jürgen Kurths2
AFFILIATIONS
1College of Artificial Intelligence, Southwest University, Chongqing 400715, People’s Republic of China
2Potsdam Institute for Climate Impact Research, 14437 Potsdam, Germany and Institute for Complex System and Mathematical
Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
a)Author to whom correspondence should be addressed: myfeng@swu.edu.cn
ABSTRACT
Evolutionary game on complex networks provides a new research framework for analyzing and predicting group decision-making behavior
in an interactive environment, in which most researchers assumed players as profiteers. However, current studies have shown that players are
sometimes conformists rather than profit-seeking in society, but most research has been discussed on a simple game without considering the
impact of multiple games. In this paper, we study the influence of conformists and profiteers on the evolution of cooperation in multiple games
and illustrate two different strategy-updating rules based on these conformists and profiteers. Different from previous studies, we introduce
a similarity between players into strategy-updating rules and explore the evolutionary game process, including the strategy updating, the
transformation of players’ type, and the dynamic evolution of the network structure. In the simulation, we implement our model on scale-
free and regular networks and provide some explanations from the perspective of strategy transition, type transition, and network topology
properties to prove the validity of our model.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0081954
The study of network evolutionary games can provide a new per-
spective for explaining cooperation in society. Our task is to
incorporate conformists and multigames into the traditional evo-
lutionary game, which are more consistent with reality. Based on
this model, this paper proposes two different strategy-updating
rules and investigates their impact on the evolution of coopera-
tion in the network. In addition, we make an interpretation of
the simulation results in terms of strategy transition, type tran-
sition, and network topology properties. Our work may shed
some new light on the study of network evolutionary games with
conformists and multigames.
I. INTRODUCTION
In the whole development process of human civilization,
although there are all kinds of conflicts and struggles, cooperative
behavior is still ubiquitous in the real world and can be found in
natural and social systems.1Cooperation is the motive force for the
stable development and progress of human society, and its scope
and depth are unmatched by any other animal. Using game the-
ory, it has been tried to explain human cooperative behavior from
various angles, and five basic theories have been formed, includ-
ing kin selection, direct reciprocity, indirect reciprocity, network
reciprocity, and group selection.2In addition, complex networks
have also made rapid progress, and various novel network mod-
els have been proposed in recent years.43–45 Meanwhile, numerous
game models have been widely employed as paradigms to describe
pairwise or group interactions. Among them, the most commonly
mentioned ones are the prisoner’s dilemma game (PDG)3–5and the
snowdrift dilemma game (SDG).6–8In the classical PDG, two play-
ers simultaneously decide whether to cooperate (C) or defect (D)
and will receive the reward R (punishment P) if both players choose
cooperation (defection). While a player chooses cooperation and the
other one chooses defection, the cooperator will get a sucker’s pay-
off S and the defector will receive a temptation to defect T. These
payoffs satisfy the ranking: T>R>P>Sand 2R>T+S.9–11 As
is well known, for a rational individual, the defect is the best choice
regardless of which strategy the opponent chooses, and only two
sides choose to cooperate to maximize the collective income. Obvi-
ously, players will inevitably fall into the social dilemma of pursuing
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-1
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individual and collective benefits, and we note that different param-
eter values will lead to different social dilemmas. In the classical
SDG, players interact and accumulate payoffs in the same way, but
the order of payoffs changes to T>R>S>P. It is worth not-
ing that (D, D) is the only Nash equilibrium for the PDG, while
the SDG has two equilibriums comprising (C, D) and (D, C) strat-
egy combinations; i.e., a player chooses the defection strategy when
the opponent chooses cooperation strategy and chooses cooperation
strategy when the opponent chooses defection strategy. In addition
to the two classical game models introduced above, there are the
public goods game (PGG),12,13 the stag-hunt game (SHG),14,15 and
the ultimatum game (UG),16,17 to name but a few.
