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IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS 1
Evolutionary Dynamics of Preguidance
Strategies in Population Games
Linjie Liu and Xiaojie Chen
Abstract—Promoting cooperation among conflicting entities in
human society and intelligent systems is a formidable task. One
potential solution could involve the formulation of incentives
designed to decrease the benefits of noncooperators and/or
increase the rewards for cooperators. We put forth a novel
incentive approach, specifically, a guidance strategy where certain
cooperators willingly bear a cost to alter the actions of agents who
intend to defect prior to the actual commencement of a game.
We introduce an innovative game-theoretical framework that
sheds light on the dynamics of guidance strategies, encompassing
both peer guidance and pool guidance. Under the peer guidance
scheme, each guider independently incurs the cost to influence
agents intending to defect, whereas in the pool guidance scheme,
guiders organically establish an institution to influence agents
prone to free riding. Regardless of whether a peer or pool
guidance scheme is utilized, the implementation of a guidance
strategy has proven to be remarkably effective in reducing the
instances of pure cooperation, also known as second-order free
riding. Intriguingly, our result suggests that the pool guidance
strategy demonstrates a more potent deterrent effect on second-
order free-riding behavior than the peer guidance strategy,
particularly when the cost of guidance is exceptionally high.
These findings underscore the significance of preguidance in
fostering cooperation in human and multiagent AI systems and
could offer valuable insights for the development of a regulatory
mechanism for preemptive guidance and subsequent punishment.
Index Terms—Cooperation, decision-making, evolutionary
game theory, peer guidance, pool guidance.
I. INTRODUCTION
FROM the inception of artificial systems to the emergence
of multifaceted and intricate complex systems, the essence
of cooperation assumes an indispensable role in molding the
Manuscript received 27 February 2024; revised 2 April 2024; accepted
3 April 2024. This work was supported in part by the National Natural
Science Foundation of China under Grant 62306243, Grant 62036002, and
Grant 61976048; in part by the Natural Science Foundation of Shaanxi under
Grant 2023-JC-QN-0791 and Grant 23JK0693; and in part by the Fundamental
Research Funds of the Central Universities of China under Grant 2452022012.
(Corresponding author: Xiaojie Chen.)
Linjie Liu is with the College of Science, Northwest A & F University,
Yangling 712100, China, and also with the College of Economics & Man-
agement, Northwest A & F University, Xianyang 712100, China (e-mail:
linjieliu1992@nwafu.edu.cn).
Xiaojie Chen is with the School of Mathematical Sciences, University of
Electronic Science and Technology of China, Chengdu 611731, China (e-mail:
xiaojiechen@uestc.edu.cn).
Digital Object Identifier 10.1109/TCSS.2024.3386501
very essence of our surrounding world [1],[2],[3],[4],[5],
[6],[7]. Nevertheless, the establishment and preservation of
cooperation within multiagent systems often present intricate
challenges that demand careful consideration and strategic ap-
proaches [8],[9],[10],[11],[12],[13],[14],[15],[16].Inmany
cases, agents may be better off pursuing their own self-interest
rather than cooperating with others. This tension between self-
interest and cooperation has been a topic of study for many
years and has led to the development of theories such as evo-
lutionary game theory [17],[18].
Evolutionary game theory offers a robust framework for ex-
amining the dynamics of cooperation, elucidating its origins
and long-term viability in various social systems, including nat-
ural environments, human societies, and artificial intelligence
(AI) communities [19],[20]. In recent scholarly endeavors, a
plethora of mechanisms encompassing direct reciprocity, indi-
rect reciprocity, and reputation have been postulated to elucidate
the origin and enduring nature of cooperative conduct [21],
[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32].In
addition to the mechanisms mentioned above, incentive strate-
gies have attracted increasing attention from researchers across
multiple domains, including economics, psychology, biology,
and computer science, as it holds the promise of reshaping
the evolutionary pathways of cooperation [33],[34],[35].For
instance, punishment can reduce the occurrence of free riding
by imposing sanctions on noncooperators, while reward can
increase agents’ motivation to participate in cooperative ac-
tions and strengthen their willingness to cooperate by providing
positive feedback [36],[37],[38],[39],[40],[41]. In recent
times, there has been a growing interest in the concept of social
exclusion strategy, which represents a more potent form of pun-
ishment [42],[43],[44],[45]. The main idea behind this strategy
is to expel noncooperative agents from groups or prevent them
from accessing benefits that arise from group cooperation. So-
cial exclusion has attracted the attention of many scholars due
to its potential effectiveness in promoting cooperative behavior.
While social exclusion can be regarded as an effective means
for promoting cooperation, it may also result in negative out-
comes, such as reduced social trust, decreased group cohesion,
and impaired cooperative capacity of excluded agents [46],
[47]. In contrast to social exclusion strategy, social guidance
is a positive approach aimed at guiding agent behavior and
attitudes through various means, including education, public-
ity, and advocacy [48],[49],[50],[51],[52]. Official media,
advertising, public services, and social organizations can all be
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2IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS
utilized to conduct social guidance activities. The phenomenon
of social guidance is very common in the real world. For exam-
ple. the “Grüner Punkt” program in Germany guides people to
recycle and dispose of waste properly in daily life by providing
waste management knowledge and waste classification con-
tainers, thereby reducing environmental pollution [53]. Amidst
the global battle against COVID-19, various governments and
organizations have been endeavoring to steer the populace to-
ward adopting essential preventative measures, such as donning
face masks, practicing frequent hand hygiene, and maintaining
physical distancing, with the aim of curbing and forestalling
the transmission of the epidemic [54],[55]. “National Le-
gal Awareness Day” is held on September 15 every year in
China, aiming to promote legal education, enhance citizens’
legal awareness, and prevent crime through various forms of
publicity and educational activities [56]. During the 2015 Paris
Climate Conference, many national leaders delivered speeches
calling for joint efforts to reduce emissions, especially to guide
countries that were not reducing their emissions [57].
