Combined effect of incentives and coupling in multigames in two-layer networks

TL;DR

This study introduces an incentive mechanism based on individual strategies and incomes, wherein a portion of the income from defectors is allocated to reward low-income cooperators, aiming to enhance cooperation by improving the equitable distribution of wealth across the entire population.

Abstract

The lack of cooperation can easily result in inequality among members of a society, which provides an increasing gap between individual incomes. To tackle this issue, we introduce an incentive mechanism based on individual strategies and incomes, wherein a portion of the income from defectors is allocated to reward low-income cooperators, aiming to enhance cooperation by improving the equitable distribution of wealth across the entire population. Moreover, previous research has typically employed network structures or game mechanisms characterized by homogeneity. In this study, we present a network framework that more accurately reflects real-world conditions, where agents are engaged in multiple games, including prisoner's dilemma games in the top-layer and public good games in the down-layer networks. Within this framework, we introduce the concept of ``external coupling'' which connects agents across different networks as acquaintances, thereby facilitating access to shared datasets. Our results indicate that the combined positive effects of external coupling and incentive mechanism lead to optimal cooperation rates and lower Gini coefficients, demonstrating a negative correlation between cooperation and inequality. From a micro-level perspective, this phenomenon primarily arises from the regular network, whereas suboptimal outcomes are observed within the scale-free network. These observations help to give a deeper insight into the interplay between cooperation and wealth disparity in evolutionary games in large populations.

Figures (7)

• Figure 1: Topological structure of our model. It involves a two-layer network with external coupling (red dashed lines), where the upper layer is a scale-free network and the lower layer is a regular network. Each node represents an agent where the size is proportional to its degree (the number of neighbors). A solid line between two nodes indicates the neighbor relationship. All lines are undirected.
• Figure 2: The diagram of the evolution of cooperation rate and Gini coefficient under the combined effect of external coupling and incentives. (a) $\alpha$-$p$ color-coded plot of cooperation level. (b) Gini coefficient as a function of $p$ for different $\alpha$. $r=2.5$ for all panels.
• Figure 3: Cooperation level on $\alpha-p$ parameter plane for different layers of the network. Panel (a) depicts the portion of cooperators on the regular network (a) while panel (b) shows the same value on the scale-free graph. $r=2.5$ for both panels.
• Figure 4: Gini coefficient on the dependence of external coupling for each layer of the network. Panels show $G$ value for $\alpha=0.23$ (a), $\alpha=0.25$ (b), and $\alpha=0.27$ (c). $r=2.5$ for all panels.
• Figure 5: A micro perspective on the cooperation of agents on the regular network layer. Snapshots of the spatial distribution evolving over time. Rows from top to bottom: $p=0$, $p=0.2$, $p=0.8$, and columns from left to right: $t=1$, $t=10$, $t=100$, $t=1000$, $t=10000$. Initially, each agent was randomly distributed in the network. When the first step was over, the agents formed four different attributes depending on their strategies and benefits: low-income cooperators (light blue), low-income defectors (light red), high-income cooperators (dark blue), and high-income defectors (dark red). The common parameters used in these experiments are $r=2.5$ and $\alpha=0.25$.