In recent years, evolutionary multigame or hybrid game, which
can use different payoff matrices in the field of social dilemmas, has
attracted extensive attention and achieved fruitful results. For exam-
ple, Huang et al. introduced multigame with the aspiration-driven
updating rule and investigated the evolution of cooperation.18 Li
et al. proposed a multigame (composed of the prisoner’s dilemma
game and the snowdrift game) and its coevolution mechanism in
networked populations.19 Han et al. incorporated the aging prop-
erty to evaluate the effect of age on cooperation in the investigated
spatial multigame.20 Liu et al. proposed the coevolution setup of
strategy and multigame based on a memory step.21 Deng et al. intro-
duced the mechanism of multi-games on interdependent networks
and explored the evolution of cooperation.22 Recently, Szolnoki and
Chen extended the prisoner’s dilemma game and the public goods
game by allowing players not simply change their strategies but also
let them vary their attitudes for a higher individual income to reveal
the possible advantage of a certain attitude.39 Besides, players not
only learn the most successful strategies, but also choose the most
popular strategies.23–29 We call this type of player conformist, and
there are many conformists in real life. For example, we have noticed
that when consumers shop online, they tend to buy mass-market
goods. In fact, a person’s point of view is often strongly influenced
by most people around him or her. Furthermore, there are also
researchers who have investigated the effect of different strategy-
updating rules on cooperation. For example, Szolnoki and Danku
studied that players may use two updating rules simultaneously,
which are imitation and death–birth rule.38 In addition, Szolnoki
et al. first proposed the idea of changing strategy and interaction
simultaneously.41,42
In previous works, the Fermi updating rule is the most widely
used strategy-updating rule, which is based on the pairwise com-
parison by using a Fermi-function-like probability function.30 Nev-
ertheless, it does not consider the impact of similarity between
players, which can play a significant role in real-life interactions.
For instance, studies have shown that people who are from the
same social class are often more similar in aspects, such as generos-
ity toward others31 or the willingness to take risks.32 In this paper,
we incorporate the similarity parameter into the strategy-updating
rule and assume that individuals are more likely to adopt strategies
similar to their counterparts. If a player tends to pursue the high
payoff, we call him/her a profiteer and his/her rule payoff-driven
strategy-updating rule. On the other hand, as we have mentioned
above, a player might adopt the strategy more frequent in his neigh-
borhood. In this case, we call him/her a conformist and his/her
rule conformity-driven strategy-updating rule. In addition to the
evolution of the strategy, we also allow for the evolution of the
type, namely, the alternation of the player’s type between profiteers
and conformists with time, which will lead to the transformation of
the strategy-updating rule between payoff-driven and conformity-
driven rules. Moreover, in the process of evolution, individuals with
lower income will be more likely to change their types and game
opponents. Because for a player with low income, this is not only
related to his/her strategy-updating rule, but it may also be rele-
vant to his/her game opponents. In order to improve the income,
he/she will try to change his/her game opponents and type to alter
the strategy-updating rule. Therefore, it is worth noting that the
whole concept used in this work belongs to the family of co-called
coevolutionary models.40
The remainder of this paper is organized as follows. In Sec. II,
we describe the model. Then, we show our numerical simulation
results in Sec. III. Finally, we summarize the conclusion and show
the outlook in Sec. IV.
II. MODEL
In this section, we study an evolutionary multigame in a pop-
ulation of Nplayers sitting on a scale-free network proposed by
Barabási–Albert (BA) and an L×Lsquare lattice network (RE), in
which each node or each lattice site denotes a game player. In our
model, each individual will be divided into two categories, one-half
of the players will participate in SDG and the other half will play
PDG. All of them are randomly distributed on the network, which
indicates that there randomly exists an interaction between different
categories of players. The different payoff matrices for the SDG and
PDG players can be described as M1 and M2, respectively,
M1=1δ
b0,M2=1−δ
b0, (1)
where 1 ≤b≤2 represents the temptation to defect. In particular,
there are two types of players in the network, including type C (ratio-
nal conformists: players with rational conformity behavior) and type
P (profit-seeking players). Additionally, we introduce a parameter
ρ∈[0, 1] that indicates the proportion of type C players, while the
rest of the population are type P players. Obviously, when ρ=0,
all players belong to type P, it becomes the traditional prisoner’s
dilemma game or the snowdrift dilemma game. Besides, a larger
value of ρindicates a greater proportion of type C and more ratio-
nal conformists in the group. In this model, we mainly consider
two kinds of strategies: cooperation C and defection D, in which
we assume that each player in the population holds only one single
strategy at a certain step.