Social guidance, also known as pre-guidance, aims to funda-
mentally prevent the occurrence of undesirable behavior, assist
agents or groups in adapting to various situations and environ-
ments, and actively take actions that are beneficial to oneself
or others [58]. Despite the recognized significance of guidance
strategies in real-world scenarios, there remains a dearth of the-
oretical studies exploring their impact on the evolution of coop-
eration within competitive agent populations. Consequently, the
quantitative effects of guidance strategies on cooperative evo-
lution from a theoretical standpoint remain largely uncharted,
leaving ample room for further investigation and understanding.
Considering the aforementioned assertions, we present
a comprehensive endeavor wherein we develop a game-
theoretical model rooted in the framework of public goods
games, wherein agents engage in a preliminary negotiation or
discussion phase before the actual gameplay, allowing them to
converse and gain insights into each other’s decision-making
tendencies. We put forth a pair of distinctive social guidance
strategies, namely peer guidance and pool guidance. In the
former, agents who opt for the guidance strategy bear the cost
of guiding agents in the game group who have a tendency to
defect during the actual game. In the latter scenario, participants
adopting the guidance strategy willingly contribute a designated
cost to the incentive pool, while the institutional framework
takes on the responsibility of guiding and deterring free riders.
The guided agents may reverse their tendency to free ride in the
actual game. Our primary focus lies in investigating the pro-
found implications of these two guidance strategies on the evo-
lutionary dynamics of cooperation, encompassing both infinite
and finite populations. By means of rigorous theoretical analysis
and meticulous numerical simulations, we unveil a compelling
finding: the strategic integration of guidance strategies serves
as a potent mechanism for curbing the prevalence of second-
order free riding. Furthermore, pool guidance strategy exhibits
greater tolerance for guidance costs compared to peer guidance
strategy, indicating that pool guidance is more advantageous in
attenuating the advantage of free riding.
Now the primary contributions and unique aspects of our
work are summarized as follows.
1) We propose a novel incentive strategy, the guidance strat-
egy, which is designed to reverse the behavior of agents
intending to defect before an actual game is played. This
preemptive approach is a unique aspect of our work that
differentiates it from existing methods that typically react
to defection after it has occurred.
2) We present a game-theoretic model that illuminates the
dynamics of two types of guidance strategies: peer guid-
ance and pool guidance. This model provides a theoreti-
cal foundation for understanding how guidance strategies
operate and affect cooperation, filling a significant gap in
the current literature.
3) Our findings indicate that the pool guidance strategy has
a stronger inhibitory effect on second-order free-riding
behavior than the peer guidance strategy, especially when
the cost of guidance is extremely high. This insight could
help inform the design of more effective strategies for
fostering cooperation in various contexts.
A comprehensive model encompassing both peer guidance and
pool guidance is delineated in Section II. Subsequently, we
delve into a comprehensive exploration of the evolutionary dy-
namics underlying population games, meticulously examining
their intricate mechanisms within the realms of both infinite and
finite populations in Sections III. The conclusion is presented
in Section IV.
II. SYSTEM MODEL
A. Public Goods Games
We consider that Nagents are randomly sampled to play the
public goods game. Each agent must choose between cooper-
ation (contributes c) or defection. By implementing a growth
factor denoted as r(where 1 <r<N), the cumulative contri-
bution within the pool undergoes multiplication, subsequently
leading to an equitable distribution among all participating
agents within the group. Consequently, in the absence of al-
ternative mechanisms, cooperators consistently endure lower
payoffs in comparison to defectors. In order to reverse this
unfavorable situation, we introduce social guidance strategies.
In the realm of the one-shot interaction public goods game,
an intriguing prelude unfolds as agents engage in a pregame
negotiation, enabling agents to engage in dialogue and gain
insights into the decision tendencies of their counterparts within
the group [59].
B. Peer Guidance
We then introduce a third strategy, peer guidance, whereby
agents adopting this particular approach demonstrate their com-
mitment to the common good by contributing a cost of cto
the shared pool. Additionally, they invest an additional cost,
denoted as γ, to guide and influence the behavior of defectors.
Accordingly, cooperators without doing social guidance are
the second-order free riders, also called as pure cooperators.
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LIU AND CHEN: EVOLUTIONARY DYNAMICS OF PREGUIDANCE STRATEGIES IN POPULATION GAMES 3
TAB L E I
PARAMETERS INVOLVED IN THE GAME MODEL
Parameter Description
NNumber of agents in the group
cContribution of a cooperator
rGrowth factor of the common pool
γAdditional cost invested by peer guiders
πCPayoff for cooperators
πDPayoff for defectors
πIEPayoff for those engaged in peer guidance
πIFPayoff for those engaged in pool guidance
NCNumber of C-players in the group
NDNumber of D-players in the group
NIENumber of IE-players in the group
NIFNumber of IF-players in the group
δGuidance cost for pool guidance strategy
Here, we consider perfect guidance, where guidance cannot fail.
Then, the guided defectors will reverse their inclination toward
defection and choose cooperative behavior. Accordingly, the
payoffs attributed to cooperators (C), defectors (D), and those
engaged in peer guidance (IE) can be formulated as follows:
πC=⎧
⎨
⎩
rc −c, if NIE=0
rc(NC+1)
N−c, otherwise (1)
πD=⎧
⎨
⎩
rc −c, if NIE=0
rcNC
N,otherwise (2)
πIE=rc −c−γND(3)
where the symbols NCand NDrepresent the quantities of
C-players and D-players within the group, respectively.
