Cooperative emergence of spatial public goods games with reputation discount accumulation

To address issues like the tragedy of the commons, reputation mechanisms prove effective in promoting the emergence of cooperative behavior in social dilemmas. Previous research assumed that the increment of reputation is a fixed constant, however, the historical behavior of individuals with time preferences affects their reputation fluctuations on different scales. Inspired by this, we propose a framework for spatial public goods game that incorporates a reputation discount accumulation model with time preferences. In this model, players are classified as either long-sighted players or short-sighted players based on the significance of their historical behavior. Compared with short-sighted players, long-sighted players are more concerned with the impact of historical behaviors on the next game. Simulation results show that long-sighted cooperative players can resist the invasion of short-sighted defectors, and short-sighted defection strategies are eventually replaced by long-sighted cooperative strategies. This indicates that a higher discount factor facilitates the generation and maintenance of cooperation.


Introduction
The emergence of cooperation is a fundamental theme in the analysis of complex interactive systems, encompassing multiple fields such as biology, sociology, and economics [1][2][3][4].Unfortunately, the pursuit of self-interest by individuals often leads to conflict with the public interest, posing a significant obstacle to the emergence and maintenance of cooperation [5][6][7][8].Evolutionary game theory is an effective method to solve cooperative dilemmas and has been extensively used in studying the prisoner's dilemma game (PDG) [9][10][11], snowdrift game (SDG) [12][13][14], and public goods game (PGG) [15,16].It has advanced the study of the evolution of social behavior [17][18][19][20].
Since the introduction of the image-scoring reputation model by Nowak and Sigmund [40], reputation mechanisms have been proven effective in increasing collective cooperation levels in various social dilemmas [41,42].In recent years, scholars have suggested several social dilemma models that rely on reputation mechanisms.On the one hand, several studies have shown that second-order evaluation of reputation [43], co-evolution of reputation [44], and reputation consistency strategy [45] can promote cooperative behavior in the PDG.On the other hand, strategy persistence [46] and second-order evaluation of reputation [47] improve the level of cooperation in the SDG.Furthermore, relevant mechanisms under the PGG have been established to counteract temptation to defect and promote cooperation, such as reputation inference [48], heterogeneous investment [49], probabilistic punishment [50], historical donations [51], and payoff transfers [52], etc.In particular, Perc et al summarize the research on cooperative dilemmas under the mechanism of reputation and reciprocity [53], and study the emergence of cooperative behaviors under reputational preferences [54], moral mechanisms [55] and second-order reputational evaluations [43], which provides a good guideline for our work.
Reputation as an evaluation of individual historical behavior can effectively promote the generation and maintenance of cooperation.In most studies, changes in reputation depend only on an individual's current strategy: cooperation or defection, without regard to his historical behaviors.That is, the players' reputation at the current moment is formed by accumulating the reputation of each past period after full discounting, and the player has a complete memory of the behavior of each past period.However, such an assumption is too ideal and absolute, which is far from reality.Specifically, the increment of reputation is set to a fixed value of 1. Choosing a cooperative strategy increases it by 1, while choosing defection decreases it by 1.In other words, the scale of reputation fluctuation in reality will fluctuate with the individual's behavior and experience.
Specifically, according to the principle of bounded rationality, the player's memory of past periods' behavior is time-preferred.Namely, an individual's reputation in the present moment is a time discount on historical behavior, with the closer to the current moment the greater the discount value and vice versa.For this reason, we classify participants into long-sighted and short-sighted players according to the importance they attach to their historical behaviors.Long-sighted players have a strong time preference for historical behaviors, and they care about the impact of their historical behaviors on their reputation.Meanwhile, short-sighted players tend to only consider the impact of their recent behaviors on their reputation.The following example illustrates all this more vividly.In real life, we judge a person's reputation by his past historical actions.Because people have a time preference for historical behavior, he attaches the most importance to the behavior of the last period (equivalent to multiplying by the discount factor δ); he attaches importance to the behavior of the penultimate period (equivalent to multiplying the discount factor δ 2 ), and one more time forward, his importance continues to decline (equivalent to multiplying the discount factor δ i ); by analogy, the farther the game period, the smaller the discount factor multiplied by that period.
In the proposed model, we introduce a reputation discount factor δ to evaluate how much players value their historical behavior.Participants identify their imitated neighbors by comparing the differences in income and reputation with the interaction objects.Specifically, in a reputation-based game environment, players are penalized for adopting short-sighted strategies, while long-sighted players form clusters due to the cumulative advantage of reputation, effectively resisting the invasion of defectors.Simulation results show that the proportion and payoff of players will increase with the increase of the reputation discount factor.
The paper is structured as follows.Section 2 presents a model of reputation discounting in spatial PGGs with time preferences.Section 3 outlines simulation experiments that analyze the cooperation phenomenon under varying reputation discounting values.Finally, section 4 provides an overview of the conclusions of the paper.