A. Strategy evolution
In this subsection, we illustrate the strategy-updating rules
based on the different types of players. In the proposed model, we
suppose that players asynchronously update their strategy; i.e., only
one player’s strategy will be updated at each round of the game.
Besides, a player is randomly chosen and accumulates his/her pay-
offs by interacting with his/her own neighbors depending on his/her
payoff matrix. Then, the payoff-driven players update strategies
relying on the payoff difference, while the conformity-driven players
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-2
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FIG. 1. An example for strategy evolution. (a) represents an initial status of a
network, where green nodes are profiteers and red nodes are conformists, while
C marks cooperators and D marks defectors. In (b), V1is randomly chosen and
updates its strategy from C to D. In (c) and (d), V2and V5are following the same
steps mentioned above.
choose the most popular strategy among their neighbors. Referred
to the strategy evolution of the model, we illustrate the strategy evo-
lution as an instant in Fig. 1. C corresponds to cooperators in the
network and D corresponds to defectors, while green nodes rep-
resent profiteers and red nodes conformists. At each step, a player
will be randomly selected and update his/her strategy according to
his/her type. We note that if a player is a conformist, e.g., V1, he/she
is more likely to choose the most popular strategy among his/her
neighbors. While if a player is a profiteer, he/she is more likely to
choose the strategy of the player with the highest payoff among
his/her neighbors.
First of all, according to the random sequential update protocol,
a randomly selected player xacquires his/her payoff 5xby play-
ing the game with all his/her neighbors. Next, player xrandomly
chooses one neighbor y, who also acquires his/her payoff 5yin the
same way as previous player x. To avoid payoff-related effects that
are due to heterogeneous interaction topologies, we normalize the
payoff with the degree of the corresponding player; we, therefore,
have 5i=5i/ki, where kidenotes the degree of player i.
After both players acquire their payoffs, if player xis a profiteer,
then he/she adopts the strategy syfrom player ywith a probability
determined by the Fermi updating rule
P(sx←sy)=1
1+e(5x−5y)/sxy , (2)
where sxy quantifies the similarity between players xand ydenoted
as
sxy =kxky+cxcy
pk2
x+c2
xqk2
y+c2
y
, (3)
where kiindicates the degree of player iand cirepresents his/her
local clustering coefficient. In this proposed rule, we have the payoff-
driven rule by Eq. (2). Different from the previous game strategy-
updating rule, our model utilizes the similarity between two players
instead of a noise factor κ. The reason is that, in the actual game
process, each player should be affected by different external influ-
ences, and the constant κis impracticable. Therefore, we employ the
similarity sxy to better express this feature.
On the other hand, if player xis a conformist, we use the Fermi
updating rule
P(Nsx−khx)=1
1+e(Nsx−khx)/sxy , (4)
where Nsxis the number of players adopting strategy sxwithin
the interaction range of player x, while khx=kx/2, which indi-
cates one-half of the degree of player x. Analogously, we propose
the conformity-driven rule by Eq. (4). Through Eq. (4), player xis
more likely to adopt the strategy more frequently in his neighbor-
hood. It is worth noting that, in this study, a conformist obtains
only local information. In particular, a conformity-driven player,
namely, a conformist, simply tends to follow the majority in its local
neighborhood.
