Edge-based reputation promotes cooperation in simplicial complexes

TL;DR

This study introduces an edge-based reputation mechanism, incorporating both direct and indirect reputation, to investigate the evolution of cooperation in simplicial complexes and reveals a nonlinear interplay between network topology and reputation mechanisms, highlighting how multi-level structures shape collective outcomes.

Abstract

Understanding how cooperation emerges and persists is a central challenge in the evolutionary dynamics of social and biological systems. Most prior studies have examined cooperation through pairwise interactions, yet real-world interactions often involve groups and higher-order structures. Reputation is a key mechanism for guiding strategic behavior in such contexts, but its role in higher-order networks remains underexplored. In this study, we introduce an edge-based reputation mechanism, incorporating both direct and indirect reputation, to investigate the evolution of cooperation in simplicial complexes. Our results show that coupling reputation mechanisms with higher-order network structures strongly promotes cooperation, with direct reputation exerting a stronger influence than indirect reputation. Moreover, we reveal a nonlinear interplay between network topology and reputation mechanisms, highlighting how multi-level structures shape collective outcomes. These findings provide a novel theoretical framework for understanding cooperation in complex social systems.

Figures (6)

• Figure 1: Social dilemma games diagram. Schematic representation of the games with free parameters $(T, S)$. The games are categorized into four quadrants based on the conditions $0 \leq T \leq 2$ and $-1 \leq S \leq 1$. From left to right and top to bottom: the Harmony game, the Snowdrift game, the Stag Hunt game, and the Prisoner's Dilemma game. For illustrative purposes, the reward (R) and punishment (P) values have been fixed to be $1$ and $0$, respectively. The annotations in parentheses represent the strategies corresponding to the Nash equilibria of the $2$-player games, where $C$ denotes cooperation and $D$ denotes defection. We also show the ordering of the entries $(R,S,T,P)$ of the payoff matrix for each game.
• Figure 2: Interaction topology schematics. Dyadic closure (a), triadic interactions (b), and edge reputation among nodes-players $i$, $j$, and $k$ (c). Dyadic closure results in a triangle devoid of higher-order properties (1-simplex), whereas triadic interactions give rise to a filled triangle (2-simplex). The entry $s_{ij}$ denotes the strategy adopted by node $i$ toward node $j$. The strategy set is binary, comprising $0$ ($\equiv C$, for cooperation) and $1$ ($\equiv D$, for defection). Edge reputation (c) is represented by $R_{\vec{ij}}$, node $i$’s assessment of node $j$, and $R_{\vec{ji}}$, node $j$’s assessment of node $i$.
• Figure 3: Reputation assessment mechanism. Left: Player $i$'s direct reputation assessment of neighbor $j$, $R^{\mathrm{direct}}_{\vec{ij}}$, derived from $j$'s historical behavior toward $i$. Right: Indirect reputation assessment, $R^{\mathrm{indirect}}_{\vec{ij}}$, obtained by averaging assessments of $j$ provided by common neighbors $k'$, $k"$, and $k"'$.
• Figure 4: Cooperation fraction diagrams. Average cooperation fraction $f_C$ in $(\gamma,T_2)$ control parameter space across different higher-order interaction ratios: $\rho= 0.1$ (left), $0.5$ (center), and $1.0$ (right). In these panels, Game 1 and Game 3 are configured as Prisoner's Dilemma games, with parameters fixed at $T_1=T_3=1.2$ and $S_1=S_3=-0.2$. Game 2 transitions between a Prisoner's Dilemma, for $T_2\geq 1$, to a Stag Hunt game, when $T_2<1$, with fixed $S_2=-0.5$.
• Figure 5: Cooperation fraction diagrams for varied game scenarios. Contour plots of the cooperation fraction $f_C$ in the asymptotic state under the indirect reputation mechanism $\gamma=0$ (panels a and b), hybrid $\gamma=0.5$ (panels c and d), and direct $\gamma=1$ (panels e and f) are presented as a function of the $2$-simplex fraction $\rho$ and the payoff parameter $T_2$. Other parameters are specified as follows: the first column corresponds to the Harmony Game (H, with $T_1=T_3=0.8$, $S_1=S_3=0.2$); the second column corresponds to the Stag Hunt (SH, with $T_1=T_3=0.8$, $S_1=S_3=-0.2$); the third column corresponds to the Snowdrift Game (SD, with $T_1=T_3=1.2$, $S_1=S_3=0.2$); and the fourth column corresponds to the Prisoner's Dilemma (PD, with $T_1=T_3=1.2$, $S_1=S_3=-0.2$). Labels at the top of each column denote the respective types of Game 1 and Game 3. Additionally, each subfigure is divided into two regions by a horizontal dashed line at $T_2=1$, with labels on the right side indicating the corresponding type of Game 2 in each subpanel. For the upper panels, $S_2=0.5$, while for the lower panels, we set $S_2=-0.5$.