Cooperation in Public Goods Games over Uniform Random Hypergraphs with Game Transitions

Abstract

The evolution of cooperation is a central enigma in evolutionary game theory. Traditionally, the combination of pairwise networks and repeated Public Goods Games with a single state fails to adequately describe realistic group interaction scenarios. On the one hand, pairwise networks lack clear group definitions. On the other hand, a participant's decision affects not only competitors' fitness but also the state of the surrounding environment. To address this problem, we propose a Public Goods Game with game transition mechanisms based on Uniform Random Hypergraphs. In our model, game groups formed by hyperedges transition between two types of games, one with abundant public resources and the other with scarce public resources. The transition probability is closely related to the strategies of players within the hyperedges. By developing a Monte Carlo simulation framework that incorporates payoff accumulation, strategy imitation, and game state transitions, we aim to reveal the coevolutionary patterns of strategies and game states in group interactions. Our study highlights a nonlinear relationship between defection sensitivity and cooperation frequency under game transitions, as well as the asymmetric effects of the two sensitivities in state-dependent transitions. These observations open new directions for how to approach social dilemmas.

Figures (5)

• Figure 1: Strategy update and game transition process. Nodes represent agents. Blue nodes are cooperators and red nodes are defectors. Hyperedge colors correspond to game states. Green hyperedges indicate high-value game $g_1$ with $r_1$ and blue hyperedges indicate low-value game $g_2$ with $r_2$, and $r_1 > r_2$. The dynamics contain two steps. (a) Strategy Imitation: An agent $i$ is selected randomly and one of its hyperedges $l$ (solid border) is chosen randomly. Agents in $l$ accumulate payoffs based on the current game state. Agent $i$ then imitates the strategy of the highest-earning agent $j$ in $l$ with probability $W_{s_i \rightarrow s_j}$. (b) Game Transition: Following the strategy update step for player $i$, regardless of whether its strategy is changed or not, all hyperedges containing $i$ (dashed borders) undergo transitions. These transitions rely on their cooperator fractions and follow the rules at the bottom right. Cooperation promotes transitions to $g_1$ while defection leads to $g_2$. These two transitions alter payoff structures and produce distinct cooperation dynamics compared to single Public Goods Games.
• Figure 2: Cooperation frequency $f_C$ as a function of $r_2$ under different sensitivity levels. The x-axis represents the low-value synergy factor $r_2$, with curves for different hypergraph orders: yellow circles ($g=5$), green squares ($g=4$), blue diamonds ($g=3$), and purple triangles ($g=2$). Panels (\ref{['fig:exp1_single_high']})-(\ref{['fig:exp1_alpha_5']}) set $\delta=0.05$ with $\alpha=0, 0.5, 1, 2, 5$, respectively. Panel (\ref{['fig:exp1_single_low']}) shows the single low-value game without game transitions ($\delta=0$). From Figs. (\ref{['fig:exp1_single_high']})-(\ref{['fig:exp1_alpha_5']}), as the system' s sensitivity to defection increases, this transition mechanism raises the cooperation threshold while amplifying $f_C$ growth to boost cooperation for high $r_2$. This compresses the $r_2$ interval required for $f_C$ to rise from 0 to 1, leading to a steeper curve trend.
• Figure 3: Heatmaps of cooperation frequency $f_C$ as functions of $\delta$ and $\alpha$ for three representative $r_2$ values. The first row corresponds to $g=3$ with $r_2=0.775, 0.825, 0.875$. The second row corresponds to $g=4$ with $r_2=0.72, 0.77, 0.82$. Heatmaps are mildly smoothed with a Gaussian filter ($\sigma=1$) to suppress numerical fluctuations, and bilinear interpolation is used for visualization. At the boundaries between pure cooperation and pure defection for different hypergraph orders, three consistent phenomena are observed as the sensitivity $\alpha$ increases: cooperation is inhibited, the cooperation level remains stable, and cooperation is promoted.
• Figure 4: $f_{g_1}$ and $\overline{\pi}$ dependence of $\alpha$ for three representative $r_2$ values.$g=3$ is fixed for all panels while the different values of $\delta$ are indicated in the legend. Left, middle, and right columns show low, intermediate, and high $r_2$ values, which are 0.775, 0.825, and 0.875, respectively. In each column, panels exhibit cooperative positive feedback when $\delta$ is large. For small $\delta$, in the first column, $g_2$ cannot sustain cooperation independently, causing $f_{g_1}$ and $\overline{\pi}$ to decline to 0. In other panels, $f_{g_1}$ decreases but remains non-zero, while $\overline{\pi}$ stays stable.
• Figure 5: Heatmaps of cooperation metrics under different $\delta$ values. Left to right in each row, panels show cooperation frequency $f_C$, $g_1$ proportion $f_{g_1}$, defector-containing hyperedges $f_{g_1}^D$, and average payoff $\overline{\pi}$ respectively. Heatmaps are mildly smoothed with a Gaussian filter ($\sigma=1$) to reduce numerical noise, and bilinear interpolation is used for visualization clarity. The first row corresponds to the changes in various indicators when cooperation forms positive feedback, while the second row depicts the scenario where cooperation goes extinct as sensitivity increases. Results in the first column directly demonstrate the asymmetric variation of $f_C$ with $\alpha_1$ and $\alpha_2$ under the state-dependent game transition scenario, and analyses of other indicators also exhibit a similar asymmetric phenomenon.