Spatial public goods game
Spatial public goods game (SPGG for short) is a two-strategy game with multiple participants, which is used to describe the cooperative dilemma of spatially structured populations.We first establish an evolutionary SPGG in a structured population: all players are at a position in an L × L regular square lattice with periodic boundary conditions.When starting a strategy interaction, each player is assigned the same probability of being a cooperator S i = C or a defector S i = D.The payoffs of each player (P i ) in the SPGG follow a specific rule.According to this rule, each player is part of G i = k i + 1 PGG groups.One group revolves around the player (known as the focal player), while the other k i groups focus on their k i closest neighbors (k i = 4).All participants in a public goods game must make a simultaneous decision on investing a fixed share in the public resource pool.For simplicity, players adopting a cooperative strategy (S i = C) contribute one unit; otherwise, they contribute no units (S i = D).We calculate the total contribution within the PGG group by counting the number of participants and multiplying it by the enhancement factor γ > 1. Subsequently, we distribute the total investment equally among all participants in the group, regardless of their individual contribution.Consequently, player i receives the following cumulative return: where γ represents the enhancement factor, N c denotes the number of collaborators, G i reflects the total number of SPGG groups to which the player belongs, and S i indicates the strategy adopted by player i; subsequently, P i stands for the payoff centered on i.Each player plays solely with its neighboring nodes that are connected by edges, and a common pool is formed by that player and its neighbors.For each player, a cooperator pays more than a defector in each round of the game.A player's payoffs for each round of the game come from separate groups centered on his and his neighbors.
In each round, the player's total payoff π i is the sum of the payoffs from each SPGG group in which he participates, where PN j i is the payoffs from the game centered on neighbor j in which i participated.Player i has M i neighbors, here M i > 0.
From the equations ( 1) and ( 2), it can be seen that the cumulative payoff π iC for choosing a cooperative strategy when a player participates in a single game can be expressed as: where N i icc represents the number of players choosing cooperative strategy, and here G i = G j i = 5.By equation (3), the cumulative gain for a player who chooses the defection strategy is:

