The impact of heterogeneous interpersonal relationships on promoting cooperation under the reputation mechanisms in public goods game

Zhenghong Wu; Yining Mu; Zihan Wang

doi:10.1371/journal.pone.0332474

Abstract

Although the principle of “case-by-case analysis” is widely endorsed, achieving complete rationality in the real world continues to be fraught with difficulties. Interpersonal relationships are heterogeneous, and the influence of social relationships and worldly wisdom on reputation evaluation should not be overlooked. Therefore, based on indirect reciprocity theory, this paper constructs a public goods game model with strategy update rules driven by reputation mechanism, aiming to investigate the impact of heterogeneous interpersonal relationships on the promotion of cooperative public goods provision among residents. The paper categorizes interpersonal relationships into three types, and proposes three corresponding reputation evaluation rules. Simulation results demonstrate that varying intensities of interpersonal relationships result in different levels of cooperation. When conducting public activities, externalities and the organizational efficiency of managers must be considered. Meanwhile, cooperation is difficult to sustain if the reputation mechanism fails to function effectively.

Citation: Wu Z, Mu Y, Wang Z (2025) The impact of heterogeneous interpersonal relationships on promoting cooperation under the reputation mechanisms in public goods game. PLoS One 20(9): e0332474. https://doi.org/10.1371/journal.pone.0332474

Editor: The Anh Han, Teesside University, UNITED KINGDOM OF GREAT BRITAIN AND NORTHERN IRELAND

Received: June 16, 2025; Accepted: August 31, 2025; Published: September 25, 2025

Copyright: © 2025 Wu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript.

Funding: This research was supported by the Fundamental Research Funds for the Central Universities (No. 3122024024).

Competing interests: The authors have declared that no competing interests exist.

Introduction

Public goods, including clean air, public parks, and national defense, bestow extensive societal benefits. Investigating how these goods are provided and maintained is essential for the well-being of communities and nations. Public goods are defined by non-excludability (no individual can be excluded from using them) and non-rivalry (one person’s consumption does not reduce another’s). These characteristics often result in the free-rider problem, where individuals consume the good without contributing to its provision. Therefore, allowing government to provide public goods is a suitable option [1–4]. However, government provision alone often results in low efficiency and a significant financial burden [5].

To address the insufficiency of government-provided public goods, some scholars have proposed a diversified provision model involving social organizations [6,7], enterprises [8], and residents [9]. This approach not only leverages market mechanisms for efficient resource allocation, but also alleviates the financial burden on the government [10–15]. Particularly with resident participation, provision can better align with demand and enhance public satisfaction with public goods [16]. Therefore, guiding residents to participate in the provision of public goods has become a key strategy for alleviating inefficiency and the fiscal burden. Due to the non-excludability and non-rivalry of public goods, free-riding becomes the optimal strategy, making it difficult for cooperative provision to be sustained among rational individuals. Moreover, Darwin’s theory of evolution also posits that selfish behavior tends to be more advantageous in competitive contexts [17]. In view of this, how to enable residents to cooperate sustainably in provision has become an urgent problem to be solved in the field of public goods provision.

Fortunately, individuals in the real world do not always act with complete rationality when making decisions. Cooperative behavior is prevalent, and individuals often coordinate their actions for the common good in daily life [18–22]. However, evolutionary game theory explains how cooperation evolves and is sustained in different environments [23,24]. Specifically, the public goods game (PGG) offers a robust theoretical framework for analyzing complex human social behavior and explaining the dynamics and barriers to cooperation [25–28]. At present, several mechanisms have been identified as promoting cooperative behavior, including kin selection [29,30], group selection [31,32], rewards [33–35], punishment [36–38], tolerance [39–41], heuristics [42–44], heterogeneity [45–47], direct reciprocity [48,49], and indirect reciprocity [50–52].

Most studies have shown that the reputation mechanism in indirect reciprocity is a key factor in promoting cooperation in PGG. In the context of the reputation mechanism, various factors have been identified that affect cooperative provision of public goods among residents, including visibility and observability of individual contributions [53], social norms [54], ethnic and social diversity [55], economic incentives and sanctions [34,38], historical factors [56], and behavioral traits [57,58]. Such as, Zhou et al., found that as the reputation factor increased, the level of facilitated cooperation also rose [59]. Quan et al., examined the relative and absolute value of reputation and its discount effect on strategic imitation [60]. Wang et al., proposed a novel tolerance-based reputation scoring mechanism based on historical donations and found that strict scoring rules could enhance cooperation in most cases [56]. Many other scholars have also investigated the role of reputation in promoting cooperation [61–66].

Furthermore, some scholars have started to focus on the impact of interpersonal relationships on reputation evaluation [67–69]. Previous studies have shown that fostering trust [57], facilitating the exchange of reputational information [70], and enhancing social bonds [71] can significantly improve cooperative behaviors driven by reputation systems through interpersonal interactions. However, most studies assumed that interpersonal relationships are homogeneous, with even fewer categorizing them based on their level of intimacy. Different types of interpersonal relationships involve varying levels of trust, communication frequency, and social bonds. In general, strong relationships can strengthen the role of reputation mechanisms in promoting cooperation through factors such as high trust levels [57,72]. It is also important to note that overly intimate relationships may lead to biased behavior, favoritism, or protectionism, which can undermine cooperation. Therefore, exploring the impact of heterogeneous interpersonal relationships on promoting cooperation within the context of reputation mechanisms is valuable. To achieve this, this paper extracts key elements of residents’ provision of public goods from real cases and constructs a group evolution model based on the PGG to explore the impact of varying degrees of interpersonal closeness on residents’ cooperative provision of public goods under the reputation mechanism. This paper findings offer valuable insights for organizers to guide residents in the cooperative provision of public goods across varying interpersonal relationship contexts.

The rest of this paper is structured as follows. The Model section presents the PGG model. The Simulation results and discussion section discusses the results of the simulations. The Conclusion and implication section concludes the paper with key findings, implications, and directions for future research.

Model

Problem description

Inspired by Hwang [73], this paper proposes reputation evaluation criteria based on three key types of interpersonal relationships: emotional relationship (ER), mixed relationship (MR), and instrumental relationship (IR). ER refers to a close, interdependent relationship typically observed between individuals such as spouses, parents and children, or close friends. MR is a relatively harmonious relationship between individuals who know each other and have regular interactions, such as ordinary friends, neighbors, classmates, or colleagues. IR refers to a social relationship where both parties engage in fair trade based on a specific purpose. In summary, the evaluator will assess reputation not only based on an individual’s contribution but also considering the heterogeneous interpersonal relationship between the parties involved.

This paper is inspired by village opera in Jiangsu and Zhejiang provinces, where funding primarily relies on villagers’ voluntary and unpaid donations. However, interpersonal relationships in villages are complex and heterogeneous, making it difficult for individuals to judge things according to situations. The influence of social relationships and worldly wisdom on residents’ reputation evaluations cannot be overlooked. Previous studies rarely considered the role of heterogeneous interpersonal relationships in PGG. Therefore, This paper investigates the impact of heterogeneous interpersonal relationships within the reputation mechanism on promoting residents’ cooperative provision of public goods. If there is an impact, how the scale and structure of interpersonal relationships in different contexts influence the level of cooperation and the role of the reputation mechanism in this process.

Public goods game model

The PGG model is constructed on a scale-free network where individuals are randomly distributed. Each agent has a distinct number of neighbors, reflecting the heterogeneity of the real world. We assume that the average degree of the scale-free network is k, and each agent participates in a number of PGGs equal to the number of neighbors plus one. Initially, all participants select their strategies with equal probability, becoming either cooperators or defectors. Cooperators contribute one unit cost to the public pool (c_i = 1), while defectors contribute nothing (c_i = 0). Thus, the total payoff for each agent i is calculated, as shown in Equation (1):

(1)

where Ω_i is the set of PGG groups in which agent i participates. r is synergy factor. c_i is the contribution of agent i, which is 0 or 1. c_j is total contribution of all cooperators in group . k_j is the number of neighbors of agent i in group j.

Reputation update rule

After the donation, the donation information is made public and serves as the basis for others to evaluate individual reputation. At step t, each individual’s reputation comprises two components: the previous round’s reputation R_i(t-1) and the current evaluation based on donation behavior ∆R_i(t). The reputation update rule is given in Equation (2):

(2)

where ∆R_i(t) is the sum of R_iER(t), R_iMR(t) and R_iIR(t), which will be explained in detail in the following section.

The rules of heterogeneous reputation evaluation

Each agent is initially assigned a random reputation value. Specifically, the initial reputation R_i(0) of agent i is randomly drawn from the interval [0, 200]. If an agent’s reputation exceeds 200, it is capped at 200. Similarly, if it falls below 0, it is set to 0.