C. Pool Guidance
We then consider pool guidance strategy, where agents opt-
ing for this strategy have the opportunity to allocate resources
to the esteemed guidance pool δbefore making contributions
to the public pool. The guidance pool is administered by the
institution and is designed to facilitate the guidance of agents
who exhibit a tendency towards defection in the negotiation.
Similarly, pool guidance will not fail. In the present scenario,
the payoffs acquired by cooperators, defectors, and pool guiders
engaged in the public goods game can be conceptually delin-
eated as follows:
πC=⎧
⎨
⎩
rc −c, if NIF=0
rc(NC+1)
N−c, otherwise (4)
πD=⎧
⎨
⎩
rc −c, if NIF=0
rcNC
N,otherwise (5)
πIF=rc −c−δ. (6)
For a quick understanding of our game model, we present the
model parameters and their meanings in Table I.
Social learning is a powerful mechanism that illuminates the
intricate process of strategy selection. Agents naturally tend to
imitate the strategies of their peers that lead to higher fitness
outcomes [61]. Next, we will delve into the replicator dynamics
and stochastic dynamics of cooperation, defection, and social
guidance strategies.
III. RESULTS
A. Evolutionary Dynamics in Infinite Populations
When investigating evolutionary dynamics in an infinite pop-
ulation, a common approach is to analyze the replicator equa-
tion [60],[61], which is given as
⎧
⎪
⎨
⎪
⎩
˙x=x(fC−¯
f)
˙y=y(fD−¯
f)
˙z=z(fIi−¯
f)
(7)
where the frequencies of C, D, and Iiare represented by x, y,
and zcorrespondingly. Furthermore, the average payoffs of
C, D, and Iiare denoted by fC,f
D,and fIi, respectively. The
average payoff of the entire population is symbolized by ¯
f, and
it is calculated as the sum of the products of the respective
frequencies and average payoffs: xfC+yfD+zfIi, and i=E
or F. In the case of a well-mixed population, where every agent
has the opportunity to interact with each other, we can calculate
the average payoffs of C,D, and Iiagents using the follow-
ing formula:
fC=
N−1
NC=0
N−NC−1
ND=0N−1
NCN−NC−1
ND
xNCyNDzN−NC−ND−1πC
fD=
N−1
NC=0
N−NC−1
ND=0N−1
NCN−NC−1
ND
xNCyNDzN−NC−ND−1πD
fIi=
N−1
NC=0
N−NC−1
ND=0N−1
NCN−NC−1
ND
xNCyNDzN−NC−ND−1πIi
where πC,π
D, and πIiare respectively presented in (1)–(6).
Next, we will analyze the replicator dynamics of peer guidance
and pool guidance, respectively.
1) Type A: Peer Guidance
In the context of incorporating peer guidance, the average
payoffs of C,D, and IEcan be derived as follows:
fC=rc −c−(1−z)N−1rc(N−1)y
N(1−z)
fD=rc −c−(1−z)N−1rc(N−1)y
N(1−z)−(N−r)c
N
fIE=rc −c−(N−1)yγ.
Considering that fC<f
D, it can be concluded that system (7)
does not possess any interior equilibrium points. We then pro-
ceed to analyze the boundary equilibrium points. In the absence
of defectors and the associated guidance cost, a continuum
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4IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS
(a) (b)
Fig. 1. Evolutionary dynamics of C,D,andIEin public goods games. (a) Evolutionary pathways of the system for different initial conditions in simplex S3.
Unstable equilibrium points are represented by empty circles, while stable equilibrium points are denoted by solid dots. (b) Temporal dynamics of cooperation,
defection, and pool guidance frequencies throughout the evolutionary process. The frequency of peer guidance is depicted by a black dashed line, while the
frequency of cooperation is represented by a blue dotted line. Conversely, the frequency of defection is illustrated by a solid red line. The initial frequencies
are (x0,y
0,z
0)=(1/3,1/3,1/3). Parameter: N=5,c=1,r =3, and γ=0.05.
of equilibrium points emerges along the boundary separating
cooperators and peer guiders when the population comprises
solely these two types of individuals. Each of these equilibrium
points, as expounded in greater detail in Appendix, is character-
ized by instability, rendering them susceptible to perturbations
and subsequent deviations from the equilibrium state. When the
population is composed solely of Cand D, the prevalence of the
latter leads to social dilemma, since cooperators receive lower
payoffs than defectors. In a population comprising defectors
and peer guiders, the replicator equation is given as
˙z=z(1−z)(fIE−fD).
By solving fIE=fD, we can derive z∗=1−[(N−1)γ/
(r−1)c]1/(N−2). Through analysis, we can determine that
this boundary equilibrium point is stable. Furthermore, sys-
tem (7) has three vertex equilibrium points: (x, y, z)=
(1,0,0),(0,1,0),and (0,0,1). Theoretical analysis reveals that
all three equilibrium points are unstable when the condition
rc −c>(N−1)γis met (It can be further explored in the
Appendix for a more detailed theoretical analysis).
To validate the aforementioned theoretical analysis, we
present numerical calculations in Fig. 1. Our results indicate
that the equilibrium point located on the DIEboundary is
stable when parameter values satisfy rc −c>(N−1)γand all
trajectories within the system converge toward this equilibrium
point. The CIEboundary is composed of unstable equilibrium
points, and on the CD boundary, the evolutionary direction is
from Cto D[see Fig. 1(a)]. In Fig. 1(b), we present the tem-
poral evolution of cooperative individuals, defectors, and peer
guiders. Our findings demonstrate that the initial frequencies of
all agents are identical, and defectors and peer guiders gradually
stabilize over time, with the frequency of peer guiders being
higher than that of defectors at equilibrium. Meanwhile, the
frequency of cooperators converges to zero.