B. Network evolution
Along with the above strategy evolution, the scale-free and the
L×Lsquare lattice networks also keep evolving over time. In the
last stage of each step, on the basis of the payoff of all players in the
network, we randomly choose a player according to the payoff of
anti-selection; i.e., the lower payoff is the higher probability of the
player being selected. The probability that a profiteer (conformist)
is selected is as follows:
Wi=(5i−5min +α)−1
Pj∈(5j−5min +α)−1, (5)
where 5irepresents the payoff of player i,5min is the minimum pay-
off of the network in the current step, indicates the player set of
the network, and αis a smoothing coefficient. Although the player’s
payoff may be negative, we can still guarantee that the probability Wi
is positive through Eq. (5). After that, the selected player will discon-
nect all the existing edges from his/her neighbors and then randomly
choose players to reconnect from the network. Thus, the number of
connected players equals the number of edges before disconnection.
Furthermore, the selected player will change his/her type; namely, if
he/she is a conformist in the current step, he/she will become a prof-
iteer in the next step and vice versa. We hereby illustrate the network
evolution as an instant in Fig. 2 to better describe the network evo-
lutionary steps. C corresponds to cooperators in the network and
D corresponds to defectors, while green nodes represent profiteers
and red nodes conformists. At each step, a player will be selected by
anti-preference and then change his/her type and reconnect existing
edges.
In brief, we first introduce our multiple game model, including
PDG and SDG, then propose two different strategy-updating rules
based on conformists and profiteers, and finally explain the evolu-
tion rules of the network and the type-updating of a player to perfect
the shortcomings of some previous studies.
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-3
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FIG. 2. An example for network evolution. (a) represents an initial status of a net-
work, where green nodes are profiteers and red nodes are conformists, while C
marks cooperators and D marks defectors. In (b), V1is chosen by an anti-prefer-
ence law; hence, type transition occurs; this makes V1becoming a profiteer from
a conformist and its connections are reconnected to V6and V7. In (c) and (d), V4
and V8are following the same steps mentioned above.
III. SIMULATION RESULTS AND DISCUSSIONS
In this section, with the purpose of confirming the previous
theory, we hereby provide the simulation method and also present
the simulation results along with their analyses. Specifically, we
analyze the changes of the topological structure before and after net-
work evolution through degree distribution first and then observe
the influence of band δon the fraction of network cooperators
fc. Next, we study the strategy and type distribution on the regu-
lar square lattice network at the micro-level and finally discuss the
quantitative evolution of conformist cooperators (CCs), conformist
defectors (CDs), profiteer cooperators (PCs), and profiteer defectors
(PDs) in the evolution process.
A. Method
The simulation is carried out in Python. At the beginning of
each simulation, unless otherwise specified, N=900 players are
embedded into a BA network or a 30 ×30 square lattice network.
BA is generated by growth and preferential attachment, as proposed
by Barabási and Albert in 1999. When a node is newly added to
the network, it connects to m=3 existing nodes with the probabil-
ity defined by Πi=ki
Pjkj. The initial strategies, types, and games of
the players are randomly selected from the spaces S ={C, D}, T = {P,
C}, and G = {PDG, SDG}respectively, which is implemented by the
roulette algorithm. Specifically, we first generate a random number
that obeys a (0, 1) uniform distribution and then let it compare to
the threshold 0.5. If it is larger than 0.5, we let the player choose to
cooperate; otherwise, we let the player choose to defect. For the state
space T and G, we use the same method to carry out. Subsequently,
a player is randomly selected from non-isolated players in the net-
work to evolve his strategy. If his type is P, he updates the strategy
by Eq. (2). On the contrary, he updates the strategy with Eq. (4).
Next, another player is chosen by an anti-preference law denoted by
Eq. (5) to reconnect his edges and change his type. In this paper,
we have a constant smoothing coefficient α=1 for Eq. (5). We
implement the evolution of cooperation in a simulation with length
T=7000 steps. Moreover, to avoid additional disturbances, the
final results were averaged over up to ten independent realizations
for each set of parameter values to assure suitable accuracy.