Reputation update
Maintaining a good reputation is crucial for every player.In real life, people are more inclined to cooperate with those who have a good reputation.Moreover, enhancing one's reputation leads to increased revenue.Reputation updating involves a two-round discounting process that takes place over consecutive games, characterized by the strategies employed by both players and their neighbors which affect the change in reputation.
To consider the reputation effects in strategy selection, we design a parameter R i (t) that represents the reputation of player i at time step t, where the initial value is set as 0. A player's reputation at time step t(t ⩾ 1) relies on both the player's historical memory and their current strategy choice.
In reality, a player's reputation is influenced by his prior behavioral decisions, which is a process of discounting and aggregating the effects of past reputation.Antecedent decisions affect the player's reputation, reflecting some time preference of the participant, which discounts and accumulates the effects of the participant's historical behavior.This implies that the player's reputation at the current moment is equal to the reputation resulting from the player's current choice of strategy plus the discounting of past reputation.Each player has a reputation in each game period, and it is necessary to aggregate the player's reputation in different game periods.At moment t, player i attaches the most importance to the impact of the current moment's decision on reputation (equivalent to multiplying by δ 0 ), the next most importance to the impact of reputation at moment t − 1 (equivalent to multiplying by δ 1 ), the second most importance to moment t − 2 (equivalent to multiplying by δ 2 ), and so on, with the reputations multiplying by a smaller impact factor the earlier they are from the current game period.Specifically, the discounted summation method is used to calculate.
First, we define a symbolic function h i (t): Second, the reputation score of player i at moment t = k is: where T is the basic unit of reputation discounting for each game, which can be interpreted as the amount of units of reputation discounting for the players' decisions in each period, and is noted here as T = 1.
Finally, the reputation score of player i at the moment t = k + 1 is as follows: From equation ( 7), it can be seen that the reputation value of player i at moment t = k + 1 is a discount of the reputation value at moment t = k plus the effect of the current strategy on the reputation, i.e.
We discuss the range of values of reputation according to equation ( 7).
(1) If player i always chooses a cooperative strategy, then For the change of δ, the above conclusion can be further discussed: (1) When δ → 0, players who persistently cooperate get a maximum reputation T and players who persistently defect get a minimum reputation −T.The reputation of the strategy-adjusted player is at (−T, T).This means that when the discount factor is smaller, past behaviors have little effect on current reputation, and players are overly myopic, considering only the role of behaviors closest to the current moment.
(2) When δ → 1, the reputation of a consistently cooperative player increases infinitely and the reputation of a consistently defector player decreases infinitely.The reputation of the strategy adjustment player is at (−∞, ∞).This indicates that as the discount factor gets larger, the current reputation is profoundly affected by past behavior.Players have memory and fully take into account the impact caused by past strategies on reputation.(3) When δ = 0, R i (k + 1) = h i (k + 1) T = ±T, this indicates that the player only considers the role of the current strategy in each decision.That is, choosing a cooperative strategy results in a reputation of T, and choosing a defecting strategy results in a reputation of −T.(4) When δ = 1, the degradation of R i (k) follows the common incremental form T. This indicates that players take into account the impact of past strategies for each decision, with reputation increasing by T for cooperation and decreasing by −T for defector.

Strategy update
Many participants, including enterprises, governments, and social organizations, view reputation as an asset equally important as earnings, with its significance even surpassing that of earnings.A researcher's personal career growth depends on their academic reputation, and the future prospects for advancement are directly influenced by the reputation level of the journal.It can be seen that the player's reputation is an important basis for exchanging trust, and the level of the player's reputation directly reflects his trustworthiness, and the player continues to adopt cooperative behaviors in order to obtain longer-term payoffs.
During interactions among players, personal reputation strongly affects individual decision-making, and players adjust their strategies accordingly.Those who apply the Fermi update rule imitate the strategy of the most profitable player, but this rule is too rational to satisfy certain personal preferences.Consequently, we slightly modify the Fermi rule to integrate game payoffs and personal reputation.This indicates that players update both their reputation and strategy based on their own and their neighbors' reputation and payoff.Player i selects a neighbor j randomly and imitates the strategy of j with probability P ( S j (t) → S i (t) ) as shown in equation (8).
Based on the discussion of reputation accumulation in section 2.2, it is argued that reputation is of equivalent importance to payoff.In most cases, discounting with a time preference does not lead to excessive absolute values of reputation.To say the least, to avoid potential deviations from reality caused by reputation accumulation, the players' reputations are limited to the interval [R min , R max ].It is also crucial to compare the impact of reputation value and payment value on the strategy update function during actual model calculations.This is to avoid the player's reputation value range being too large due to reputation accumulation, thus making the contribution of the payoff to the strategy update negligible.
Players update their strategies asynchronously, and player i prefers to imitate the strategy S j (t) that yields a higher fitness.Then, the probability of adopting the strategy S j (t) is given by equation ( 8) where the noise intensity K quantifies the degree of rationality in the strategy selection process.When K → 0 the perfectly rational player j fully imitates the strategy of the more profitable player j.Conversely, when K → ∞ , W j →i (t) → 0.5, the player chooses his strategy completely at random and learns little from his neighbors.Here K = 0.5 .

