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

TL;DR

The paper addresses the emergence of cooperation in spatial public goods games by coupling reputation-based tolerance with a punishment mechanism. It introduces a reputation threshold $R_0$ that governs third-party punishment and a fitness function $f_i(\Pi_i,R_i)=\delta\Pi_i+(1-\delta)\frac{R_i-R_0}{\lambda}$ to bias imitation toward high payoff and high reputation. Simulations on a square lattice show that higher $R_0$ and stronger punishment environments substantially increase cooperation, with cooperation clustering and eventual dominance in many regimes. The work highlights the synergistic effect of reputation and punishment in sustaining cooperation and suggests directions for more nuanced, group-specific or non-linear extensions.

Abstract

Reputation and punishment are significant guidelines for regulating individual behavior in human society, and those with a good reputation are more likely to be imitated by others. In addition, society imposes varying degrees of punishment for behaviors that harm the interests of groups with different reputations. However, conventional pairwise interaction rules and the punishment mechanism overlook this aspect. Building on this observation, this paper enhances a spatial public goods game in two key ways: 1) We set a reputation threshold and use punishment to regulate the defection behavior of players in low-reputation groups while allowing defection behavior in high-reputation game groups. 2) Differently from pairwise interaction rules, we combine reputation and payoff as the fitness of individuals to ensure that players with both high payoff and reputation have a higher chance of being imitated. Through simulations, we find that a higher reputation threshold, combined with a stringent punishment environment, can substantially enhance the level of cooperation within the population. This mechanism provides deeper insight into the widespread phenomenon of cooperation that emerges among individuals.

Figures (7)

• Figure 1: The node strategy update process in the network. Red nodes represent defectors and blue nodes represent cooperators. (a) All nodes are randomly assigned reputations and strategies. (b) All groups of public goods as circled by the dotted line interact. (c) Punishment is carried out according to the average reputation of the game group and update strategy using \ref{['eq:4']}. Punishment is implemented within the low-reputation group (red dashed box) but not within the high-reputation group (green dashed box). (d) The network is updated, and the next Monte Carlo step is taken.
• Figure 2: Frequency of cooperators $\rho_{C}$ as a function of damping factor $\delta$ for different reputation threshold $R_{0}$. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Note that the more an individual's fitness depends on reputation, the more it promotes the emergence of cooperation.
• Figure 3: Frequency of cooperators $\rho_{C}$ as a function of enhancement factor $r$ for different punishment of $b$. Each data is obtained by averaging the proportion of cooperators in the last 500 iterations after the system reaches evolutionary stability. Different panels display the cooperation level under different reputation thresholds, as (a) $R_{0}=1$, (b) $R_{0}=4$. The curves show the critical values $r$ at which the phase transition from defection to cooperation occurs under different punishment $b$. For instance, the phase transition points for the emergence of cooperation corresponding to $b$ = 0, 0.25, and 0.50 in (a) are $r$ = 1.1, 2.1, and 2.8, respectively.
• Figure 4: Frequency of cooperators $\rho_{C}$ varies with time for different $b$. Fixed enhancement factor $r$ = 2.5. Different panels display the cooperation level under different reputation thresholds, as (a) $R_{0}=1$, (b) $R_{0}=4$. In different punishment $b$, the system can reach evolutionarily stable in $10^{4}$ iterations. For $b$=0, 0.25, 0.5, and 0.75, the frequency of cooperators initially declines and then gradually increases towards a dynamically stable and non-zero level, whereas $b = 0$, the level of cooperation finally drops to zero.
• Figure 5: Snapshots of the spatial arrangements of strategies at four representative moments for different reputation thresholds. (a) $R_{0}$= 0; (b) $R_{0}$ = 4; (c) $R_{0}$ = 12 and (d) $R_{0}$=16 and fixed enhancement factor $r$ = 2.5 and $b$=0.2. Each row from top to bottom represents a situation corresponding to a different reputation threshold $R_{0}$. Each column from left to right represents a different time step $t$. White pixels stand for cooperators and blue pixels for defectors. It is noteworthy that defectors can survive in an environment where no punishment is introduced ($R_{0}$ = 0). However, in an environment with a higher reputation threshold ($R_{0}$ = 4, 12, and 16), after 100 time steps, cooperators gradually gather to form stable clusters. Thus, a strict reputation evaluation environment can induce the formation and development of cooperator clusters.