In this paper, a randomly selected agent i is assumed to possess a reference circle, centered at itself, with a radius λ, as illustrated in Fig 1, representing the scope of its social circle. Only evaluations from individuals within the social circle affect the reputation of agent i. For simplicity, it is assumed that all agents share reference circles of equal size in the simulation. Interpersonal relationships are classified into three types: ER, MR, and IR, each adopting distinct rules for reputation assessment. The three reputation evaluation rules are introduced below.

Download:

Fig 1. Ilustration of three types of interpersonal relationships.

The outermost circle represents the reference circle with agent i as its center and λ as its radius. In the figure, colored nodes represent agents belonging to distinct emotional relationship (ER) subgroups, while gray nodes indicate agents with no ER connections. The color coding is used solely to visualize ER subgroup structure and does not imply spatial proximity or topological organization.

https://doi.org/10.1371/journal.pone.0332474.g001

Reputation evaluation rule of ER: As is shown in Fig 1, the gray nodes indicate that they don’t have ERs with others, while the other colored nodes have. In reality, the former resemble ordinary individuals, while the latter resemble extended families with blood ties. Specifically, in the model, a stable relationship between individuals sharing the same color (except gray) within the social circle is defined as an ER, regardless of the presence of a link. For example, a₁ and a₂ both have emotional relationships with agent i. Under ER, participants will refrain from making negative judgments, even if agent i chooses to be a free rider. The ER formula for reputation evaluation as shown in Equation (3):

(3)

where R_iER(t) denotes the reputation evaluated by an individual within the ER of agent i.

Reputation evaluation rule of MR: Consider the green agent i in Fig 1 as an example. In the model, MR represents the interpersonal relationship between agent i and a link-neighbor of a different color, such as between agent i and b₁, or between agent i and c1. The number of serial betrayals by agent i is considered when determining the individual’s reputation. If the number of betrayals is within the tolerance threshold α, the individual will tolerate agent i’s defection at step t. Accordingly, once it exceeds α, other participants will focus on agent i’s contribution and make negative judgments about the reputation. The MR formula for reputation evaluation as shown in Equation (4):

(4)

where R_iMR(t) indicates the reputation given by an individual following MR with agent i. F is the number of consecutive defections by agent i.

Reputation evaluation rule of IR: The IR represents the relationship between agent i and nodes that are neither connected to, nor share the same color as, agent i in the reference circle shown in Fig 1, such as between agent i and b₂, or agent i and c₂. If the two are in an IR, they evaluate each other’s reputation based on their contributions to the public pool. They think that contributions increase reputation, while defections decrease it. The IR formula for reputation evaluation as shown in Equation (5):

(5)

where R_iIR(t) is the reputation evaluated by an individual conforming to IR with agent i.

Strategy update rules based on reputation mechanism

Most individuals care about face, which is represented by reputation in this paper. If an individual receives numerous negative evaluations, causing the reputation to fall below the threshold R_T, embarrassment and shame will result [53]. Therefore, when R_i(t)<R_T, the agent must contribute in the next step to improve the reputation and save face. Conversely, when R_i(t)≥R_T, individuals follow the principle of primacy of returns, aiming to optimize their returns by randomly copying a neighbor’s strategy from the previous round with a given probability, as shown in Equation (6):

(6)

where κ is the noise figure, representing the uncertainty in the participant’s strategy choice. S_i and S_j represent strategies of agent i and j, respectively. p_i and p_j refer to the total payoffs of agent i and j, respectively.

Simulation results and discussion

Although ER, MR, and IR relationships commonly coexist in real-world networks, we adopt an experimental control strategy in our simulations to isolate the marginal influence of each relationship type on cooperative dynamics. Specifically, we systematically vary the parameters of one relationship type at a time while keeping the others constant, in order to evaluate their respective effects on cooperation.

To minimize stochastic errors in Monte Carlo simulation (MCS), multiple independent simulations were performed to investigate evolutionary dynamics on a scale-free network. Initially, 10% of individuals are assigned ERs and randomly grouped into three groups (g) to reflect realistic conditions. The proportion of cooperation (ρ_c) is set to 0.5. The other key parameters are set as follows: N = 200, k = 4, r = 2, λ = 6, α = 2, κ = 0.1, R_T = 20. All results are calculated by 20 independent simulations within a total of 10000 steps. Each data point is calculated by averaging the cooperation proportion over the last 1,000 steps after the system reaches equilibrium. The parameter Settings are shown in Table 1.

Download:

Table 1. Parameters setting.

https://doi.org/10.1371/journal.pone.0332474.t001

To improve the clarity of the model design, we provide additional explanations for several key parameters listed in Table 1:

ρ_E (emotional relationship intensity): This parameter denotes the proportion of agents involved in emotional relationships (ERs) within the network. Specifically, ρ_E × N agents are randomly assigned as ER members and further divided into subgroups to form stable emotional clusters.
g (number of emotional subgroups): This denotes the number of subgroups into which ER agents are partitioned. All ER-designated agents are evenly allocated to g subgroups, simulating multiple “families” or “close-knit social circles.” Emotional ties are assumed to exist only within each subgroup; no ER connections are assumed between agents from different subgroups.
λ (social radius): This parameter defines the range within which an agent can receive reputation information. In the scale-free network used in our simulations, node distance is measured by the number of hops (i.e., shortest path length), rather than by geometric distance. An agent i receives reputation evaluations only from nodes whose shortest path distance to i is less than or equal to λ. Thus, a larger λ expands the agent’s reputation horizon, increasing the effective influence range of the reputation mechanism.

The impact of emotional relationships

In the experiment assessing the impact of ER on cooperation, we fixed the number of ER agents (ρ_E > 0) and maintained the influence mechanisms of MR and IR relationships at their default settings, without targeted manipulation. This setup allowed us to isolate the marginal effects of ER intensity and intra-group structure on cooperative behavior.

First, the effect of the individual aggregate with ER on the evolution of cooperation is investigated. Fig 2 illustrates the proportion of cooperation ρ_c as a function of synergy factor r for different values of ρ_E. As shown in Fig 2, variations in ρ_E influence the proportion of cooperation ρ_c, and this effect is constrained by the synergy factor r. That is, there exists a critical threshold r’. When r < r’, the proportion of cooperation ρ_c decreases as ρ_E increases. When r ≥ r’, the proportion of cooperation ρ_c is not affected by the change of ρ_E. According to Fig 2, when r < 2.7, the proportion of cooperation ρ_c declines with increasing ρ_E. When ρ_E is 10%, it provides only a slight improvement in the overall proportion of cooperation. For r > 2.7, further increases in the synergy factor r result in no significant change the proportion of cooperation in the equilibrium.

Download:

Fig 2. The proportion of cooperation ρ_c as a function of synergy factor r for different values of ρ_E.

Other parameters are set as follows: k = 4, λ = 6, α = 2, κ = 0.1, and R_T = 20.

https://doi.org/10.1371/journal.pone.0332474.g002

A larger value of ρ_E implies that more individuals will shield non-cooperative agents. A smaller value of ρ_E suggests that most individuals act independently, leading to reduced favoritism within the system. Greater intimacy in relationships within a society fosters more favoritism and protective behavior, thereby hindering cooperation efficiency. The results also suggest that the nature of the activity or project should be considered when organizing a public event.

Next, the impact of heterogeneity in the distribution within the ER population on the evolution of cooperation is investigated. The parameter g is introduced to represent the number of groups with distinct affective relationships within the population. For example, in a village, individuals a1 to a10 belong to one family, while b1 to b10 belong to another. As g increases, the number of groups with distinct affective types in the village increases, and conversely, this number reduces.

The relationship between the parameter g and the proportion of cooperation ρ_c is presented in Fig 3. Figs 3(a), (b) and (c) correspond to ρ_E = 10%, ρ_E = 20% and ρ_E = 30%, respectively. The curves are generated under different synergy factors: r = 1.5, 2, 2.4 and 2.5. In Fig 3(a), the cooperative evolution results are basically similar. A similar trend is observed in Figs 3(b) and (c). These findings indicate that variations in g exert minimal influence on the proportion of cooperation. Additionally, the results validate the robustness of the model against changes in group number. Consequently, g = 3 is adopted in subsequent analysis, as it does not significantly affect the final outcomes.

Download:

Fig 3. The proportion of cooperation ρ_c as the internal structure of the group with ER changes for different values of r.

Here, ρ_E = 10% in (a), ρ_E = 20% in (b) and ρ_E = 30% in (c).

https://doi.org/10.1371/journal.pone.0332474.g003

The impact of mixed relationships

In the experiment investigating the effect of MR on cooperation, we varied the network’s average degree (k) to modulate MR intensity. According to the model design, an MR exists between two nodes if they are directly connected (i.e., neighbors) but do not share an ER. In this set of simulations, the proportion of ER agents was held constant. As a result, variations in k did not alter the absolute number of ER agents or generate new ER ties. Instead, increasing k led to more non-ER neighbors, thereby enhancing both the quantity and density of MR connections in the network.