B. Structural properties
In this simulation, we present the degree distribution, the most
significant topological property of the network, to analyze the fol-
lowing conclusions. We use the two networks mentioned above
(BA and RE) and then record the degree distribution at the begin-
ning and end of their evolution, respectively. As a result, Fig. 3
shows the degree distribution before and after the evolution of the
BA network and the regular network, respectively. From Figs. 3(a)
and 3(c), it is obvious that the degree distribution obeys the power-
law characteristics of BA networks and the uniformity characteris-
tics of regular networks. Since there will be random reconnection of
player at each step, the degree distribution shown in Figs. 3(b) and
3(d) has changed into a near normal distribution, which means that
the network at this time is no longer a BA network or a regular net-
work. Moreover, we can see that the degree distributions of the two
evolved networks are very similar.
C. Cooperative behaviors
For a population, the cooperation level is one of the most con-
cerning indicators to people and is commonly characterized by the
cooperation frequency fc, denoting the fraction of cooperators to a
population. Hereby, we mainly investigate the fluctuation of fcin
various scenarios.
1. fcvs b
For the purpose of exploring the relationship between the frac-
tion of cooperators fcand b, we proceed by examining the impact of
the value of temptation to defect bunder different values of δ, which
measures the payoff of a sucker to PDG and SDG. The results are
shown in Fig. 4, from which we can clearly see how the frequency of
cooperators fcvaries in dependence on bfor different values of δon
the BA network and the regular network, respectively. First, it can
be observed that when the value of bis small, the number of coop-
erators is slightly lower than the number of defectors. Then, fcwill
decline with the increase of b. However, cooperators are not extinct
when the value of bis large enough, and there still exist a few coop-
erators in the network. In particular, as simulation, it also shows that
the fraction of cooperators fcis independent of the value of δ(Fig. 4).
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-4
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FIG. 3. Degree distribution of networks before and after evolution. BA is generated by the growth and preferential attachment, where the parameters are set as N=900 and
m=3. RE is generated by a square lattice network with 900 nodes. In the game model, we set b=1.1 and δ=0.8. Subplots (a) and (b) show the degree distribution before
and after the evolution of the BA network, respectively. Subplots (c) and (d) show the degree distribution before and after the evolution of the regular network, respectively.
In addition, we can see that the changing trend of fcin the BA and the
regular network is similar from Figs. 4(a) and 4(b); i.e., the change
of the network will not lead to too much variation in the number of
network cooperators.
2. fcvs
δ
In order to better observe the relationship between the frac-
tion of cooperators fcand δ, we let the cooperation frequency fc
be a function of the independent variable δfor different values
of b, where both the BA and the regular network size N=900.
The experimental results are shown in Fig. 5. From both Figs. 5(a)
and 5(b), we can see that fchas slight fluctuations with the increase
of δ, and fcis always stable around 0.3. Besides, it can be observed
that fcwill be inhibited with the increase of bwhen δis moder-
ate, e.g., 0.70 < δ < 0.90, which proves the correctness of our above
experiment indirectly.
To summarize, we study the fluctuation of the fraction of
cooperators fcunder different values of band δon the scale-free and
regular network and find that fcdecreases with the increase of band
is independent of the transformation of δ.
D. Snapshots on the regular network
A question worthy of consideration has emerged, i.e., why did
the above results appear? As is well known, cooperation can survive
by means of forming clusters, which is known as network reci-
procity. Therefore, we subsequently concentrate on snapshots that
describe the evolution of the strategy and type at the micro-level. For
the sake of clarity, cooperators and profiteers are colored by green,
defectors and conformists are colored by red, and isolated players
are colored by black. From top to bottom, the parameter pairs (b,δ)
are set as (1.2, 0.7), (1.3, 0.8), and (1.5, 0.8), while the time steps from
left to right are equal to 0, 2500, and 7000, respectively.
Snapshots visualize the evolutionary game in network and are
more understandable and demonstrate the evolution of the strate-
gies and types of players with different parameter pairs (b,δ) intu-
itively. The simulation results are shown in Fig. 6. In the beginning,
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-5
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FIG. 4. Plots of cooperator fraction against defection temptation. BA is generated by the growth and preferential attachment, where the parameters are set as N=900
and m=3. RE is generated by a square lattice network with 900 nodes. We set δ=[0.7, 0.75, 0.8, 0.85, 0.9, 0.95], and the defection temptation bis set from 1.1 to 1.7.
