Cooperation in public goods game on regular lattices with agents changing interaction groups

TL;DR

This paper investigates how cooperation emerges in a Public Goods Game when agents can change their interaction groups on regular lattices. It extends prior spatial interaction-diversity models by allowing a subpopulation of roaming agents to reevaluate their interaction neighborhoods, in both local and global settings. The results show that interaction-group diversity generally promotes cooperation and that there is an optimal roaming level $\delta$ that minimizes the synergy factor $r$ required for high cooperation; excessive roaming can degrade cooperation and reduce stability. The findings offer a mechanism to understand cooperation in systems without reputation or punishment mechanisms, with implications for social networks and ecological interaction networks.

Abstract

The emergence of cooperation in the groups of interacting agents is one of the most fascinating phenomena observed in many complex systems studied in social science and ecology, even in the situations where one would expect the agent to use a free-rider policy. This is especially surprising in the situation where no external mechanisms based on reputation or punishment are present. One of the possible explanations of this effect is the inhomogeneity of the various aspects of interactions, which can be used to clarify the seemingly paradoxical behavior. In this report we demonstrate that the diversity of interaction networks helps to some degree to explain the emergence of cooperation. We extend the model of spatial interaction diversity introduced in [L. Shang et al., Physica A, 593:126999 (2022)] by enabling the evaluation of the interaction groups. We show that the process of the reevaluation of the interaction group facilitates the emergence of cooperation. Furthermore, we also observe that a significant participation of agents switching their interaction neighborhoods has a negative impact on the formation of cooperation. The introduced scenario can help to understand the formation of cooperation in the systems where no additional mechanisms for controlling agents are included.

Figures (5)

• Figure 1: Impact of roaming agents on the formation of cooperation in the local scenario. The interaction group is selected from von Neumann (left panels) and Moore (right panels) neighbourhoods. Plots represent the average number of cooperators after $2^{10}$ steps, on $d=L\times L = 64\times64$ grid with periodic boundary conditions. The case $\delta=0$ in \ref{['fig:random-local-roaming-fd']} corresponds to the results obtained in shang2022cooperation for the situation where the interaction neighbourhoods are random and fixed. In the case \ref{['fig:random-local-roaming-diff']} the participation of roaming agents is even more important than in the case of the Fermi imitation function and it is necessary to increase the fraction of cooperators in the population. One can also note that in the asynchronous update mode \ref{['fig:random-local-roaming-fd-ascync']} the effect of the roaming agents is still present, albeit it is lees visible and almost vanishes for the Moore neighbourhood in the considered range of parameters.
• Figure 2: Impact of the roaming agents in the global scenario with the fixed number of interacting agents. In this case the effect of boosting cooperation by introducing roaming agents is clearly visible. However, for the larger values of $k$ the system is struggling to achieve a high rate of collaboration. Results presented in the figure were obtained for the model with $d=L\times L=32\times 32$ and periodic boundary conditions, averaged over 50 realizations, each consisting of $2^{11}$ steps, synchronous update mode.
• Figure 3: Impact of the increasing participation of roaming agents on the formation of cooperative behaviour in the case of random interaction groups. The results represent values of the average in last $2^{10}$ steps, averaged over 50 realizations, for $K=4,6,8,10$ and for $0\leq\delta\leq0.4$. Plots were obtained for square $L\times L$ grid, with $L=64$ and periodic boundary conditions. Each configuration was evaluated for $2^{11}$ steps. Average value of cooperators in the last $10^3$ steps was calculated over $50$ realizations, synchronous update mode.
• Figure 4: Optimal participation of roaming agents minimizing the synergy factor required for achieving the cooperation with $0.90$ (blue crosses) and $0.98$ (red circles) for increasing values of maximal interaction neighbourhood size $K$. In the left plot the data were obtained for case with $L=16$ using the averaging over 150 realizations. In the middle plot the data were obtained using $L=32$ and in the right plot data were obtained for $L=32$. Data for $L=32$ and $L=64$ were obtained from averaging over 50 realizations.
• Figure 5: Comparison of the static model from shang2022cooperation and the model with the roaming agents in the local and the global variant. The plots where prepared for $\delta=0$ (black dotted line), $\delta=0.1$ (green squares), and $\delta=0.4$ (red crosses). In two top plots and for the case $\delta=0$, the base model from shang2022cooperation is obtained. Plots were obtained for square $L\times L$ grid, with $L=64$ and periodic boundary conditions. Each configuration was evaluated for $2^{11}$ steps. Average value of cooperators at the final step, after $2^{10}$ steps, calculated over $50$ realizations, synchronous update mode.