MR connections refer to interpersonal links between agents who are connected but not involved in ERs. To examine the effect of MR intensity on cooperation, the average number of neighbors k is varied. The evolutionary results are illustrated in Fig 4 for network average degrees of 4, 5, 6, 7, and 8. In general, MR intensity exhibits a negative correlation with ρ_c, that is, lower values of k tend to yield higher levels of cooperation. As observed in Fig 4, the red curve representing k = 4 lies above the other curves in most cases. However, the negative correlation between MR intensity and ρ_c is affected by r, and it can be seen from Fig 4 that all lines show a trend of convergence, divergence and re-convergence. To further quantify the influence of r, the standard deviation σ of ρ_c is computed, revealing a rise-and-fall pattern.

Download:

Fig 4. The proportion of cooperation ρ_c as a function of synergy factor r for different values of k.

Other parameters are set as follows, ρ_E = 10%, λ = 6, α = 2, κ = 0.1, R_T = 20.

https://doi.org/10.1371/journal.pone.0332474.g004

In general, the increase of MR is not conducive to the cooperative behavior of the group. This occurs because higher MR intensity fosters concealment within the group. In such social circles, a greater number of individuals are willing to tolerate defection by agent i, leading to a decrease in the proportion of cooperation. From the perspective of network structure, an influential individual in a social circle resembles a core node in a scale-free network. Core nodes are highly connected and exert dominant influence within the network. Once a central individual chooses to defect, more neighbors will follow this behavior with a certain probability. In summary, the number of cooperators declines with increasing MR intensity at equilibrium. Furthermore, higher externalities in cooperative provision encourage greater participation in cooperation. Within the scope of this study, the synergy factor r exhibits a trend where differences in cooperation across various k values initially increase and then decrease. This indicates that as r increases, the disparity in the proportion of cooperation under different MR intensities first widens, but narrows after reaching a certain threshold. This dynamic is further quantified by calculating the standard deviation σ of ρ_c, which exhibits a rising and then falling trend as r increases, reflecting the changing influence of the synergy factor on cooperation differences.

Furthermore, the influence of the tolerance threshold α on individual strategic choice in MR is examined through simulations. Fig 5 depicted how the average number of neighbors k and the tolerance threshold α affect the proportion of cooperators in the group. In Fig 5(a), with k held constant, the cooperation rate decrease as α increases. As the synergy factor r increases, the effective range of α narrows, as reflected by the expansion of the red region. Specifically, in Figs 5 (b) and (c), when k is less than approximately 5 and 7, respectively, individual strategic choices are insensitive to the tolerance threshold α. For instance, in Fig 5(b), when k = 4, a high level of cooperation is maintained regardless of the value of α. By contrast, for larger values of k becomes more pronounced in the evolution of cooperation, as shown in Figs 5 (b) and (c).

Download:

Fig 5. The contour map of the proportion of cooperation ρ_c as a function of average degree of network k and tolerance threshold α.

Here, r = 2 in panel (a), r = 3 in panel (b) and r = 4 in panel (c).

https://doi.org/10.1371/journal.pone.0332474.g005

In social networks with a high proportion of MR, individuals expect future reciprocation. As a result, relationships tend to be preserved rather than reputations tarnished by defection within tolerance limits. This often leads to a “favor trap” within the organization. Additionally, as shown in Fig 5, a fully cooperative state can only be achieved under high α and large r. However, such conditions are relatively rare in public activities with significant externalities. Therefore, it is necessary to guide individuals to reduce their tolerance levels to better accommodate more practical scenarios.

To further investigate the influence of the tolerance threshold α on the evolution of cooperation, Fig 6 illustrates the evolutionary process of cooperation proportion over time for two different parameter sets. In Fig 6, r = 3, k = 4 in Fig 6 (a) and r = 4, k = 6 in Fig 6 (b) correspond to the critical values shown in Figs 5(b) and (c), respectively. Clearly, all curves clearly decrease initially and then increase, eventually reaching equilibrium. At the same time, as the tolerance threshold α increases, the curves reaching the fully cooperative state shift to the right. It is worth noting that although there is no significant difference in the final evolutionary result, the defection behavior cannot be blindly tolerated in the social system. As society develops, As society develops, MR, which emphasizes emotional ties and social connections, will gradually become dominant and widespread. Therefore, efforts are required to eliminate indulgent behavior and actively promote the emergence of cooperation.

Download:

Fig 6. The proportion of cooperation ρ_c changes over time for different parameter combinations at each MCS time step.

(a) Under three different tolerance thresholds α for fixed synergy factor r = 3 and average degree k = 4. (b) Under three different tolerance thresholds α for fixed synergy factor r = 4 and average degree k = 6.

https://doi.org/10.1371/journal.pone.0332474.g006

The impact of instrumental relationships

In the experiment assessing the effect of instrumental relationships (IR) on cooperation, we primarily varied the social radius (λ) to control each agent’s interaction neighborhood, thereby simulating the expansion of the reputation evaluation scope linked to IR. The intensities of emotional (ER) and mixed (MR) relationships were held constant to isolate the specific effect of IR on cooperative behavior.

The impact of IR intensity in interpersonal relationships on the evolution of cooperation is subsequently examined. In real life, most interpersonal connections are relatively weak and distant, characterized as IR. To represent increased IR intensity, the agents’ reference radius λ is enlarged. Fig 7 illustrates the effect of λ when other parameters are constant expect the synergy factor r. Overall, all the curves exhibit a similar trend and appear to have a critical value r. For example, when λ = 4, the proportion of cooperation remains relatively stable for r < 2, followed by a sharp increase. The curve then increases more rapidly until full cooperation is achieved. After r = 3, the proportion of cooperation is not affected by λ when the system achieves equilibrium. However, for r < 3, variations in λ significantly influence the evolution of cooperation. A larger λ more effectively suppresses defection within the group.

Download:

Fig 7. The proportion of cooperation ρ_c as a function of λ with the variation of synergy factor r.

Other parameters are set as follows, ρ_E = 10%, k = 4, α = 2, κ = 0.1, R_T = 20.

https://doi.org/10.1371/journal.pone.0332474.g007

It can be deduced that the larger λ is, the greater the proportion of IR in social interactions, which makes individuals’ reputations more reliant on rational judgments based on contributions. Individuals in IR pay more attention to facts. Reputation is more widely evaluated based on actions rather than interpersonal connections. Undeniably, an increase in spontaneous rational decision-making is advantageous for the emergence of cooperators. Although achieving a purely rule-based society in real life is challenging, fairness in the process must be emphasized to promote the development of social civilization.

Reputation mechanism analysis

The above analysis mainly discuss the degree of tolerance towards others in interpersonal interactions. However, it is essential to emphasize the role of self-discipline in social cooperation, specifically the reputation threshold R_T addressed in this paper. The effect of the reputation threshold R_T on the proportion of cooperation ρ_c under different synergy factors r is further examined, as shown in Fig 8. In this paper, the reputation accumulated through an agent’s contributions is used to represent “face”, a concept highly valued in society. When R_T ≠ 0, the curves exhibit similar trends. It is obvious that ρ_c increases as R_T enlarges. Notably, when R_T = 0 and r < 2.8, the cooperation rate remains zero. A small number of cooperators appear when r = 2.8.

Download:

Fig 8. The proportion of cooperation ρ_c as a function of synergy factor r for different values of R_T.

Other parameters are set as follows, ρ_E = 10%, k = 4, λ = 6, α = 2, κ = 0.1.

https://doi.org/10.1371/journal.pone.0332474.g008

The simulation results indicate that when R_T ≠ 0, maintaining face is conducive to promoting cooperation. That is, when individuals are sensitive to others’ evaluations, they are motivated to contribute to maintain a good reputation. The higher R_T is, the more concerned individuals become with their image and status in social interactions. Additionally, the phenomenon when R_T = 0 illustrates that maintaining cooperation is challenging if individuals ignore external institutional norms, rendering the reputation mechanism ineffective.

To further explore the role of the reputation mechanism in PGG, the dynamic evolution of the cooperation proportion is plotted at each MCS step for different values of r and R_T. The reputation threshold R_T is set to 0, 20 and 80. The evolutionary curves are shown in Figs 9(a) and (b), corresponding to r = 2 and r = 3, respectively. It can be observed that when RT = 0, the proportion of cooperation decreases monotonically at the beginning of the evolution until to 0 or a minimal value. When R_T = 20, the proportion of cooperation initially decreases and then gradually increases around step 5. For R_T = 80, the proportion of cooperation initially increases. As shown in Fig 9(b), the curve for R_T = 80 continues to rise, eventually reaching full cooperation. Consequently, higher R_T leads to more cooperative behaviors in the early stages. In contrast, when R_T is low, individuals are motivated by self-interest and tend to defect. The result suggests that cooperative behavior emerges and eventually stabilizes at equilibrium under the influence of the reputation mechanism.