2) Type B: Pool Guidance
We turn next to pool guidance strategy. Through the compu-
tation of the average payoffs associated with C, D, and IF,we
derive the following outcomes:
fC=rc −c−(1−z)N−1rc(N−1)y
N(1−z)
fD=rc −c−(1−z)N−1rc(N−1)y
N(1−z)−(N−r)c
N
fIF=rc −c−δ.
We can observe that the average payoffs of Cand Dare
consistent with those under peer guidance. Therefore, there
still exists no interior equilibrium point for system (7) since
fC<f
Dholds. Moreover, on the boundary CD, the evolu-
tionary direction is from point Ctoward point D, while the
evolutionary direction on the CIFboundary is from IFto C.
On the boundary of DIF, the frequency of cooperators is zero,
resulting in the replicator equation becoming
˙z=z(1−z)(fIF−fD).
Solving fIF=fD, we can obtain z1=1−(δ/(rc
−c))1/(N−1). Thus, there exists one boundary fixed point
along the DIFedge when 0 <1−(δ/(rc −c))1/(N−1)<1,
and it is stable. Furthermore, the three existing equilibrium
points at the vertices exhibit instability when rc −c−δ>0.
To validate our theoretical analysis, we provide numerical
examples in Fig. 2. The phase plane exhibits four equilibrium
points, including three unstable vertex equilibrium points and
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LIU AND CHEN: EVOLUTIONARY DYNAMICS OF PREGUIDANCE STRATEGIES IN POPULATION GAMES 5
(a) (b)
Fig. 2. Evolutionary dynamics of cooperation, defection, and pool guidance in public goods games. (a) Evolutionary trajectories of the system for different
initial conditions in simplex S3. Unstable equilibrium points are represented by empty circles, while stable equilibrium points are denoted by solid dots.
(b) Temporal dynamics of cooperation, defection, and pool guidance frequencies throughout the evolutionary process. The frequency of pool guidanceis
depicted by a black dashed line, while the frequency of cooperation is represented by a blue dotted line. Conversely, the frequency of defection is illustrated
by a solid red line. The initial frequencies are (x0,y
0,z
0)=(1/3,1/3,1/3). Parameter values: N=5,c=1,r =3, and δ=0.05.
one stable equilibrium point located on the DIFboundary.
All trajectories within the system converge toward the stable
fixed point [see Fig. 2(a)]. In Fig. 2(b), we present a concrete
example where the initial frequencies of the three strategists are
the same. We find that the frequencies of IFand Dstabilize
over time, with pool guiders having a higher advantage than
defectors, while the frequency of Cdecreases to zero.
B. Evolutionary Dynamics in Finite Populations
In the previous section, we study the replicator dynamics
of peer guidance and pool guidance strategies in an infinite
well-mixed population. However, considering the finite and
relatively small population sizes in the real society, the evo-
lutionary dynamics of the system can be easily influenced by
stochastic factors such as behavioral mutations. Consequently,
due to the altered dynamics of the system, the applicability
of replicator equations diminishes, prompting us to investigate
the evolutionary dynamics through an examination of the sta-
tionary distribution of the Markov process and the gradient of
selection [62].
Let’s consider a finite, well-mixed population with a total
size of Z. Within this population, there are iCcooperators,
iIguiders, and Z−iC−iIdefectors. By employing the hy-
pergeometric sampling technique, we can compute the average
payoffs associated with cooperators, defectors, and guiders as
ΠC=
N−1
NC=0
N−NC−1
NI=0iC−1
NCiI
NI Z−iC−iI
N−NC−NI−1
Z−1
N−1πC
ΠD=
N−1
NC=0
N−NC−1
NI=0iC
NCiI
NIZ−iC−iI−1
N−NC−NI−1
Z−1
N−1πD
ΠI=
N−1
NC=0
N−NC−1
NI=0iC
NCiI−1
NI Z−iC−iI
N−NC−NI−1
Z−1
N−1πI
where πC,π
D, and πI, respectively, denote the payoffs of coop-
erators, defectors, and guiders obtained form the public goods
game, which are presented in (1)–(6). The hypergeometric dis-
tribution (iC
NCiI
NIZ−iC−iI
N−NC−NI)/Z
Nrepresents the proba-
bility of selecting NCcooperators from iCpotential coopera-
tors, NIguiders from iIpotential guiders, and N−NC−NI
defectors from the remaining Z−iC−iIindividuals in the
population.
We then combine the pairwise comparison rules and mutation
to describe the evolution of the number of agents adopting a
given strategy in a finite well-mixed population. Concretely, at
each time step, an agent Lis selected randomly and with a prob-
ability μ, it mutates, changing its current strategy to another one
from the available strategy space. If the mutation does not occur
(with a probability of 1 −μ), agent Ladopts the strategy of a
randomly paired agent R, a process governed by a probability
function, denoted as p=1/(1+exp[β(ΠL−ΠR)]), where β
represents the intensity of selection, which characterizes how
likely an individual is to imitate the strategy of another indi-
vidual based on their payoff differences. When β→∞,the
better strategy is always adopted. Conversely, when β→0,
the choice of strategies becomes random, regardless of their
payoff differences [63]. Based on the above description, we can
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6IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS
derive the transition probabilities between any pair of strategies.
Concretely,
TC→D=(1−μ)iC
Z
iD
Z−1
1
1+exp[β(ΠC−ΠD)] +μiC
2Z
TD→C=(1−μ)iD
Z
iC
Z−1
1
1+exp[β(ΠD−ΠC)] +μiD
2Z
TC→I=(1−μ)iC
Z
iI
Z−1
1
1+exp[β(ΠC−ΠI)] +μiC
2Z
TI→C=(1−μ)iI
Z
iC
Z−1
1
1+exp[β(ΠI−ΠC)] +μiI
2Z
TD→I=(1−μ)iD
Z
iI
Z−1
1
1+exp[β(ΠD−ΠI)] +μiD
2Z
TI→D=(1−μ)iI
Z
iD
Z−1
1
1+exp[β(ΠI−ΠD)] +μiI
2Z.