Subplots (a) and (b) show the cooperator fraction against defection temptation on the BA and regular network, respectively.
FIG. 5. Plots of cooperator fraction against sucker’s payoff. BA is generated by the growth and preferential attachment, where the parameters are set as N=900 and
m=3. RE is generated by a square lattice network with 900 nodes. We set b=[1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7], and the sucker’s payoff δis set from 0.6 to 1. Subplots (a)
and (b) show the cooperator fraction against sucker’s payoff on the BA and regular network, respectively.
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-6
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FIG. 6. Characteristic snapshots for different parameter pairs (b,δ). From top to bottom, the parameter pairs (b,δ) are set as (1.2, 0.7), (1.3, 0.8), and (1.5, 0.8), respectively.
From left to right, the time step is fixed to be T = 0, 2500, and 7000. Here, green denotes profiteers and cooperators, red denotes conformists and defectors, and black
denotes isolated players. Other parameters include the scale of the lattice network N=100 ×100 and the smoothing coefficient α=1. As evolution proceeds, the profiteers
eventually dominate the network, with slightly more defectors than cooperators and a small number of isolated players emerging in the network.
FIG. 7. Plots of fractions of conformist cooperators (CCs), conformist defectors (CDs), profiteer cooperators (PCs), and profiteer defectors (PDs) against MCS. BA is
generated by the growth and preferential attachment, where the parameters are set as N=900 and m=3. RE is generated by a square lattice network with 900 nodes.
In the game model, b=[1.1, 1.3, 1.5] and δ=[0.7, 0.8, 0.9] are set for cross experiments. Subplots (a) and (b) show the network evolutionary game with conformists and
profiteers on the BA and regular network, respectively.
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-7
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the same number of cooperators and defectors and profiteers and
conformists is located on the lattice randomly, and there exist no
isolated players in the network at this time. We observe that sev-
eral isolated players appear on the network when time reaches 2500.
Moreover, we see from Fig. 6(a) that with the evolution of the
network game, profiteers are gradually increasing and eventually
dominate the whole network. Figure 6(b) shows that the number
of defectors slowly increases along with the evolution proceeding.
Ultimately, the cooperators are not annihilated, and the fraction of
cooperators fcis stable at about 0.3.
Next, we analyze the reasons for the results in Figs. 4 and 5
based on the degree distribution in Fig. 3 and snapshots in Fig. 6.
Due to the existence of network reciprocity, the BA network will
promote the evolution of cooperation. However, previous research
has also shown that when we normalize payoffs on a heterogeneous
interaction network, we destroy the heterogeneity among players
and drastically weaken the positive impact of the enhanced network
reciprocity.33–35 In addition, there will be random reconnection of
players in the network at each step, which further reduces the degree
of heterogeneity between the players. As we have shown in Fig. 3(b),
the degree distribution of the evolved BA network not following the
power-law distribution means that it is no longer a BA network.
Therefore, it does not promote the emergence of cooperation in our
model. From Fig. 3, we know that although there are great differ-
ences in the degree distribution between the BA network and the
regular network at the beginning, the degree distributions of them
are very similar after the evolution, and the evolution of the frac-
tion of cooperators fcwill consequently have similar results on the
BA and regular network. As demonstrated in Fig. 6(a), there are
only a few conformists in the network at the end of the evolution
of the network game. Previous studies have shown that appro-
priate conformists in the network can promote the emergence of
cooperation.36,37 Since the number of conformists is too small, it does
not promote the emergence of network cooperation. On the other
hand, it is the existence of these conformists that makes the network
cooperators not annihilated.