Cooperative emergence with different enhancement factors
We will conduct a systematic simulation experiment to investigate the effects of a reputation-based discounted accumulation mechanism on facilitating the evolution of cooperation.We first describe a population of size N = 10 000 as a 100 × 100 square lattice.Each player is randomly assigned to one of two strategies {C, D} with equal probability.To ensure accuracy, the evolution process is iterated in 10 4 time steps.We describe the underlying logic of strategy change over time in SPGG in terms of a 5 × 5 network, as shown in figure 1. Player i invests in a total of five groups centered on himself and his neighbor nodes j, k, s, t, respectively.First, we examine the effect of the enhancement factor γ on the proportion of collaborators ρ C , where δ = 0.3 and 0.5.When δ is constant, the proportion of cooperators increases as the enhancement factor increases, showing an increasing trend from bottom to top.In figure 2, the proportion of cooperators ρ C shows an increasing trend as γ increases from 1.5 to 3.5.In other words, the greater the proportion of reward for cooperative behavior, the more conducive to promoting cooperation.
Figure 3 provides a snapshot for the parameter δ = 0.3, with the yellow and blue patch areas representing cooperators and defectors, respectively.We set the time steps M to 5, 50, 500, 1000, 2000, and 10 000, and the enhancement factors γ = 1.5, 2.0, 2.5, 3.0, and 3.5.At the beginning of the strategy interaction we set ρ C = ρ D = 0.5.In the first column of subgraphs where M = 5, ρ C is proportional to γ, and defectors are dominant.When M = 50, ρ C is 0, 0.4, 0.8, 0.9, and 0.97 from top to bottom.At this point, ρ C grows rapidly with γ.When M = 500 and γ = 3.5, ρ C almost reaches 100%.Additionally, after M = 500, cooperative evolution essentially enters a stationary phase.Specifically, the state of population evolution approaches full cooperation (ρ C = 1) when M = 1000, γ = 3.0 and 3.5.
Compared with figure 3, when δ = 0.5, the proportion of cooperators increases rapidly with γ.In figure 3, when δ = 0.3, γ = 1.5, all participants eventually defected.Whereas in figure 4, when δ = 0.5, γ = 1.5, the group of cooperators and defectors were almost equally divided, achieving coexistence.It should also be noted that γ = 2.0 can achieve full cooperation at around M = 2000.Therefore, we conclude that the enhancement factor promotes the creation and maintenance of cooperation from the perspective of player revenue, and the increase of the discount factor also enhances the effect of γ on cooperation.This is manifested in two ways: on the one hand, it accelerates the time for the generation of cooperative groups.For example, when M = 50, the proportion of cooperation in the five subgraphs increases rapidly from the top down.On the other hand, it also increases the proportion of cooperators, and cooperation dominates the entire network even if the value of γ is small.