Download:

Fig 9. The time evolution of the proportion of cooperation ρc for R_T = 0, 20 and 80.

The synergy factor r is set to be 2 in (a) and 3 in (b).

https://doi.org/10.1371/journal.pone.0332474.g009

Robustness of the model

Finally, to validate the robustness of the model, Fig 10 depicts the relationship between the proportion of cooperation ρ_c and the noise figure κ. As observed, the proportion of cooperation remains relatively stable across different values of κ. Specifically, ρ_c reaches a maximum of 0.584 at κ = 0.5, and a minimum of 0.570 at κ = 0.2. The fluctuation range of ρ_c does not exceed 0.02, confirming the robustness of the proposed model.

Download:

Fig 10. The bar graph of the proportion of cooperation ρc with noise figure κ set at 0.1,0.2,0.3,0.4 and 0.5.

https://doi.org/10.1371/journal.pone.0332474.g010

Conclusion and implication

Conclusions

Given the influence of social relationships and worldly wisdom, it is often difficult to judge the facts from the facts. Based on existing research, this paper classifies interpersonal relationships into three types to examine how heterogeneous relationships under the reputation mechanism affect cooperation in PGG. Through simulation experiment and analysis, the following conclusions are drawn.

(ⅰ) Changes in the distribution structure within the ER group have minimal impact on the proportion of cooperation. Furthermore, a smaller ρ_E promotes cooperation provision.
(ⅱ) An increase in MR reduces the proportion of cooperation. The tolerance threshold α positively influences cooperation, but its effective range diminishes as r increases.
(ⅲ) Increasing IR helps suppress defection under the reputation mechanism, allowing individuals’ reputations to be based more on rational evaluations of contributions.
(ⅳ) Without a functioning reputation mechanism, maintaining cooperation is challenging. High R_T levels promote the diffusion of cooperative behavior in public goods.

Implications and future research

Theoretical implications

(ⅰ) Studying residents’ cooperative provision of public goods from the perspective of bounded rationality offers a more realistic framework. Although Darwin’s evolutionary theory emphasizes competition as the norm, the rational man hypothesis suggests that non-excludability leads to free-riding, rendering voluntary cooperation by residents unsustainable. However, cooperative behaviors—such as subsidizing disadvantaged children and donating blood without compensation—are prevalent in real-world societies. The village opera studied here is a typical example. Findings based on the village opera further support the rationale for exploring how to guide residents in providing public goods under bounded rationality, consistent with the principles of behavioral economics.
(ⅱ) This paper incorporates varying intensities of interpersonal relationships, thereby broadening the scope of research on residents’ cooperative provision of public goods. At the same time, the results confirm that heterogeneous interpersonal relationships significantly affect cooperation, as different relationship types yield varying impacts on residents’ participation.
(ⅲ) The positive externality associated with cooperative provision of public goods is an important factor in encouraging residents’ cooperation. This study shows that the synergy factor r has a significant impact on the proportion of cooperation, with its range of variation significantly affecting the influence of other parameters on cooperation. Synergy factor r can be regarded as the positive externality of cooperative provision of public goods. As r increases, individuals benefit more per capita by choosing to cooperate. While this paper emphasizes cooperation from a bounded rationality perspective, it acknowledges the significant role of rational payoffs in fostering cooperation. Future research should classify public goods according to the positive externality of their cooperative provision and further advance studies to promote resident cooperation in public goods provision.

Practical implications

(ⅰ) Analyzing promotional effects under varying scenarios helps organizers select appropriate public goods. This paper demonstrates that interpersonal relationship intensity, reputation threshold R_T, and synergy factor r influence residents’ cooperative provision of public goods, providing a quantitative relationship based on village opera. This offers strategic insights for organizers to effectively guide cooperation and ensure sustainability.
(ⅱ) To improve residents’ cooperative provision, it is essential to actively promote social morality and ethics, as civilization should not equate to blind tolerance. The study finds that α and R_T, representing heteronomy and self-discipline, respectively, are positively correlated with the proportion of cooperation. In other words, higher parameter values correspond to a higher proportion of cooperation. The study concludes that the government should actively publicize positive initiatives, enforce moral and ethical constraints, and avoid excessive tolerance of gray-area behaviors that, while not illegal, are immoral.
(ⅲ) The resident self-provision pattern is an effective way to reduce the government’s financial burden and improve the alignment of supply and demand. This study demonstrates, based on real cases, the feasibility of mobilizing residents to cooperate in public goods provision from a bounded rationality perspective. On one hand, residents raise funds to provide public goods, reducing the government’s fiscal expenditure. On the other hand, as consumers of public goods, residents become suppliers after self-provision, effectively bridging the supply and demand sides. Therefore, resident self-provision not only reduces the government’s financial burden but also improves the alignment of public goods supply and demand, ultimately enhancing residents’ satisfaction.

Future research

Although the current model classifies interpersonal relationships into fixed categories—emotional (ER), mixed (MR), and instrumental (IR) ties—real-world relationships are often dynamic and evolve through repeated interactions. Emotional bonds may strengthen or weaken, instrumental ties can evolve into trust-based relationships, and reputation assessments adjust accordingly. Future research should consider co-evolutionary dynamics among social ties, strategic behaviors, and reputation updates, enabling relationship types to shift based on behavioral history and mutual feedback.

Recent developments in evolutionary game theory highlight the effectiveness of adaptive feedback mechanisms in promoting cooperation in social dilemmas. For example, adaptive rewards or penalties responsive to population-level cooperation have been shown to produce stable coexistence or evolutionary cycles [74,75]. Similarly, studies show that in collective-risk games, risk levels and social behavior mutually influence each other, with feedback loops playing a key role in shaping cooperation outcomes [76]. These findings suggest that incorporating feedback-driven mechanisms into interpersonal and reputational dynamics could yield a more realistic framework for modeling cooperation.

Furthermore, the study by Krellner M et al. emphasizes the role of joint commitments in reinforcing cooperation through indirect reciprocity [77]. This supports our findings, indicating that reputation systems embedded in relationships—whether based on interpersonal types or shared commitments—can serve as informal enforcement mechanisms in the absence of formal institutions. Future extensions of our model could investigate how public commitments, trust-building behaviors, and repeated cooperation signals reshape relationship structures and reputation thresholds over time.

Collectively, these directions suggest a more integrative modeling framework that captures the co-adaptation of individual behaviors, reputational dynamics, and social structures—an essential step toward understanding long-term cooperation in complex social systems.