E. Evolution of the number of different categories
Eventually, we present the number of conformist cooperators
(CCs), conformist defectors (CDs), profiteer cooperators (PCs), and
profiteer defectors (PDs) to be dependent on the Monte Carlo Step
(MCS) with different parameter pairs (b,δ) on the BA and regular
network, respectively. The simulation results are shown in Fig. 7.
In each subplot, despite the network type, the number of con-
formist defectors and conformist cooperators converges to zero after
MCS = 4000, while the quantity of profiteer cooperators and profi-
teer defectors gradually increases as time goes by and finally reaches
a steady value. It is observed that there is little difference in the
number of conformist cooperators and conformist defectors, while
the quantity of profiteer defectors is always greater than profiteer
cooperators in the process of evolution, which shows that the phe-
nomenon demonstrated in Fig. 7 is consistent with our previous
analysis. It is worth noting that although our model has more defec-
tors than cooperators in the network when the evolution reaches
a steady state, defection does not dominate the network. Besides,
in contrast to previous studies, when the temptation to defection
bis large, the number of cooperators in the network is not extinct
in our model and just slightly lower than the number of defectors.
Therefore, our model still has a positive effect in preventing the
annihilation of cooperative behavior.
Then, we analyze the causes of the phenomenon in Fig. 7. As
shown in Figs. 3(b) and 3(d), players have random reconnection at
each step, and the degree distribution of the BA and regular net-
work after evolution is similar and is no longer the same as the initial
degree distribution; i.e., the topology of both networks is changing in
the evolution process and becomes more and more similar. There-
fore, we find that the network has had little effect on the evolution
of the quantity of CC, CD, PC, and PD. In addition, as is mentioned
in the model, a player will be selected to change his type at each
step, and when the type transition occurs, players with lower payoff
are more likely to change their strategy-updating rules. Concretely,
when a profiteer is chosen by the anti-preference law in Eq. (5), he
obtains a relatively low payoff than other players; namely, defectors
account for a large proportion in his neighbors. Thus, the chosen
player will become a conformist at the next step and may become
a defector with a higher probability than a cooperator according
to Eq. (4). Otherwise, when a player changes his type from a con-
formist to a profiteer, its major neighbors are more likely to be
defectors as well. In spite of the fact that defectors are still dominant
in the chosen player’s neighbors, its next strategy totally depends on
neighbors’ payoff.
IV. CONCLUSION AND OUTLOOK
In this paper, we propose a multigame model with transfor-
mations between conformists and profiteers on dynamic networks.
The multigame evolution process is introduced, including strategy
evolution and network evolution. In the simulation, we analyze the
changes of the topological structure before and after the network
evolution through degree distribution, additionally study the coop-
eration sensitivity to the parameters band δ, and then observe the
strategy and type distribution on the square lattice network at the
micro-level. Moreover, we show the evolution process of conformist
cooperators (CCs), conformist defectors (CDs), profiteer coopera-
tors (PCs), and profiteer defectors (PDs) with different parameter
pairs (b,δ). Finally, we provide some explanations from the per-
spective of strategy transition, type transition, and network topology
properties for the simulation results.
In our work, the similarity is only conducted based on the
cosine similarity of the players’ degree and clustering coefficient.
Different similarity rules, however, may lead to different results. In
addition, players’ choice of game opponents for reconnection is ran-
dom in this paper. Different results may also appear when we give
players a certain rule for selecting game opponents. Eventually, we
hope that our work will stimulate further research in the multigame
in structured populations to address the social dilemmas, especially
from the perspective of empirical experiments.
ACKNOWLEDGMENTS
This work was supported in part by the Ministry of Education
in China (MOE) Project of Humanities and Social Sciences under
Grant No. 21YJCZH028 and in part by the Ministry of Science and
Chaos 32, 023117 (2022); doi: 10.1063/5.0081954 32, 023117-8
Published under an exclusive license by AIP Publishing
Chaos ARTICLE scitation.org/journal/cha
Higher Education of the Russian Federation within the framework
of state support for the creation and development of World-Class
Research Center’s Digital Biodesign and Personalized Healthcare
under Grant No. 075-15-2020-926.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts of interest to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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