For arbitrarily mutation rate, the evolutionary dynamics
among these three strategies can be modeled as a two-
dimensional Markov process. The evolutionary dynamics of
the system can then be analyzed by studying the probabil-
ity distribution function pi(t), which provides information on
the pervasiveness of each configuration at time t, satisfies the
discrete-time Master equation [64]:
pi(t+τ)−pi(t)=
i{Tiipi(t)−Tiipi(t)}
where Tiiand Tiirepresent the transition probabilities between
configuration iand configuration i. Technically, by solving
for the eigenvector of the probability transition matrix with
eigenvalue of 1 [64], the stationary distribution ¯pican be
obtained. Given the finite population size Z, the number of
individuals using a particular strategy can range from 0 to Z.
Therefore, the dimensions of the resulting Markov matrix
would be ((Z+1)(Z+2))/2.
Furthermore, we can use the gradient of selection to describe
the evolutionary path of the system after departing from con-
figuration i, which can be written as
∇i=(TC+
i−TC−
i)uC+(TI+
i−TI−
i)uI(8)
where uCand uIare basis vectors. TC+
i(TC−
i)and TI+
i(TI−
i)
respectively denote the probabilities that the numbers of Cand
Iagents increase (decrease) one
TC+
i=TD→C+TI→C
TC−
i=TC→D+TC→I
TI+
i=TC→T+TD→T
TI−
i=TI→C+TI→D.
In addition, we offer a significant metric for assessing the
average proficiency of each strategy in steady state, in order to
study which strategy is favored more by natural selection. The
specific form is as follows:
¯ρC=
i
iiC¯pi
Z
¯ρI=
i
iiI¯pi
Z
¯ρD=
i
iiD¯pi
Z
where the symbols iiC,iiI, and iiDrepresent the counts of
agents with strategies C, I,and Dagents in the configuration
i, respectively.
In order to explore the resilience of our findings in the
face of random influences, we additionally employ an agent-
based simulation incorporating a pairwise comparison mech-
anism. In this simulation, each agent is randomly chosen and
assigned an average payoff based on their interactions within
the game. The evolution of strategies takes place through a
mutation-selection process, allowing for the exploration of
different strategy combinations over time. Specifically, with
probability μ, agents Lrandomly selects one of the available
strategies, and with probability 1 −μ, agent Ladopts the strat-
egy of randomly selected agent R, known as a role model with
a probability of 1/(1+exp(β(ΠL−ΠR))).
Fig. 3illustrates the stochastic dynamics of C,D, and IE
within finite well-mixed populations. The triangular simplex
illustrated in Fig. 3(a) encompasses all configurations of a finite
population, with each configuration being depicted by a small
circle in the representation. The stationary distributions can be
visualized using a grey scale to represent their magnitudes.
The darker regions correspond to configurations that have been
encountered more frequently. To depict the most probable di-
rection of evolution from a specific configuration, the visu-
alization employs red arrows. These red arrows represent the
gradient of selection, which can be calculated using (8). We
find that the system predominantly resides in configurations
that are located near the DIEboundary for the majority of
the time. The findings from the gradient of selection further
validate this observation, as all interior trajectories converge
to the region near the DIEboundary. Importantly, we present
the average levels of each strategy in a 3-D pie chart when the
system reaches steady state, and we find that the proportion
of peer guiders is the highest, followed by defectors, with the
proportion of cooperators being the lowest.
To explore the influence of peer guidance costs on evolution-
ary dynamics, we examine the changes in the average levels of
each strategy as the costs of peer guidance vary. As illustrated
in Fig. 3(b), we find that the average frequency of defectors
increases as the cost of guidance increased, while the average
frequencies of cooperators and peer guiders decrease when the
cost of peer guidance is below the median value (γ<0.5).
However, when the cost of guidance exceeds the median value,
the average frequency of defectors approaches one, while the
frequencies of cooperators and guiders both tend towards zero.
Furthermore, we conduct agent-based simulations to inves-
tigate the evolutionary dynamics for different peer guidance
cost. When the guidance cost is low (γ=0.05), our findings
demonstrate a stable coexistence of peer guiders and cooper-
ators within the population, where the frequency of guiders
surpasses that of defectors, while the frequency of cooperators
tends toward zero [see Fig. 3(c)]. For higher guidance cost
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LIU AND CHEN: EVOLUTIONARY DYNAMICS OF PREGUIDANCE STRATEGIES IN POPULATION GAMES 7
(a) (b)
(c) (d)
Fig. 3. Stochastic dynamics of C,D,andIEin finite populations. (a) Stochastic dynamics of the system, where the arrows in simplex S3depict the gradient
of selection, revealing the most probable evolutionary trajectory for the system as it departs from the current state. The color bar illustrates the stationary
distribution values for each configuration, with darker colors indicating longer durations of population occupancy in these states. The 3-D pie chartchart
showcases the steady-state average levels of each strategy, symbolized by colors. Cooperation is represented by black, defection by red, and peer guidance by
green. (b) Average frequency of each strategy in the steady state, displaying the relationship between strategy distribution and the guidance cost parameter γ.
(c) and (d) Temporal evolution of strategies in agent-based simulations for different γvalues. Parameters are Z=100,N=5,r =3,c =1,μ=1/Z, γ =0.05,
and β=5 in panel (a); Z=100,N =5,r=3,c =1,μ=1/Z, and β=5in(b);Z=100,N =5,r=3,c=1,μ=10−6,γ =0.05,and β=500 in panel
(c); Z=100,N =5,r=3,c =1,μ=10−6,γ =0.6,and β=500 in (d).