References

1. Pavón N, Rizzi LI. Road infrastructure and public bus transport service provision under different funding schemes: A simulation analysis. Transportation Research Part A: Policy and Practice. 2019;125:89–105.
- View Article
- Google Scholar
2. Chih Y-Y. Social network structure and government provision crowding-out on voluntary contributions. Journal of Behavioral and Experimental Economics. 2016;63:83–90.
- View Article
- Google Scholar
3. Florenzano M. Government and the provision of public goods:from equilibrium models to mechanismdesign. The European Journal of the History of Economic Thought. 2010;17(4):1047–77.
- View Article
- Google Scholar
4. Banerjee A, Somanathan R. The political economy of public goods: Some evidence from India. Journal of Development Economics. 2007;82(2):287–314.
- View Article
- Google Scholar
5. Hörcher D, De Borger B, Graham DJ. Subsidised transport services in a fiscal federation: Why local governments may be against decentralised service provision. Economics of Transportation. 2023;34:100312.
- View Article
- Google Scholar
6. Besley T, Ghatak M. Public–private partnerships for the provision of public goods: Theory and an application to NGOs. Research in Economics. 2017;71(2):356–71.
- View Article
- Google Scholar
7. Hirshleifer J. From weakest-link to best-shot: The voluntary provision of public goods. Public Choice. 1983;41(3):371–86.
- View Article
- Google Scholar
8. Vagropoulos SI, Biskas PN, Bakirtzis AG. Market-based TSO-DSO coordination for enhanced flexibility services provision. Electric Power Systems Research. 2022;208:107883.
- View Article
- Google Scholar
9. Chen F, Sun Z, Zhao Y. The effects of social capital and family income on farmers’ participation in rural public goods provision. Journal of Rural Studies. 2024;109:103332.
- View Article
- Google Scholar
10. Gu Y, Rasch A, Wenzel T. Consumer salience and quality provision in (un)regulated public service markets. Regional Science and Urban Economics. 2022;93:103754.
- View Article
- Google Scholar
11. Coate S, Anderson SP. Market Provision of Public Goods: The Case of Broadcasting. Cambridge: National Bureau of Economic Research; 2000.
12. Blöchliger H. Market mechanisms in public service provision. OECD Economics Department Working Papers 626. Paris: OECD; 2008.
13. Chan KS, Mestelman S, Moir R, Muller RA. Heterogeneity and the voluntary provision of public goods. Exp Econ. 1999;2(1):5–30.
- View Article
- Google Scholar
14. Freundt J, Lange A. On the voluntary provision of public goods under risk. Journal of Behavioral and Experimental Economics. 2021;93:101727.
- View Article
- Google Scholar
15. Que W, Zhang Y, Liu S. The spatial spillover effect of fiscal decentralization on local public provision: Mathematical application and empirical estimation. Applied Mathematics and Computation. 2018;331:416–29.
- View Article
- Google Scholar
16. Castelnovo W. Citizens coproduction, service self-provision and the state 2.0. Network, Smart and Open: Three Keywords for Information Systems Innovation. Cham: Springer International Publishing. 2018;109-125.
17. Darwin C. On the origin of species. Cambridge: Cambridge University Press; 2009.
18. Ma Z-Q, Xia C-Y, Sun S-W, Wang L, Wang H-B, Wang J. Heterogeneous link weight promotes the cooperation in spatial prisoner’s dilemma. Int J Mod Phys C. 2011;22(11):1257–68.
- View Article
- Google Scholar
19. Nowak MA, Sigmund K. Evolution of indirect reciprocity by image scoring. Nature. 1998;393(6685):573–7. pmid:9634232
- View Article
- PubMed/NCBI
- Google Scholar
20. Ohtsuki H, Hauert C, Lieberman E, Nowak MA. A simple rule for the evolution of cooperation on graphs and social networks. Nature. 2006;441(7092):502–5. pmid:16724065
- View Article
- PubMed/NCBI
- Google Scholar
21. Szolnoki A, Perc M, Danku Z. Making new connections towards cooperation in the prisoner’s dilemma game. Europhys Lett. 2008;84(5):50007.
- View Article
- Google Scholar
22. Zhang J, Zhang C, Cao M. How insurance affects altruistic provision in threshold public goods games. Sci Rep. 2015;5:9098. pmid:25765206
- View Article
- PubMed/NCBI
- Google Scholar
23. Smith JM. Evolution and the theory of games. Am Sci. 1976;64(1):41–5. pmid:1247210
- View Article
- PubMed/NCBI
- Google Scholar
24. Von Neumann J, Morgenstern O. Theory of games and economic behavior: 60th anniversary commemorative edition. Theory of games and economic behavior. Princeton university press;2007.
25. Peng D, Yang H-X, Wang W-X, Chen GR, Wang B-H. Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game. Eur Phys J B. 2010;73(3):455–9.
- View Article
- Google Scholar
26. Rand DG, Dreber A, Ellingsen T, Fudenberg D, Nowak MA. Positive interactions promote public cooperation. Science. 2009;325(5945):1272–5. pmid:19729661
- View Article
- PubMed/NCBI
- Google Scholar
27. Traulsen A, Hauert C, De Silva H, Nowak MA, Sigmund K. Exploration dynamics in evolutionary games. Proc Natl Acad Sci U S A. 2009;106(3):709–12. pmid:19124771
- View Article
- PubMed/NCBI
- Google Scholar
28. Chen Q, Chen T, Wang Y. How the expanded crowd-funding mechanism of some southern rural areas in China affects cooperative behaviors in threshold public goods game. Chaos, Solitons & Fractals. 2016;91:649–55.
- View Article
- Google Scholar
29. Griffin AS, West SA. Kin selection: fact and fiction. Trends in Ecology & Evolution. 2002;17(1):15–21.
- View Article
- Google Scholar
30. Hamilton WD. The genetical evolution of social behaviour. II. J Theor Biol. 1964;7(1):17–52. pmid:5875340
- View Article
- PubMed/NCBI
- Google Scholar
31. Boyd R, Richerson PJ. Group selection among alternative evolutionarily stable strategies. J Theor Biol. 1990;145(3):331–42. pmid:2232821
- View Article
- PubMed/NCBI
- Google Scholar
32. Traulsen A, Nowak MA. Evolution of cooperation by multilevel selection. Proc Natl Acad Sci U S A. 2006;103(29):10952–5. pmid:16829575
- View Article
- PubMed/NCBI
- Google Scholar
33. Choi J-K, Ahn TK. Strategic reward and altruistic punishment support cooperation in a public goods game experiment. Journal of Economic Psychology. 2013;35:17–30.
- View Article
- Google Scholar
34. dos Santos M. The evolution of anti-social rewarding and its countermeasures in public goods games. Proc Biol Sci. 2015;282(1798):20141994. pmid:25429015
- View Article
- PubMed/NCBI
- Google Scholar
35. Szolnoki A, Perc M. Reward and cooperation in the spatial public goods game. EPL. 2010;92(3):38003.
- View Article
- Google Scholar
36. Boyd R, Gintis H, Bowles S, Richerson PJ. The evolution of altruistic punishment. Proc Natl Acad Sci U S A. 2003;100(6):3531–5. pmid:12631700
- View Article
- PubMed/NCBI
- Google Scholar
37. Brandt H, Hauert C, Sigmund K. Punishing and abstaining for public goods. Proc Natl Acad Sci U S A. 2006;103(2):495–7. pmid:16387857
- View Article
- PubMed/NCBI
- Google Scholar
38. Szolnoki A, Perc M. Effectiveness of conditional punishment for the evolution of public cooperation. J Theor Biol. 2013;325:34–41. pmid:23485452
- View Article
- PubMed/NCBI
- Google Scholar
39. Gao J, Li Z, Cong R, Wang L. Tolerance-based punishment in continuous public goods game. Physica A: Statistical Mechanics and its Applications. 2012;391(16):4111–20.
- View Article
- Google Scholar
40. Ren T, Zheng J. Evolutionary dynamics in the spatial public goods game with tolerance-based expulsion and cooperation. Chaos, Solitons & Fractals. 2021;151:111241.
- View Article
- Google Scholar
41. Zhang S, Zhang Z, Wu Y, Yan M, Xie Y. Tolerance-based punishment and cooperation in spatial public goods game. Chaos, Solitons & Fractals. 2018;110:267–72.
- View Article
- Google Scholar
42. Capraro V, Jordan JJ, Rand DG. Heuristics guide the implementation of social preferences in one-shot Prisoner’s Dilemma experiments. Sci Rep. 2014;4:6790. pmid:25348470
- View Article
- PubMed/NCBI
- Google Scholar
43. Wang Y, Chen T, Chen Q, Si G. Emotional decisions in structured populations for the evolution of public cooperation. Physica A: Statistical Mechanics and its Applications. 2017;468:475–81.
- View Article
- Google Scholar
44. Wang Y, Chen T. Heuristics guide cooperative behaviors in public goods game. Physica A: Statistical Mechanics and its Applications. 2015;439:59–65.
- View Article
- Google Scholar
45. Fan R, Zhang Y, Luo M, Zhang H. Promotion of cooperation induced by heterogeneity of both investment and payoff allocation in spatial public goods game. Physica A: Statistical Mechanics and its Applications. 2017;465:454–63.
- View Article
- Google Scholar
46. Lei C, Wu T, Jia J-Y, Cong R, Wang L. Heterogeneity of allocation promotes cooperation in public goods games. Physica A: Statistical Mechanics and its Applications. 2010;389(21):4708–14.
- View Article
- Google Scholar
47. Zhang Y, Wang J, Ding C, Xia C. Impact of individual difference and investment heterogeneity on the collective cooperation in the spatial public goods game. Knowledge-Based Systems. 2017;136:150–8.
- View Article
- Google Scholar
48. Axelrod R, Hamilton WD. The evolution of cooperation. Science. 1981;211(4489):1390–6. pmid:7466396
- View Article
- PubMed/NCBI
- Google Scholar
49. Trivers RL. The Evolution of Reciprocal Altruism. The Quarterly Review of Biology. 1971;46(1):35–57.
- View Article
- Google Scholar
50. Alexander R. The biology of moral systems. New York: Routledge. 2017.
51. Saavedra S, Smith D, Reed-Tsochas F. Cooperation under indirect reciprocity and imitative trust. PLoS One. 2010;5(10):e13475. pmid:21048950
- View Article
- PubMed/NCBI
- Google Scholar
52. Wedekind C, Milinski M. Cooperation through image scoring in humans. Science. 2000;288(5467):850–2. pmid:10797005
- View Article
- PubMed/NCBI
- Google Scholar
53. Chen Q, Chen T, Wang Y. Publishing the donation list incompletely promotes the emergence of cooperation in public goods game. Applied Mathematics and Computation. 2017;310:48–56.
- View Article
- Google Scholar
54. Podder S, Righi S, Pancotto F. Reputation and punishment sustain cooperation in the optional public goods game. Philos Trans R Soc Lond B Biol Sci. 2021;376(1838):20200293. pmid:34601913
- View Article
- PubMed/NCBI
- Google Scholar
55. Romano A, Giardini F, Columbus S, de Kwaadsteniet EW, Kisfalusi D, Triki Z, et al. Reputation and socio-ecology in humans. Philos Trans R Soc Lond B Biol Sci. 2021;376(1838):20200295. pmid:34601915
- View Article
- PubMed/NCBI
- Google Scholar
56. Wang L, Chen T, Wu Z. Promoting cooperation by reputation scoring mechanism based on historical donations in public goods game. Applied Mathematics and Computation. 2021;390:125605.
- View Article
- Google Scholar
57. Jehn KA, Shah PP. Interpersonal relationships and task performance: An examination of mediation processes in friendship and acquaintance groups. Journal of Personality and Social Psychology. 1997;72(4):775–90.
- View Article
- Google Scholar
58. Zhang H, An T, Wang J, Wang L, An J, Zhao J, et al. Reputation-based adaptive strategy persistence can promote cooperation considering the actual influence of individual behavior. Physics Letters A. 2024;508:129495.
- View Article
- Google Scholar
59. Zhou T, Ding S, Fan W, Wang H. An improved public goods game model with reputation effect on the spatial lattices. Chaos, Solitons & Fractals. 2016;93:130–5.
- View Article
- Google Scholar
60. Quan J, Tang C, Wang X. Reputation-based discount effect in imitation on the evolution of cooperation in spatial public goods games. Physica A: Statistical Mechanics and its Applications. 2021;563:125488.
- View Article
- Google Scholar
61. Brandt H, Sigmund K. Indirect reciprocity, image scoring, and moral hazard. Proc Natl Acad Sci U S A. 2005;102(7):2666–70. pmid:15695589
- View Article
- PubMed/NCBI
- Google Scholar
62. Nowak MA, Sigmund K. Evolution of indirect reciprocity. Nature. 2005;437(7063):1291–8. pmid:16251955
- View Article
- PubMed/NCBI
- Google Scholar
63. Wang X, Chen X, Gao J, Wang L. Reputation-based mutual selection rule promotes cooperation in spatial threshold public goods games. Chaos, Solitons & Fractals. 2013;56:181–7.
- View Article
- Google Scholar
64. Liu Y, Chen T. Sustainable cooperation based on reputation and habituation in the public goods game. Biosystems. 2017;160:33–8. pmid:28827161
- View Article
- PubMed/NCBI
- Google Scholar
65. Han W, Zhang Z, Sun J, Xia C. Emergence of cooperation with reputation-updating timescale in spatial public goods game. Physics Letters A. 2021;393:127173.
- View Article
- Google Scholar
66. Quan J, Cui S, Chen W, Wang X. Reputation-based probabilistic punishment on the evolution of cooperation in the spatial public goods game. Applied Mathematics and Computation. 2023;441:127703.
- View Article
- Google Scholar
67. Mui L. Computational models of trust and reputation: Agents, evolutionary games, and social networks. PhD thesis, Massachusetts Institute of Technology. 2002.
- View Article
- Google Scholar
68. van Prooijen A, Ellemers N. Does It Pay to Be Moral? How Indicators of Morality and Competence Enhance Organizational and Work Team Attractiveness. British J of Management. 2014;26(2):225–36.
- View Article
- Google Scholar
69. Grunig J E, Hung-Baesecke C F. The effect of relationships on reputation and reputation on relationships: A cognitive, behavioral study. New York: Routledge; 2015.
70. Diekmann A, Jann B, Przepiorka W, Wehrli S. Reputation Formation and the Evolution of Cooperation in Anonymous Online Markets. Am Sociol Rev. 2013;79(1):65–85.
- View Article
- Google Scholar
71. Butler EA. Temporal interpersonal emotion systems: the “TIES” that form relationships. Pers Soc Psychol Rev. 2011;15(4):367–93. pmid:21693670
- View Article
- PubMed/NCBI
- Google Scholar
72. Vanneste BS. From interpersonal to interorganisational trust: The role of indirect reciprocity. Journal of Trust Research. 2016;6(1):7–36.
- View Article
- Google Scholar
73. Hwang K. Face and Favor: The Chinese Power Game. American Journal of Sociology. 1987;92(4):944–74.
- View Article
- Google Scholar
74. Hua S, Liu L. Coevolutionary dynamics of population and institutional rewards in public goods games. Expert Systems with Applications. 2024;237:121579.
- View Article
- Google Scholar
75. Hua S, Liu L. Facilitating the evolution of cooperation through altruistic punishment with adaptive feedback. Chaos, Solitons & Fractals. 2023;173:113669.
- View Article
- Google Scholar
76. Liu L, Chen X, Szolnoki A. Coevolutionary dynamics via adaptive feedback in collective-risk social dilemma game. Elife. 2023;12:e82954. pmid:37204305
- View Article
- PubMed/NCBI
- Google Scholar
77. Krellner M, Han TA. Words are not wind - how public joint commitment and reputation solve the Prisoner’s Dilemma. Applied Mathematics and Computation. 2025;506:129530.
- View Article
- Google Scholar