Cooperative emergence with different reputation discounting factors
In addition to analyzing the impact of the enhancement factor, we further analyze the discount factor for reputation.In figure 5, we denote ρ C as a function of δ over time t.In this paper, we consider the parameters γ = 2 (figure 5(a)) and γ = 2.5 (figure 5(b)) , with δ = 0.1, 0.3, 0.5, 0.7, and 0.9.In figure 5(a), ρ C first decreases significantly with increasing δ and starts to rise after t = 10.After t = 100, the population gradually enters an evolutionary stable state regardless of the value of δ.It can be seen that the reputation of early players has not yet been effectively accumulated, and most defectors take advantage of their collaborators to maximize their personal interests.Before t = 10, a large number of defectors gradually dominate the network due to free riding on the contributions of collaborators.And after t = 10, the cumulative effect of reputation   induces surviving cooperators to reverse the situation and recover their share of the space.This suggests that reputation plays a key role in players' strategic choices, prompting defectors to gradually choose to cooperate, and the proportion of cooperative players will be substantially increased over time.Compared with figures 5(a) and (b) shows that a larger enhancement factor accelerates the time to evolve to the steady state.In conclusion, reputation discounting with time preference has a significant contribution to the emergence of cooperation, and players who pay more attention to the influence of historical behaviors accumulate more benefits.To better demonstrate the evolutionary pattern 5 and 6 shows the spatiotemporal distribution of cooperation and defection in the SPGG at different MCSs when γ = 2.0.The five sets of subgraphs are the experiment results with respect to δ = 0.1, 0.3, 0.5, 0.7 and 0.9, respectively.
In the first row, reputation accumulation of cooperators is insufficient to withstand invasion by defectors, and cooperative clusters shrink to small clusters and disappear in subsequent time steps.In the second row, scattered cooperators banded together to form clusters against the invasion of defectors.Then, the cooperative clusters expand rapidly, while the cooperative clusters do not eliminate the defectors and coexist with them.In the third row, the power of reputation becomes stronger, prompting the cooperators to gradually form huge clusters and exclude the defectors.Then, the system eventually reaches a state of full cooperation.When δ approaches 1, the cooperative state can be achieved quickly, and it can be seen that the full cooperative state is finally achieved at t = 100.After continuous evolution, finally, the fraction of cooperation in the whole population keeps stable, and the larger δ is, the larger ρ C is.Further, corresponding to the evolution trend in figure 6, cooperative clusters form faster when γ increases from 2.0 to 2.5.
To summarize, if we divide players into long-sighted and short-sighted players, the discounted value of long-sighted players' reputations for their historical behaviors tends to 1, while short-sighted players tend to 0. As shown in figures 6 and 7, when the discount factor is greater than 0.5, the group eventually converges to a fully cooperative state.Long-sighted cooperative players organize large compact clusters to protect themselves from defectors.

