Effects of non-uniform number of actions by Hawkes process on spatial cooperation

TL;DR

Cooperation evolves more effectively in scenarios with even slight non-uniformity in action frequency compared to completely uniform cases, and scenarios where agents' actions are primarily triggered by their own previous actions more effectively support cooperation, whereas those triggered by others' actions are less effective.

Abstract

The emergence of cooperative behavior, despite natural selection favoring rational self-interest, presents a significant evolutionary puzzle. Evolutionary game theory elucidates why cooperative behavior can be advantageous for survival. However, the impact of non-uniformity in the frequency of actions, particularly when actions are altered in the short term, has received little scholarly attention. To demonstrate the relationship between the non-uniformity in the frequency of actions and the evolution of cooperation, we conducted multi-agent simulations of evolutionary games. In our model, each agent performs actions in a chain-reaction, resulting in a non-uniform distribution of the number of actions. To achieve a variety of non-uniform action frequency, we introduced two types of chain-reaction rules: one where an agent's actions trigger subsequent actions, and another where an agent's actions depend on the actions of others. Our results revealed that cooperation evolves more effectively in scenarios with even slight non-uniformity in action frequency compared to completely uniform cases. In addition, scenarios where agents' actions are primarily triggered by their own previous actions more effectively support cooperation, whereas those triggered by others' actions are less effective. This implies that a few highly active individuals contribute positively to cooperation, while the tendency to follow others' actions can hinder it.

Figures (9)

• Figure 1: An example where $\lambda(t)$ fluctuates over time according to event occurrences. We use the following setting: $\rho=0.5$, $\alpha=0.5$, $\nu=1$, and $\beta=1$.
• Figure 2: Effects of the parameters on the exponential kernel (Eq. \ref{['eq:kernel']}). Plots in the top row compare the effect of $\alpha$ (left: $\alpha=0.1$; right: $\alpha=0.9$). $\nu=1$ and $\beta=1$ are constant in both. Plots in the middle compare the effect of $\beta$ (left: $\beta=1$; right: $\beta=2$). $\alpha=0.5$ and $\nu=1$ are constant in both. Plots in the bottom compare the effect of $\nu$ (left: $\nu=1$; right: $\nu=5$). $\alpha=0.5$ and $\beta=1$ are constant in both. Each red broken line denotes a peak of $\lambda(t)$ that is raised by an event occurrence. Each blue broken line represents the half-life period of $\lambda(t)$ after it rises, affected by an occurrence of an event.
• Figure 3: Flow of our model.
• Figure 4: Changes in the final average fraction of cooperators ($f_{\rm C}$) for each case. The horizontal axis represents the advantage of defectors ($b$), and the line colors distinguish between different models/cases: gray dashed line for the standard model, gray for Poisson, red for Endo, and blue for Exo cases.
• Figure 5: Changes in $f_{\rm C}$ with various values of $\alpha$ and $\nu$. The upper row represents the variation with $\alpha_n$ ($\alpha_x$), while the lower row shows the variation with $\nu$. The left column (red series) corresponds to the Endo case, and the right column (blue series) corresponds to the Exo case.