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The evolution of cooperation in spatial public goods game with tolerant punishment based on reputation threshold

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Gui Zhang
Gui Zhang
Gui Zhang
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Yichao Yao at Southwest University
Yichao Yao
Ziyan Zeng at Southwest University
Ziyan Zeng
Minyu Feng at Southwest University
Minyu Feng
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Abstract and Figures

Reputation and punishment are significant guidelines for regulating individual behavior in human society, and those with a good reputation are more likely to be imitated by others. In addition, society imposes varying degrees of punishment for behaviors that harm the interests of groups with different reputations. However, conventional pairwise interaction rules and the punishment mechanism overlook this aspect. Building on this observation, this paper enhances a spatial public goods game in two key ways: (1) We set a reputation threshold and use punishment to regulate the defection behavior of players in low-reputation groups while allowing defection behavior in high-reputation game groups. (2) Differently from pairwise interaction rules, we combine reputation and payoff as the fitness of individuals to ensure that players with both high payoff and reputation have a higher chance of being imitated. Through simulations, we find that a higher reputation threshold, combined with a stringent punishment environment, can substantially enhance the level of cooperation within the population. This mechanism provides deeper insight into the widespread phenomenon of cooperation that emerges among individuals.
The node strategy update process in the network. Red nodes represent defectors and blue nodes represent cooperators. (a) All nodes are randomly assigned reputations and strategies. (b) All groups of public goods as circled by the dotted line interact. (c) Punishment is carried out according to the average reputation of the game group and update strategy using Eq. (5). Punishment is implemented within the low-reputation group (red dashed box) but not within the high-reputation group (green dashed box). (d) The network is updated, and the next Monte Carlo step is taken.
The node strategy update process in the network. Red nodes represent defectors and blue nodes represent cooperators. (a) All nodes are randomly assigned reputations and strategies. (b) All groups of public goods as circled by the dotted line interact. (c) Punishment is carried out according to the average reputation of the game group and update strategy using Eq. (5). Punishment is implemented within the low-reputation group (red dashed box) but not within the high-reputation group (green dashed box). (d) The network is updated, and the next Monte Carlo step is taken.
… 
Frequency of cooperators ρ C as a function of damping factor δ for different reputation threshold R 0. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Note that more an individual’s fitness depends on reputation, the more it promotes the emergence of cooperation.
Frequency of cooperators ρ C as a function of damping factor δ for different reputation threshold R 0. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Note that more an individual’s fitness depends on reputation, the more it promotes the emergence of cooperation.
… 
Frequency of cooperators ρ C as a function of enhancement factor r for different punishment of b. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Different panels display the cooperation level under different reputation thresholds, as (a) R 0 = 1 and (b) R 0 = 4. The curves show the critical values r at which the phase transition from defection to cooperation occurs under different punishment b. For instance, the phase transition points for the emergence of cooperation corresponding to b = 0, 0.25, and 0.50 in (a) are r = 1.1, 2.1, and 2.8, respectively.
Frequency of cooperators ρ C as a function of enhancement factor r for different punishment of b. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Different panels display the cooperation level under different reputation thresholds, as (a) R 0 = 1 and (b) R 0 = 4. The curves show the critical values r at which the phase transition from defection to cooperation occurs under different punishment b. For instance, the phase transition points for the emergence of cooperation corresponding to b = 0, 0.25, and 0.50 in (a) are r = 1.1, 2.1, and 2.8, respectively.
… 
Frequency of cooperators ρ C varies with time for different b. Fixed enhancement factor r = 2.5. Different panels display the cooperation level under different reputation thresholds, as (a) R 0 = 1 and (b) R 0 = 4. In different punishment b, the system can reach evolutionarily stable in 10 4 iterations. For b=0, 0.25, 0.5, and 0.75, the frequency of cooperators initially declines and then gradually increases toward a dynamically stable and non-zero level, whereas b = 0, the level of cooperation finally drops to zero.
Frequency of cooperators ρ C varies with time for different b. Fixed enhancement factor r = 2.5. Different panels display the cooperation level under different reputation thresholds, as (a) R 0 = 1 and (b) R 0 = 4. In different punishment b, the system can reach evolutionarily stable in 10 4 iterations. For b=0, 0.25, 0.5, and 0.75, the frequency of cooperators initially declines and then gradually increases toward a dynamically stable and non-zero level, whereas b = 0, the level of cooperation finally drops to zero.
… 
Snapshots of the spatial arrangements of strategies at four representative moments for different reputation thresholds. (a) R 0 = 0, (b) R 0 = 4, (c) R 0 = 12, and (d) R 0 = 16, and fixed enhancement factor r = 2.5 and b = 0.2. Each row from top to bottom represents a situation corresponding to a different reputation threshold R 0. Each column from left to right represents a different time step t. White pixels stand for cooperators and blue pixels for defectors. It is noteworthy that defectors can survive in an environment where no punishment is introduced ( R 0 = 0). However, in an environment with a higher reputation threshold ( R 0 = 4, 12, and 16), after 100 time steps, cooperators gradually gather to form stable clusters. Thus, a strict reputation evaluation environment can induce the formation and development of cooperator clusters.