Correlation analysis and strategy evolution
Since both the reputation factor and the enhancement factor have a facilitating effect on cooperation emergence, we analyze the evolutionary pattern of the cooperation ratio ρ C with simultaneous changes of the  The values of δ from top to bottom are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively.Comparing with figure 6, the level of cooperators is greatly improved when the enhancement factor increases from γ = 2.0 to γ = 2.5.two parameters through figure 8. Figure 8(a) depicts the level of cooperation when δ increases from 0.1 to 0.9 for successive values of γ.At five different values of δ, the level of cooperation shows an increasing trend with significant differences.Specifically, the level of cooperation at δ = 0.1 increases rapidly between γ = 2 and γ = 2.9.Similarly, the level of cooperation at δ = 0.3 increases significantly between γ = 1.8 and γ = 2.6, and the population enters an evolutionary steady state (ρ C = 1) after γ = 2.6.When δ > 0.5, even smaller gain factors can effectively promote the emergence of cooperation, when δ = 0.7 (δ = 0.9), and the population enters a fully cooperative state after γ = 1.5.In other words, the more players emphasize the influence of historical strategies (the larger δ is), the more favorable the emergence and maintenance of cooperation is.For comparison, figure 8(b) depicts the level of cooperation when γ increases from 1.5 to 3.5 at successive values of δ.It explores how the enhancement factor γ affects the evolution of cooperation at different values of discounting intensity δ.Taken together, the two figures show that in our proposed model of reputation discount accumulation with time preference spawns a more cooperative group, and also mitigates to some extent the side effects caused by increasing the enhancement factor.
After examining figure 8, we determine that cooperative evolution is affected by the enhancement factor and reputation adjustment.Consequently, we investigate the heat map illustrating variations in cooperative regions in the δ − γ parameter plane depicted in figure 9. From this figure, we conclude that increasing both δ and γ support the establishment of fully cooperative states.Specifically, for any fixed γ, the larger the δ, the more players value the impact of past behavior on current reputation.This implies that long-sighted players  are more likely to form tight cooperative clusters, while short-sighted players are more likely to adopt defection strategies.Additionally, the enhancement factor directly affects the player's payoff for any given δ value.As δ increases, the benefits also increase, enhancing cooperators' adaptability and the probability of creating cooperative clusters.Figure indicates that the blue area in the lower left corner denotes the group of defectors, who are completely short-sighted, the yellow area in the upper right corner denotes the players who adopt long-sighted strategies, and the intersection of the two colors indicates the coexistence of the two strategies.In the evolutionary process, what appears to be a game between cooperators and defectors is essentially a confrontation between the players between the two factors of reputation and payoff.
For different reputation players, the crucial question is how to adjust their strategies based on their time preferences of reputations.The transfer of strategy pairs includes four forms: cooperator-cooperator (C-C), defector-defector (D-D), and defector-cooperator (D-C).Figure 10 shows the size of the four strategy pairs in the population under γ = 2.5 and the heterogeneous reputation discount δ.We can draw the following interesting conclusions from the figure.First, when δ = 0 (figure 10(a)), absolutely short-sighted players focus only on the current strategy and completely ignore the impact of past behaviors on reputation, which leads to the coexistence of cooperative and defecting players in the system.When the reputation discounting factor δ = 1 (figure 10(f)), absolutely long-sighted players have a complete memory of all past historical behaviors, and full reputation discounting allows the system to quickly achieve full cooperation.Second, at the beginning of the game, all players prefer to adopt the short-sighted strategy of profiting more because of insufficient reputation accumulation.After reputation is accumulated over time, players with short-sighted strategies will have to change to cooperators due to reputation disadvantage.At the same time, the new cooperative players and the original long-sighted players stick to their own strategies because of the higher profit.Third, as can be seen from figure 10, as δ increases, the proportion of long-sighted players persistently adopting cooperative strategies increases sharply, while the proportion of short-sighted players persistently From panel (a) to (f), the discount factor are set to be 0, 0.2, 0.4, 0.6, 0.8 and 1.0, respectively.The proportion of players choosing long-sighted and short-sighted strategies eventually converges to an evolutionary stable state.here γ = 2.5.
adopting defection strategy strategies decreases rapidly.In other words, the reputation accumulation mechanism with time preference promotes more players to adopt long-sighted strategies, thus making short-sighted strategies disappear from the system.Finally, the strategies will evolve in a dynamic equilibrium with players maintaining a smooth transition between strategy pairs.Additionally, once evolution is in equilibrium, the ratio of players holding strategy pairs C-D and D-C will be essentially equal.
Specifically, the proportion of C-C strategy pairs is 0.82 and the proportion of D-C strategy pairs is 0.12 when δ = 0.2.When δ is increased to 0.4, the C-C strategy pairs take over the entire system and the proportion of D-C strategy pairs tends to 0 at MCS = 1000.Furthermore, as δ increases to 0.6 and 0.8, the system reaches equilibrium over time with a maximum density of C-C strategy pairs and disappearance of D-D strategy pairs.Meanwhile, as δ increases, short-sighted players quickly shift to long-sighted strategies, while long-sighted players prefer to adopt cooperative strategies consistently.The cumulative effect of reputation discounts can accelerate the proportion of cooperators surpassing defectors, ultimately leading to full cooperation.That is, as the cumulative utility of reputation increases, players can transition from short-sighted to long-sighted strategies more quickly, and with less time for the system to achieve global cooperation.