(γ=0.6), we find that defectors could dominate the entire pop-
ulation, while cooperators and peer guiders gradually disappear
from the population [see Fig. 3(d)].
Our focus now shifts to exploring the stochastic dynamics
of cooperation, defection, and pool guidance strategies within
finite well-mixed populations. In Fig. 4(a), we present the
numerical outcomes regarding the stationary distribution and
gradient of selection. Our findings reveal that the population
predominantly resides in configurations characterized by the
coexistence of defectors and pool guiders. Moreover, the tra-
jectories within the simplex S3converge toward the central
region near the boundary of DIF. Additionally, upon reach-
ing the steady-state, our observations indicate that the average
frequency of pool guiders is the highest, reaching an impres-
sive 56.28%. Defectors follow closely behind with an average
frequency of 33.29%, whereas cooperators exhibit the lowest
average frequency at 10.43% [see the 3-D pie chart in Fig. 4(a)].
Similarly, we further investigate how the cost of pool guid-
ance affects the evolutionary dynamics. Specifically, we exam-
ine the average frequencies of cooperation, defection, and pool
guidance strategy at steady state as a function of the guidance
cost δ, as shown in Fig. 4(b). We find that the average frequency
of defectors increases with the increase of δwhen the guidance
cost is not particularly high (δ<2), while a decline in the
average frequencies of cooperators and pool guiders. When the
value of δis large (δ>2), we find that the average frequency
of defectors approaches one, while the average frequencies of
cooperators and pool guiders tend to zero.
Finally, we present agent-based simulation results for two
different pool guidance costs. Our findings reveal a fascinating
phenomenon: under low guidance cost conditions (δ=0.05),
the population exhibits a stable coexistence of pool guiders and
defectors, with the former outnumbering the latter. Intriguingly,
the frequency of cooperators tends to diminish, nearing zero.
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8IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS
(a)
(b)
(c) (d)
Fig. 4. Stochastic dynamics of C,D,andIFin finite populations. (a) Stochastic dynamics of the system. (b) Graphical representation showcasing the
average occurrence rate of each strategy at the equilibrium state, elucidating the relationship between strategy distribution and the guidance cost parameter δ.
(c) and (d) Time evolution of strategies in agent-based simulations for different δvalues. Parameters are Z=100,N =5,r=3,c=1,μ=1/Z , δ =0.05,
and β=5in(a);Z=100,N =5,r =3,c =1,μ=1/Z, and β=5in(b);Z=100,N =5,r =3,c=1,μ=10−6,δ =0.05,and β=500 in (c); Z=
100,N =5,r=3,c =1,μ=10−6,δ=2,and β=500 in (d).
For a higher guidance cost (δ=2), we observe that defectors
almost dominate the entire population, while the frequencies of
cooperators and pool guiders tend to zero.
IV. CONCLUSIONS
Social guidance, as an incentive measure, refers to the pro-
cess of influencing and guiding the thoughts, behaviors, values,
and other aspects of members through various means and chan-
nels. In modern society, social guidance has become a common
phenomenon, with various forces such as government, media,
educational institutions, and enterprises engaging in different
forms of social guidance [65],[66],[67]. Despite the significant
role of social guidance in real-world societies, there appears to
be a dearth of theoretical research examining the influence of
such incentive measures on the evolution of cooperation.
Within the scope of our study, we have developed a sophis-
ticated game-theoretic model rooted in public goods games.
This model serves as a potent instrument for understanding
the intricate evolutionary dynamics underlying social guidance
strategies. We have proposed two different forms of social
guidance, namely peer guidance and pool guidance. Through
an examination of both deterministic and stochastic dynamics
within the game system, we have discovered that the integration
of social guidance strategies effectively alleviates the occur-
rence of second-order free riding, regardless of whether a peer
or pool guidance regime is employed. Furthermore, we have
also shown that pool guidance can maintain public cooperation
even when the guidance cost is high. An intriguing observation
is that the threshold guiding cost required for pool guidance to
overcome free-riding behavior surpasses that of peer guidance.
This suggests that pool guidance has a greater capacity to foster
cooperation across a broader spectrum of parameter values
compared to peer guidance.
As stated in the model section, our model encompasses a one-
shot public goods game and assumes a pre-game negotiation
phase, wherein agents finalize their decisions. Once this nego-
tiation concludes, agents are bound to their choices and cannot
alter them during the subsequent actual game. This can be
interpreted as players making a pregame commitment prior to
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LIU AND CHEN: EVOLUTIONARY DYNAMICS OF PREGUIDANCE STRATEGIES IN POPULATION GAMES 9
engaging in the actual gameplay [59]. In recent years, there has
been a notable upsurge in research investigating the impact of
pre-commitment mechanisms on the evolutionary dynamics of
cooperation. Remarkably, these investigations have consistently
demonstrated that pre-commitment strategies wield a signifi-
cant impact, effectively enhancing the overall level of cooper-
ation [68],[69],[70],[71],[72]. Introducing social guidance
strategies in this precommitment scenario would be meaningful,
as the social guidance needs to assess the behavioral decisions
of agents in the game group and then guide those agents with a
tendency to defect. In a similar vein, it is interesting to con-
sider the potential application of such guidance mechanisms
in improving coordination and promoting safe and responsible
development in other areas, such as AI. Given the ongoing dis-
cussions on AI regulation and governance, the implementation
of a pre-commitment strategy could serve as a valuable tool to
ensure the ethical and responsible progression of this rapidly
evolving field [73],[74].