+2
Snapshots of the spatial arrangements of strategies at four representative moments for different reputation thresholds. (a) R 0 = 0, (b) R 0 = 4, (c) R 0 = 12, and (d) R 0 = 16, and fixed enhancement factor r = 2.5 and b = 0.2. Each row from top to bottom represents a situation corresponding to a different reputation threshold R 0. Each column from left to right represents a different time step t. White pixels stand for cooperators and blue pixels for defectors. It is noteworthy that defectors can survive in an environment where no punishment is introduced ( R 0 = 0). However, in an environment with a higher reputation threshold ( R 0 = 4, 12, and 16), after 100 time steps, cooperators gradually gather to form stable clusters. Thus, a strict reputation evaluation environment can induce the formation and development of cooperator clusters.
… 
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Chaos ARTICLE pubs.aip.org/aip/cha
The evolution of cooperation in spatial public
goods game with tolerant punishment based on
reputation threshold
Cite as: Chaos 35, 013104 (2025); doi: 10.1063/5.0250120
Submitted: 22 November 2024 ·Accepted: 12 December 2024 ·
Published Online: 3 January 2025
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Gui Zhang,1Yichao Yao,1Ziyan Zeng,1Minyu Feng,1,a)and Manuel Chica2,3
AFFILIATIONS
1The College of Artificial Intelligence, Southwest University, No. 2 Tiansheng Road, Beibei, Chongqing 400715, China
2Department of Computer Science and A.I. Andalusian Research Institute DaSCI “Data Science and Computational Intelligence,”
University of Granada, 18071 Granada, Spain
3School of Information and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
a)Author to whom correspondence should be addressed: myfeng@swu.edu.cn
ABSTRACT
Reputation and punishment are significant guidelines for regulating individual behavior in human society, and those with a good reputation
are more likely to be imitated by others. In addition, society imposes varying degrees of punishment for behaviors that harm the interests
of groups with different reputations. However, conventional pairwise interaction rules and the punishment mechanism overlook this aspect.
Building on this observation, this paper enhances a spatial public goods game in two key ways: (1) We set a reputation threshold and use
punishment to regulate the defection behavior of players in low-reputation groups while allowing defection behavior in high-reputation
game groups. (2) Differently from pairwise interaction rules, we combine reputation and payoff as the fitness of individuals to ensure that
players with both high payoff and reputation have a higher chance of being imitated. Through simulations, we find that a higher reputation
threshold, combined with a stringent punishment environment, can substantially enhance the level of cooperation within the population.
This mechanism provides deeper insight into the widespread phenomenon of cooperation that emerges among individuals.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0250120
In the real world, individual behavioral choices have attracted
a large amount of attention. In complex networks, cooperation
and defection of nodes reflect real-life behaviors. Edges gener-
ate interactions, and spatial public goods games (SPGG) help to
understand social choices. Realistic social individuals are more
likely to choose bad behaviors due to the high rewards of free-
riding behaviors, and reputation and punishment mechanisms
are crucial to behavioral regulation. A good reputation leads to
trust and cooperation, while a bad reputation leads to isolation.
In addition, real societies have an assessment of individual rep-
utations, resulting in different communities and corresponding
management systems. This paper explores the evolutionary pun-
ishment mechanism of reputation and tolerance, combining rep-
utation and punishment to construct a community mechanism
that promotes cooperation, aiming to reveal behavioral patterns
and provide guidance for social cooperation.
I. INTRODUCTION
The study of cooperation in biology, sociology, and economics
is extensive.1–6When confronted with conflicts between individ-
ual interests and general interests, the selfish choices of individuals
often clash with the public interests of the group, thereby hindering
the emergence of cooperation.7–10 Consequently, evolutionary game
theory has attracted significant attention and extensive research.
Prisoner dilemma game (PDG),11–14 snow drift game (SDG),15–18
and public goods game (PGG)19–21 are used to address dilemmas
observed in evolutionary games.22,23 The resolution of the social
dilemma is noteworthy, with numerous studies proposed to elu-
cidate the emergence of cooperation. For example, Nowak and
May24 are the first to show that a spatial structure can promote
the development of cooperation in the PDG in 1992. Since then,
substantial research on cooperative behavior has been carried out
in various evolutionary game models, including those based on
Chaos 35, 013104 (2025); doi: 10.1063/5.0250120 35, 013104-1
Published under an exclusive license by AIP Publishing
... As a further step, the punishment is used as an incentive mechanism, and a new classification form of punishment is introduced into the spatial public goods game, 53 where the probability of punishment changes dynamically according to the number of consecutive defections in the game. Furthermore, Zhang et al. 46 studied the impact of the reputation threshold based tolerant punishment mechanism on the evolution of cooperation in spatial public goods games. Specifically, if an individual's reputation fell below the threshold, he would be punished after he chose to defect; whereas he was not punished even though he adopted defective strategy, if his reputation exceeded the threshold. ...
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