[ref1] 1. Pavón N, Rizzi LI. Road infrastructure and public bus transport service provision under different funding schemes: A simulation analysis. Transportation Research Part A: Policy and Practice. 2019;125:89–105.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Chih Y-Y. Social network structure and government provision crowding-out on voluntary contributions. Journal of Behavioral and Experimental Economics. 2016;63:83–90.
View Article
Google Scholar

[5] View Article

[6] Google Scholar

[ref3] 3. Florenzano M. Government and the provision of public goods:from equilibrium models to mechanismdesign. The European Journal of the History of Economic Thought. 2010;17(4):1047–77.
View Article
Google Scholar

[8] View Article

[9] Google Scholar

[ref4] 4. Banerjee A, Somanathan R. The political economy of public goods: Some evidence from India. Journal of Development Economics. 2007;82(2):287–314.
View Article
Google Scholar

[11] View Article

[12] Google Scholar

[ref5] 5. Hörcher D, De Borger B, Graham DJ. Subsidised transport services in a fiscal federation: Why local governments may be against decentralised service provision. Economics of Transportation. 2023;34:100312.
View Article
Google Scholar

[14] View Article

[15] Google Scholar

[ref6] 6. Besley T, Ghatak M. Public–private partnerships for the provision of public goods: Theory and an application to NGOs. Research in Economics. 2017;71(2):356–71.
View Article
Google Scholar

[17] View Article

[18] Google Scholar

[ref7] 7. Hirshleifer J. From weakest-link to best-shot: The voluntary provision of public goods. Public Choice. 1983;41(3):371–86.
View Article
Google Scholar

[20] View Article

[21] Google Scholar

[ref8] 8. Vagropoulos SI, Biskas PN, Bakirtzis AG. Market-based TSO-DSO coordination for enhanced flexibility services provision. Electric Power Systems Research. 2022;208:107883.
View Article
Google Scholar

[23] View Article

[24] Google Scholar

[ref9] 9. Chen F, Sun Z, Zhao Y. The effects of social capital and family income on farmers’ participation in rural public goods provision. Journal of Rural Studies. 2024;109:103332.
View Article
Google Scholar

[26] View Article

[27] Google Scholar

[ref10] 10. Gu Y, Rasch A, Wenzel T. Consumer salience and quality provision in (un)regulated public service markets. Regional Science and Urban Economics. 2022;93:103754.
View Article
Google Scholar

[29] View Article

[30] Google Scholar

[ref11] 11. Coate S, Anderson SP. Market Provision of Public Goods: The Case of Broadcasting. Cambridge: National Bureau of Economic Research; 2000.

[ref12] 12. Blöchliger H. Market mechanisms in public service provision. OECD Economics Department Working Papers 626. Paris: OECD; 2008.

[ref13] 13. Chan KS, Mestelman S, Moir R, Muller RA. Heterogeneity and the voluntary provision of public goods. Exp Econ. 1999;2(1):5–30.
View Article
Google Scholar

[34] View Article

[35] Google Scholar

[ref14] 14. Freundt J, Lange A. On the voluntary provision of public goods under risk. Journal of Behavioral and Experimental Economics. 2021;93:101727.
View Article
Google Scholar

[37] View Article

[38] Google Scholar

[ref15] 15. Que W, Zhang Y, Liu S. The spatial spillover effect of fiscal decentralization on local public provision: Mathematical application and empirical estimation. Applied Mathematics and Computation. 2018;331:416–29.
View Article
Google Scholar

[40] View Article

[41] Google Scholar

[ref16] 16. Castelnovo W. Citizens coproduction, service self-provision and the state 2.0. Network, Smart and Open: Three Keywords for Information Systems Innovation. Cham: Springer International Publishing. 2018;109-125.

