Supervised cooperation on interdependent public goods games

TL;DR

This work addresses the challenge of achieving global cooperation among self-interested agents by modeling a two-layer interdependent network: a spatial public goods game layer and a monitoring referee layer with fair and corrupt referees. Cooperation and fairness coevolve through imitation-based updates in both layers, with payoffs shaped by a punishment ratio $\alpha$, bribery cost $\beta$, and supervision fee $m$, all integrated via explicit payoff formulations $P_{i,j}$ and $\Pi_i$. The key contributions show that large $\alpha$ and $\beta$ significantly reduce the defection/corruption phase and induce rapid transitions to cooperative and fair states, highlighting interdependent network reciprocity as a mechanism for sustaining cooperation. These findings offer insights for designing incentives and governance structures to promote cooperation and curb corruption in interconnected social-ecological systems.

Abstract

It is a challenging task to reach global cooperation among self-interested agents, which often requires sophisticated design or usage of incentives. For example, we may apply supervisors or referees who are able to detect and punish selfishness. As a response, defectors may offer bribes for corrupt referees to remain hidden, hence generating a new conflict among supervisors. By using the interdependent network approach, we model the key element of the coevolution between strategy and judgment. In a game layer, agents play public goods game by using one of the two major strategies of a social dilemma. In a monitoring layer, supervisors follow the strategy change and may alter the income of competitors. Fair referees punish defectors while corrupt referees remain silent for a bribe. Importantly, there is a learning process not only among players but also among referees. Our results suggest that large fines and bribes boost the emergence of cooperation by significantly reducing the phase transition threshold between the pure defection state and the mixed solution where competing strategies coexist. Interestingly, the presence of bribes could be as harmful for defectors as the usage of harsh fines. The explanation of this system behavior is based on a strong correlation between cooperators and fair referees, which is cemented via overlapping clusters in both layers.

Figures (6)

• Figure 1: Interdependent approach to model coevolution between strategies and judgments. The lower layer is the stage of social dilemma where cooperator and defector players interact via spatial public goods game. The upper layer hosts supervisors or referees who monitor the strategy evolution and mediate it indirectly. Panel (a) illustrates the case when a fair referee punishes defectors in the corresponding group. Panel (b) shows defectors' response who pay a bribe to a corrupt referee hence avoid being punished.
• Figure 2: Cooperation level and the portion of fair supervisors in dependence of synergy factor at different fine values. Panel (a) shows the fraction of cooperators in the game layer as we increase the synergy factor $r$. The values of fine for defectors are indicated in the legend. Panel (b) depicts the fraction of fair referees in the monitoring layer. Other parameters, the supervision fee $m=0.5$ and the bribery cost $\beta=0.2$ are fixed. The stationary values are calculated from the last 500 steps of 3000 total steps and averaged over 10 independent simulations. The gray dashed line in Panel (a) indicates the baseline without referees.
• Figure 3: Cooperation level and the portion of fair supervisors in dependence of synergy factor at different bribe values. Panel (a) shows the fraction of cooperators in the game layer, while panel (b) depicts the fraction of fair referees in the monitoring layer. The values of bribe paid by defectors to avoid punishment are marked in the legend. The remaining parameters, the supervision fee $m=0.5$ and the punishment level $\alpha=0.2$ are fixed. The stationary values are calculated from the last 500 steps of 3000 total steps and averaged over 10 independent simulations. The gray dashed line in Panel (a) indicates the baseline without referees.
• Figure 4: The consequence of heterogeneous supervisors on fine-bribery parameter plane. The left panels show the color-coded portion of cooperators in the game layer on $\alpha-\beta$ parameter plane. Right panels show the color coded density of fair referees in the monitoring layer at the same fine-bribery parameter pairs. Top row shows the case obtained at $m=0.3$ supervision fee, while bottom row indicates when $m=0.7$. The synergy factor was fixed $r=2.5$ for all cases. The cooperation frequency for each data point is averaged from the last 500 steps of 3000 total steps and averaged over 10 independent simulations.
• Figure 5: Coevolution of cooperation and fairness. Starting from a random initial state panels (a)-(d) show the actual distribution of strategies in the game layer obtained at $t\in\{20,50,100,1000\}$ respectively. Here blue (resp. white) cells represent cooperator (resp. defector) agents. In the bottom row panels (e)-(h) denote the distribution of competing referees in the monitoring layer obtained at the same steps specified above. Here red (resp. linen) color represents fair (resp. corrupt) referees. Other parameters are $r=3.4$, $m=0.5$, $\alpha = 0.1$, and $\beta = 0.2$. As the panels show, cooperation and fairness evolve hand in hand by supporting each other.