Costly punishment, which involves imposing sanctions on
noncooperative agents, provides a powerful deterrent against
selfish behavior and is a common form of negative incentive
used to promote the emergence of cooperation. Previous the-
oretical research has revealed two substantial challenges to
the existence of peer punishment [42]. First, a small number
of punishers are unable to overcome widespread defection,
and second, the presence of second-order free riders prevents
the emergence of peer punishment. In our current study, we
discover that the integration of a peer guidance strategy can
significantly suppress the occurrence of free riding, even with a
minimal number of peer guiders within the group. Furthermore,
previous research has revealed that defectors can dominate the
entire population when considering pool punishment [75].Our
theoretical results demonstrate that the pool guidance strategy
can effectively suppress the level of free-riding behavior to a
certain extent. In summary, social guidance strategies are more
effective than costly punishment strategies in discouraging free-
riding behavior.
In this work, we have investigated the effectiveness of dif-
ferent form of guidance strategies in promoting cooperation.
An area of future research worth exploring is the examina-
tion of how various forms of social guidance influence the
development of cooperation in repeated group interactions [76],
[77]. In the current work, we have examined a one-shot game
interaction that necessitates the reliance on pregame commit-
ments to identify free riders. Consequently, in the absence of
commitment, a repeated interactions scenario can effectively
identify defectors in a game group [78], thus enabling more
precise guidance. The potential impact of such antisocial guid-
ance on the evolution of cooperation is indeed intriguing. For
instance, antisocial guidance could potentially undermine the
effectiveness of prosocial guidance strategies, or alternatively,
it could create additional pressure for cooperation if it leads
to negative consequences for defectors. In addition, our model
assumes that guidance will not fail, which is an idealized as-
sumption. An inherent question arises regarding the effective-
ness of social guidance strategies in fostering the emergence and
sustainability of cooperation within the context of probabilistic
guidance. Moreover, it is intriguing to investigate the influ-
ence of social networks on the evolutionary dynamics of so-
cial guidance strategies, considering that real-world interactions
among agents often occur within diverse social relationship
networks rather than random matching [79],[80],[81],[82].
On top of that, an intriguing question worthy of investigation
is which strategy, peer guidance or pool guidance, can more
effectively deter the emergence of uncooperative behavior when
both strategies are concurrently employed within a population.
Finally, it would be valuable to explore the potential trade-
offs between social guidance and alternative strategies, such
as costly punishment [43], and their implications for the evo-
lution of cooperation. Investigating these tradeoffs can offer
valuable insights into the dynamics of social dilemmas and shed
light on the effectiveness of different approaches in promoting
sustained cooperation.
APPENDIX
This appendix elucidates the stability criteria for all equilib-
rium points both in peer guidance and pool guidance scenarios.
The Jacobian matrix plays a crucial role in characterizing the
local dynamics of a system around its equilibrium point. By
scrutinizing the eigenvalues of the Jacobian matrix, we can gain
valuable insights into the direction and rate of change of the
system’s state variables [83]. In particular, the stability of an
equilibrium point can be determined by examining the signs
of the eigenvalues. A set of negative real eigenvalues signifies
stability, while the presence of any positive real eigenvalues
indicates instability.
For convenience in analysis, we set f(x, y)=x(fC−¯
f)and
g(x, y)=y(fD−¯
f). Accordingly, we have
f(x, y)=x(fC−¯
f)=x[(1−x)(fC−fIi)−y(fD−fIi)]
g(x, y)=y(fD−¯
f)=y[(1−y)(fD−fIi)−x(fC−fIi)].
The Jacobian matrix corresponding to system (7) can be ex-
pressed as
J=⎛
⎜
⎜
⎝
∂f(x, y)
∂x
∂f(x, y)
∂y
∂g(x, y)
∂x
∂g(x, y)
∂y
⎞
⎟
⎟
⎠(9)
where
∂f(x, y)
∂x =[(1−x)(fC−fIi)−y(fD−fIi)]
+x−(fC−fIi)+(1−x)∂(fC−fIi)
∂x
−y∂(fD−fIi)
∂x
∂f(x, y)
∂y =x(1−x)∂(fC−fIi)
∂y −x(fD−fIi)
−xy ∂(fD−fIi)
∂y
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10 IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS
∂g(x, y)
∂x =y(1−y)∂(fD−fIi)
∂x −y(fC−fIi)
−xy ∂(fC−fIi)
∂x
∂g(x, y)
∂y =[(1−y)(fD−fIi)−x(fC−fIi)]
+y−(fD−fIi)+(1−y)∂(fD−fIi)
∂y
−x∂(fC−fIi)
∂y .
Next, we will analyze the stabilities of all equilibrium points in
peer guidance and pool guidance scenarios, respectively.
Case 1: Peer guidance
In the case of a successful peer guidance strategy, the Ja-
cobian matrix for each equilibrium point can be represented
as follows:
For the equilibrium point (x, y, z)=(0,0,1), the Jocobian is
J(0,0,1)=00
00
.
The Jacobian matrix does not provide any useful information for
determining the stability of the equilibrium point. We will use
perturbation theory to analyze the stability of this equilibrium
point [42]. Because fIE<f
Cholds in the interior space, then
if rare defectors are introduced, the evolutionary advantage of
peer guidance strategy will be lower than that of cooperation,
and therefore, this equilibrium point is not robustly stable.
For the equilibrium point (x, y, z)=(0,1,0), the corre-
sponding Jacobian matrix is as follows:
J(0,1,0)=⎛
⎜
⎝
rc
N−c0
rc −rc
N−(N−1)γrc−c−(N−1)γ⎞
⎟
⎠
when rc −c−(N−1)γ>0 the equilibrium point is deemed
unstable. Conversely, if rc −c−(N−1)γ<0 it is considered
stable. Notably, when rc −c−(N−1)γ=0, one eigenvalue
of the Jacobian becomes zero, while the other eigenvalue re-
mains negative. In such cases, the stability analysis is conducted
using the center manifold theorem [83].