Conclusion and outlooks
This paper introduces a novel reputation discount model for investigating the critical aspects related to the emergence and maintenance of cooperation in the SPGG.Specifically, we introduce the concept of reputation-based discounted accumulation in evolutionary SPGG, which emphasizes the fact that reputation discounting has a time preference for past historical behavior, i.e. the payoff of reputation is positively correlated with the reputation discount factor.The focal player i chooses a neighbor j based on his fitness and updates his current strategy using a modified Fermi rule.Numerical simulations demonstrate that the reputation discounting factor and enhancement factor in SPGG facilitate cooperation emergence and mutually reinforce each other in promoting cooperation.
Compared to existing models, our model points to the fact that players with a long-term time preference are able to accumulate reputation quickly, facilitating the emergence of cooperative group behavior.In particular, when the reputation discount factor becomes large, players can be rewarded or punished in a reasonably timely manner, which promotes the rapid emergence of cooperation.The reputation accumulation model with time preference replaces the fixed growth pattern with the nonlinear growth of reputation, reflecting the rationality of players with time preference.
Furthermore, we draw some conclusions to demonstrate the physical insights of the proposed model.Long-sighted cooperators can resist the invasion of short-sighted defectors, and short-sighted defection strategies will eventually be replaced by long-sighted cooperative strategies.Hence, a player who emphasizes its reputation is more likely to achieve a higher degree of adaptation.A larger reputation discount factor accelerates the emergence of cooperation.While the reputation mechanism can prompt cooperation, players' reputation discounting is limited by the principle of bounded rationality.Additionally, we observe a reciprocal relationship between the enhancement factor and the reputation discounting factor.Players can increase the proportion of cooperative behavior by adopting either a high reputation discounting with a low enhancement factor or a high enhancement factor with a low reputation discounting.
We upgraded the reputation mechanism in the classical model to an evaluation of players' historical behaviors, examining cooperation emergence under different reputation discounting.The model addresses the question of how to evaluate the time preference of reputation under the assumption of bounded rationality.Nevertheless, we will further improve the following five aspects of the study in future work.First, based on different network structures in reality, we will consider our model under small-world networks, scale-free networks, and even dynamic temporal networks.Second, players in the real world cannot play the game indefinitely, and we should analyze more deeply the effect of players' reputation discounts on cooperative behaviors in finite games.Third, due to the emergence of moral preference behaviors such as one-time altruistic behavior, we will consider moral mechanisms such as truth-telling, altruistic punishment, and trustworthiness in future models.Fourth, the corresponding reputation discounting problem can be considered to be analyzed in the Prisoner's Dilemma game.Finally, the model is improved based on the heterogeneity of reputation discounts among different participants.

Figure 1 .
Figure 1.Example of a 5 × 5 square lattice with N = 25 in PGG.In this model, player i invests in a total of five groups centered on himself and his neighbor nodes j, k, s, t, respectively.Yellow circles indicate cooperators (C) and blue circles indicate defectors (D).

Figure 3 .
Figure 3. Snapshots of the strategy spatial distributions for different enhancement factors at five representative time steps, where γ = 1.5, 2.0, 2.5, 3.0, 3.5.Cooperators and defectors are represented by yellow and blue dots, respectively.When δ = 0.3, players are at a lower level of reputation discounting, and the fraction of cooperative players is restricted by the increase in the enhancement factor.

Figure 6 .
Figure 6.Representative snapshots of the spatial distributions of cooperators (yellow) and defectors (blue), as obtained for different values of δ when γ = 2. From the top to the bottom the values of δ are 0.1, 0.3, 0.5, 0.7 and 0.9, and the number of MCS in the columns from left to right are 5, 50, 500, 1000, 2000, and 10 000, respectively.

Figure 7 .
Figure 7. Snapshots of the characteristics of cooperators and defectors on the spatial rule lattice at different discounted values of reputation when MCS = 5, 50, 500, 1000, 2000, and 10 000.Blue dots represent defectors and yellow dots represent collaborators.The values of δ from top to bottom are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively.Comparing with figure6, the level of cooperators is greatly improved when the enhancement factor increases from γ = 2.0 to γ = 2.5.

Figure 8 .
Figure 8. Analyzing the evolutionary trend of the proportion of cooperators under two-parameter interactions.(a)When the enhancement factor γ varies continuously in the interval [1.5,3.5], the proportion of cooperation increases with the discount factor δ. (b) When the discount factor δ varies continuously within the interval [0,1], the proportion of cooperation increases as the enhancement factor γ increases.There is mutual enhancement between parameter pairs (γ, δ).

Figure 9 .
Figure 9. Heat map of the three regions under the two-parameter δ − γ.Full cooperators are represented by Pure yellow regions.Pure blue regions represent full defectors.The remaining regions represent states where cooperators and defectors coexist.(For an explanation of the colors in the figure, see the web version of this article.).

Figure 10 .
Figure 10.Trends of four player strategy pairs transferring under different δ.The X-axis indicates MCS and the Y-axis indicates the proportion of cooperation.From panel (a) to (f), the discount factor are set to be 0, 0.2, 0.4, 0.6, 0.8 and 1.0, respectively.The proportion of players choosing long-sighted and short-sighted strategies eventually converges to an evolutionary stable state.here γ = 2.5.