[ref17] 17. Darwin C. On the origin of species. Cambridge: Cambridge University Press; 2009.

[ref18] 18. Ma Z-Q, Xia C-Y, Sun S-W, Wang L, Wang H-B, Wang J. Heterogeneous link weight promotes the cooperation in spatial prisoner’s dilemma. Int J Mod Phys C. 2011;22(11):1257–68.
View Article
Google Scholar

[45] View Article

[46] Google Scholar

[ref19] 19. Nowak MA, Sigmund K. Evolution of indirect reciprocity by image scoring. Nature. 1998;393(6685):573–7. pmid:9634232
View Article
PubMed/NCBI
Google Scholar

[48] View Article

[49] PubMed/NCBI

[50] Google Scholar

[ref20] 20. Ohtsuki H, Hauert C, Lieberman E, Nowak MA. A simple rule for the evolution of cooperation on graphs and social networks. Nature. 2006;441(7092):502–5. pmid:16724065
View Article
PubMed/NCBI
Google Scholar

[52] View Article

[53] PubMed/NCBI

[54] Google Scholar

[ref21] 21. Szolnoki A, Perc M, Danku Z. Making new connections towards cooperation in the prisoner’s dilemma game. Europhys Lett. 2008;84(5):50007.
View Article
Google Scholar

[56] View Article

[57] Google Scholar

[ref22] 22. Zhang J, Zhang C, Cao M. How insurance affects altruistic provision in threshold public goods games. Sci Rep. 2015;5:9098. pmid:25765206
View Article
PubMed/NCBI
Google Scholar

[59] View Article

[60] PubMed/NCBI

[61] Google Scholar

[ref23] 23. Smith JM. Evolution and the theory of games. Am Sci. 1976;64(1):41–5. pmid:1247210
View Article
PubMed/NCBI
Google Scholar

[63] View Article

[64] PubMed/NCBI

[65] Google Scholar

[ref24] 24. Von Neumann J, Morgenstern O. Theory of games and economic behavior: 60th anniversary commemorative edition. Theory of games and economic behavior. Princeton university press;2007.

[ref25] 25. Peng D, Yang H-X, Wang W-X, Chen GR, Wang B-H. Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game. Eur Phys J B. 2010;73(3):455–9.
View Article
Google Scholar

[68] View Article

[69] Google Scholar

[ref26] 26. Rand DG, Dreber A, Ellingsen T, Fudenberg D, Nowak MA. Positive interactions promote public cooperation. Science. 2009;325(5945):1272–5. pmid:19729661
View Article
PubMed/NCBI
Google Scholar

[71] View Article

[72] PubMed/NCBI

[73] Google Scholar

[ref27] 27. Traulsen A, Hauert C, De Silva H, Nowak MA, Sigmund K. Exploration dynamics in evolutionary games. Proc Natl Acad Sci U S A. 2009;106(3):709–12. pmid:19124771
View Article
PubMed/NCBI
Google Scholar

[75] View Article

[76] PubMed/NCBI

[77] Google Scholar

[ref28] 28. Chen Q, Chen T, Wang Y. How the expanded crowd-funding mechanism of some southern rural areas in China affects cooperative behaviors in threshold public goods game. Chaos, Solitons & Fractals. 2016;91:649–55.
View Article
Google Scholar

[79] View Article

[80] Google Scholar

[ref29] 29. Griffin AS, West SA. Kin selection: fact and fiction. Trends in Ecology & Evolution. 2002;17(1):15–21.
View Article
Google Scholar

[82] View Article

[83] Google Scholar

[ref30] 30. Hamilton WD. The genetical evolution of social behaviour. II. J Theor Biol. 1964;7(1):17–52. pmid:5875340
View Article
PubMed/NCBI
Google Scholar

[85] View Article

[86] PubMed/NCBI

[87] Google Scholar

[ref31] 31. Boyd R, Richerson PJ. Group selection among alternative evolutionarily stable strategies. J Theor Biol. 1990;145(3):331–42. pmid:2232821
View Article
PubMed/NCBI
Google Scholar

[89] View Article

[90] PubMed/NCBI

[91] Google Scholar

[ref32] 32. Traulsen A, Nowak MA. Evolution of cooperation by multilevel selection. Proc Natl Acad Sci U S A. 2006;103(29):10952–5. pmid:16829575
View Article
PubMed/NCBI
Google Scholar

[93] View Article

[94] PubMed/NCBI

[95] Google Scholar

[ref33] 33. Choi J-K, Ahn TK. Strategic reward and altruistic punishment support cooperation in a public goods game experiment. Journal of Economic Psychology. 2013;35:17–30.
View Article
Google Scholar

[97] View Article

[98] Google Scholar

[ref34] 34. dos Santos M. The evolution of anti-social rewarding and its countermeasures in public goods games. Proc Biol Sci. 2015;282(1798):20141994. pmid:25429015
View Article
PubMed/NCBI
Google Scholar

[100] View Article

[101] PubMed/NCBI

[102] Google Scholar

[ref35] 35. Szolnoki A, Perc M. Reward and cooperation in the spatial public goods game. EPL. 2010;92(3):38003.
View Article
Google Scholar

[104] View Article

[105] Google Scholar

[ref36] 36. Boyd R, Gintis H, Bowles S, Richerson PJ. The evolution of altruistic punishment. Proc Natl Acad Sci U S A. 2003;100(6):3531–5. pmid:12631700
View Article
PubMed/NCBI
Google Scholar

[107] View Article

[108] PubMed/NCBI

[109] Google Scholar

[ref37] 37. Brandt H, Hauert C, Sigmund K. Punishing and abstaining for public goods. Proc Natl Acad Sci U S A. 2006;103(2):495–7. pmid:16387857
View Article
PubMed/NCBI
Google Scholar

[111] View Article

[112] PubMed/NCBI

[113] Google Scholar

[ref38] 38. Szolnoki A, Perc M. Effectiveness of conditional punishment for the evolution of public cooperation. J Theor Biol. 2013;325:34–41. pmid:23485452
View Article
PubMed/NCBI
Google Scholar

[115] View Article

[116] PubMed/NCBI

[117] Google Scholar

[ref39] 39. Gao J, Li Z, Cong R, Wang L. Tolerance-based punishment in continuous public goods game. Physica A: Statistical Mechanics and its Applications. 2012;391(16):4111–20.
View Article
Google Scholar

[119] View Article

[120] Google Scholar

[ref40] 40. Ren T, Zheng J. Evolutionary dynamics in the spatial public goods game with tolerance-based expulsion and cooperation. Chaos, Solitons & Fractals. 2021;151:111241.
View Article
Google Scholar

[122] View Article

[123] Google Scholar

[ref41] 41. Zhang S, Zhang Z, Wu Y, Yan M, Xie Y. Tolerance-based punishment and cooperation in spatial public goods game. Chaos, Solitons & Fractals. 2018;110:267–72.
View Article
Google Scholar

[125] View Article

[126] Google Scholar

[ref42] 42. Capraro V, Jordan JJ, Rand DG. Heuristics guide the implementation of social preferences in one-shot Prisoner’s Dilemma experiments. Sci Rep. 2014;4:6790. pmid:25348470
View Article
PubMed/NCBI
Google Scholar

[128] View Article

[129] PubMed/NCBI

[130] Google Scholar

[ref43] 43. Wang Y, Chen T, Chen Q, Si G. Emotional decisions in structured populations for the evolution of public cooperation. Physica A: Statistical Mechanics and its Applications. 2017;468:475–81.
View Article
Google Scholar

[132] View Article

[133] Google Scholar

[ref44] 44. Wang Y, Chen T. Heuristics guide cooperative behaviors in public goods game. Physica A: Statistical Mechanics and its Applications. 2015;439:59–65.
View Article
Google Scholar

[135] View Article

[136] Google Scholar

[ref45] 45. Fan R, Zhang Y, Luo M, Zhang H. Promotion of cooperation induced by heterogeneity of both investment and payoff allocation in spatial public goods game. Physica A: Statistical Mechanics and its Applications. 2017;465:454–63.
View Article
Google Scholar

[138] View Article

[139] Google Scholar

[ref46] 46. Lei C, Wu T, Jia J-Y, Cong R, Wang L. Heterogeneity of allocation promotes cooperation in public goods games. Physica A: Statistical Mechanics and its Applications. 2010;389(21):4708–14.
View Article
Google Scholar

[141] View Article

[142] Google Scholar

[ref47] 47. Zhang Y, Wang J, Ding C, Xia C. Impact of individual difference and investment heterogeneity on the collective cooperation in the spatial public goods game. Knowledge-Based Systems. 2017;136:150–8.
View Article
Google Scholar

[144] View Article

[145] Google Scholar

[ref48] 48. Axelrod R, Hamilton WD. The evolution of cooperation. Science. 1981;211(4489):1390–6. pmid:7466396
View Article
PubMed/NCBI
Google Scholar

[147] View Article

[148] PubMed/NCBI

[149] Google Scholar

[ref49] 49. Trivers RL. The Evolution of Reciprocal Altruism. The Quarterly Review of Biology. 1971;46(1):35–57.
View Article
Google Scholar

[151] View Article

[152] Google Scholar

[ref50] 50. Alexander R. The biology of moral systems. New York: Routledge. 2017.