Because y=1−x−z, the replicator equation becomes
˙x=x[(1−x)(fC−fD)−z(fIE−fD)]
˙z=z[(1−z)(fIE−fD)−x(fC−fD)].(10)
The equation system (10) exhibits an equilibrium point at
(x, z)=(0,0). Then, the Jacobian is
J(0,1,0)=⎛
⎝
rc
N−c0
0rc −c−(N−1)γ⎞
⎠.
When rc −c−(N−1)γ=0, the eigenvalues of the Jaco-
bian matrix exhibit distinct characteristics. One eigenvalue is
(rc/N)−c, while the other eigenvalue is precisely zero. In
this condition, we encounter a center manifold x=h(z)in the
equation system. In an attempt to explore the behavior of the
system, we embark on a trial of the function h(z), assuming it
to be of the form h(z)=O(z2), thus the equation system can
be expressed as
˙z=z(1−z)(fIE−fD)
=z(1−z)[(1−z)N−1(rc −c)−(N−1)γ(1−z)]
+O(|z|N+2).
Since rc −c=(N−1)γ, then we have
˙z=z(1−z)2(rc −c)[(1−z)N−2−1]+O(|z|N+2).
Given that −(N−2)(rc −c)=0, we can deduce that the point
z=0 exhibits instability. Consequently, this instability extends
to the fixed point (0,1,0)as well.
For the equilibrium point (x, y, z)=(1,0,0), the corre-
sponding Jacobian matrix is
J(1,0,0)=⎛
⎝0rc
N−c
0c−rc
N⎞
⎠
we can judge that it is unstable since r<N.
For (x, y, z )=(0,1−z∗,z∗), where z∗=1−[((N−1)γ/
(r−1)c)]1/(N−2), then the Jocobian is
J(0,1−z∗,z∗)=a11 0
a21 −z∗(1−z∗)(N−1)(N−2)γ
where a11 =(N−1)γ[((N−1)γ/(r−1)c)]1/(N−2)((r−
N)c/N(r−1)c)and a21 =z∗(1−z∗)(∂(fD−fIE)/∂x)−
y(fC−fIE), we can judge that it is stable since r<N.
For (x, y, z )=(1−z0,0,z
0), where z0=1−((Nγ/
rc))1/(N−2), then the Jocobian is
J(1−z0,0,z
0)=0a12
0(N−r)c
N(1−z0)N−1
where a12 =z0(1−z0)(∂(fC−fIE)/∂y)−(1−z0)(fD−
fIE), and we can judge that it is unstable since r<N.
Case 2: Pool guidance
We turn next to pool guidance. In this case, the system (7)
exhibits a maximum of four equilibrium points: (x, y, z)=
(0,0,1),(0,1,0),(1,0,0),and (0,z
1,1−z1)where z1=1−
((δ/rc −c))1/(N−1). We then investigate the stability of each
equilibrium point.
For the equilibrium point (x, y, z)=(0,0,1), the Jocobian is
J(0,0,1)=δ0
0δ
due to the positive value of δ, this equilibrium point is deemed
unstable.
For the equilibrium point (x, y, z)=(0,1,0), the Jocobian is
J(0,1,0)=⎛
⎜
⎝
rc
N−c0
rc −rc
N−δrc−c−δ⎞
⎟
⎠.
We observe instability when rc −c−δ>0, while when rc −
c−δ<0 it is stable. Notably, when rc −c−δ=0, we en-
counter a situation where one eigenvalue of the Jacobian is
zero, while the other eigenvalue is negative. In light of this,
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LIU AND CHEN: EVOLUTIONARY DYNAMICS OF PREGUIDANCE STRATEGIES IN POPULATION GAMES 11
we proceed to analyze its stability using the center manifold
theorem [83].
Because y=1−x−z, the replicator equation becomes
˙x=x[(1−x)(fC−fD)−z(fIF−fD)]
˙z=z[(1−z)(fIF−fD)−x(fC−fD)].(11)
Considering the equation system (11), we can identify
the equilibrium point (x, z)=(0,0). Then, the corresponding
Jacobian is
J(0,1,0)=rc
N−c0
0rc −c−δ.
For the fixed point (x, y, z)=(0,1,0), when rc −c−δ=0,
the eigenvalues of the Jacobian matrix are rc/N −cand 0,
respectively. Under these circumstances, we identify a center
manifold x=u(z)in the equation system. To proceed, we con-
sider a trial function u(z)=O(z2), resulting in the following
expression for the equation system:
˙z=z(1−z)(fIF−fD)
=z(1−z)[(1−z)N−1(rc −c)−δ]+O(|z|N+2).
Since rc −c=δ, then we have
˙z=z(1−z)2(rc −c)[(1−z)N−2−1]+O(|z|N+2).
Since −(N−2)(rc −c)=0, we can ascertain the instability
of the point z=0. Consequently, this instability extends to the
fixed point (0,1,0).
For the equilibrium point (x, y, z)=(1,0,0), the corre-
sponding Jocobian is
⎛
⎝−δrc
N−c−δ
0c−rc
N⎞
⎠.
Due to the condition r<N, we can conclude that the equilib-
rium point is unstable.
For the equilibrium point (x, y, z)=(0,1−z1,z
1), the cor-
responding Jocobian is
⎛
⎝δ(r−N)c
N(r−1)c0
a21 a22⎞
⎠
where a21 =z1(1−z1)(∂(fD−fIF)/∂x)−y(fC−fIF)and
a22 =−z1(1−z1)N−1(N−1)(rc −c). Considering the con-
dition r<N, we can deduce that the point exhibits stability.
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