[ref51] 51. Saavedra S, Smith D, Reed-Tsochas F. Cooperation under indirect reciprocity and imitative trust. PLoS One. 2010;5(10):e13475. pmid:21048950
View Article
PubMed/NCBI
Google Scholar

[155] View Article

[156] PubMed/NCBI

[157] Google Scholar

[ref52] 52. Wedekind C, Milinski M. Cooperation through image scoring in humans. Science. 2000;288(5467):850–2. pmid:10797005
View Article
PubMed/NCBI
Google Scholar

[159] View Article

[160] PubMed/NCBI

[161] Google Scholar

[ref53] 53. Chen Q, Chen T, Wang Y. Publishing the donation list incompletely promotes the emergence of cooperation in public goods game. Applied Mathematics and Computation. 2017;310:48–56.
View Article
Google Scholar

[163] View Article

[164] Google Scholar

[ref54] 54. Podder S, Righi S, Pancotto F. Reputation and punishment sustain cooperation in the optional public goods game. Philos Trans R Soc Lond B Biol Sci. 2021;376(1838):20200293. pmid:34601913
View Article
PubMed/NCBI
Google Scholar

[166] View Article

[167] PubMed/NCBI

[168] Google Scholar

[ref55] 55. Romano A, Giardini F, Columbus S, de Kwaadsteniet EW, Kisfalusi D, Triki Z, et al. Reputation and socio-ecology in humans. Philos Trans R Soc Lond B Biol Sci. 2021;376(1838):20200295. pmid:34601915
View Article
PubMed/NCBI
Google Scholar

[170] View Article

[171] PubMed/NCBI

[172] Google Scholar

[ref56] 56. Wang L, Chen T, Wu Z. Promoting cooperation by reputation scoring mechanism based on historical donations in public goods game. Applied Mathematics and Computation. 2021;390:125605.
View Article
Google Scholar

[174] View Article

[175] Google Scholar

[ref57] 57. Jehn KA, Shah PP. Interpersonal relationships and task performance: An examination of mediation processes in friendship and acquaintance groups. Journal of Personality and Social Psychology. 1997;72(4):775–90.
View Article
Google Scholar

[177] View Article

[178] Google Scholar

[ref58] 58. Zhang H, An T, Wang J, Wang L, An J, Zhao J, et al. Reputation-based adaptive strategy persistence can promote cooperation considering the actual influence of individual behavior. Physics Letters A. 2024;508:129495.
View Article
Google Scholar

[180] View Article

[181] Google Scholar

[ref59] 59. Zhou T, Ding S, Fan W, Wang H. An improved public goods game model with reputation effect on the spatial lattices. Chaos, Solitons & Fractals. 2016;93:130–5.
View Article
Google Scholar

[183] View Article

[184] Google Scholar

[ref60] 60. Quan J, Tang C, Wang X. Reputation-based discount effect in imitation on the evolution of cooperation in spatial public goods games. Physica A: Statistical Mechanics and its Applications. 2021;563:125488.
View Article
Google Scholar

[186] View Article

[187] Google Scholar

[ref61] 61. Brandt H, Sigmund K. Indirect reciprocity, image scoring, and moral hazard. Proc Natl Acad Sci U S A. 2005;102(7):2666–70. pmid:15695589
View Article
PubMed/NCBI
Google Scholar

[189] View Article

[190] PubMed/NCBI

[191] Google Scholar

[ref62] 62. Nowak MA, Sigmund K. Evolution of indirect reciprocity. Nature. 2005;437(7063):1291–8. pmid:16251955
View Article
PubMed/NCBI
Google Scholar

[193] View Article

[194] PubMed/NCBI

[195] Google Scholar

[ref63] 63. Wang X, Chen X, Gao J, Wang L. Reputation-based mutual selection rule promotes cooperation in spatial threshold public goods games. Chaos, Solitons & Fractals. 2013;56:181–7.
View Article
Google Scholar

[197] View Article

[198] Google Scholar

[ref64] 64. Liu Y, Chen T. Sustainable cooperation based on reputation and habituation in the public goods game. Biosystems. 2017;160:33–8. pmid:28827161
View Article
PubMed/NCBI
Google Scholar

[200] View Article

[201] PubMed/NCBI

[202] Google Scholar

[ref65] 65. Han W, Zhang Z, Sun J, Xia C. Emergence of cooperation with reputation-updating timescale in spatial public goods game. Physics Letters A. 2021;393:127173.
View Article
Google Scholar

[204] View Article

[205] Google Scholar

[ref66] 66. Quan J, Cui S, Chen W, Wang X. Reputation-based probabilistic punishment on the evolution of cooperation in the spatial public goods game. Applied Mathematics and Computation. 2023;441:127703.
View Article
Google Scholar

[207] View Article

[208] Google Scholar

[ref67] 67. Mui L. Computational models of trust and reputation: Agents, evolutionary games, and social networks. PhD thesis, Massachusetts Institute of Technology. 2002.
View Article
Google Scholar

[210] View Article

[211] Google Scholar

[ref68] 68. van Prooijen A, Ellemers N. Does It Pay to Be Moral? How Indicators of Morality and Competence Enhance Organizational and Work Team Attractiveness. British J of Management. 2014;26(2):225–36.
View Article
Google Scholar

[213] View Article

[214] Google Scholar

[ref69] 69. Grunig J E, Hung-Baesecke C F. The effect of relationships on reputation and reputation on relationships: A cognitive, behavioral study. New York: Routledge; 2015.

[ref70] 70. Diekmann A, Jann B, Przepiorka W, Wehrli S. Reputation Formation and the Evolution of Cooperation in Anonymous Online Markets. Am Sociol Rev. 2013;79(1):65–85.
View Article
Google Scholar

[217] View Article

[218] Google Scholar

[ref71] 71. Butler EA. Temporal interpersonal emotion systems: the “TIES” that form relationships. Pers Soc Psychol Rev. 2011;15(4):367–93. pmid:21693670
View Article
PubMed/NCBI
Google Scholar

[220] View Article

[221] PubMed/NCBI

[222] Google Scholar

[ref72] 72. Vanneste BS. From interpersonal to interorganisational trust: The role of indirect reciprocity. Journal of Trust Research. 2016;6(1):7–36.
View Article
Google Scholar

[224] View Article

[225] Google Scholar

[ref73] 73. Hwang K. Face and Favor: The Chinese Power Game. American Journal of Sociology. 1987;92(4):944–74.
View Article
Google Scholar

[227] View Article

[228] Google Scholar

[ref74] 74. Hua S, Liu L. Coevolutionary dynamics of population and institutional rewards in public goods games. Expert Systems with Applications. 2024;237:121579.
View Article
Google Scholar

[230] View Article

[231] Google Scholar

[ref75] 75. Hua S, Liu L. Facilitating the evolution of cooperation through altruistic punishment with adaptive feedback. Chaos, Solitons & Fractals. 2023;173:113669.
View Article
Google Scholar

[233] View Article

[234] Google Scholar

[ref76] 76. Liu L, Chen X, Szolnoki A. Coevolutionary dynamics via adaptive feedback in collective-risk social dilemma game. Elife. 2023;12:e82954. pmid:37204305
View Article
PubMed/NCBI
Google Scholar

[236] View Article

[237] PubMed/NCBI

[238] Google Scholar

[ref77] 77. Krellner M, Han TA. Words are not wind - how public joint commitment and reputation solve the Prisoner’s Dilemma. Applied Mathematics and Computation. 2025;506:129530.
View Article
Google Scholar

[240] View Article

[241] Google Scholar

Figures

Abstract

Introduction

Model

Problem description

Public goods game model

Reputation update rule

The rules of heterogeneous reputation evaluation

Strategy update rules based on reputation mechanism

Simulation results and discussion

The impact of emotional relationships

The impact of mixed relationships

The impact of instrumental relationships

Reputation mechanism analysis

Robustness of the model

Conclusion and implication

Conclusions

Implications and future research

Theoretical implications

Practical implications

Future research

References