Evolutionary Dynamics Based on Reputation in Networked Populations with Game Transitions

TL;DR

The study investigates how reputation and changing environments shape cooperation in networked populations by integrating a stochastic game framework with DSIGT and PBIGT transitions and a reputation-based fitness update. It develops a donation-game model on networks where reputation increases with cooperative surroundings and influences imitation, incorporating biased mutation and a Geometric Brownian Motion–based environmental drift. Key findings show that endogenous and exogenous game transitions can significantly promote cooperation, especially when the difference between dilemma strengths (Δb) is large and reputation coupling (δ) is strong, with topology modulating outcomes. The work demonstrates robustness across network sizes and highlights a mutation–selection–reputation–game stationary distribution, offering insights into cooperation resilience in dynamically evolving social systems.

Abstract

The environment undergoes perpetual changes that are influenced by a combination of endogenous and exogenous factors. Consequently, it exerts a substantial influence on an individual's physical and psychological state, directly or indirectly affecting the evolutionary dynamics of a population described by a network, which in turn can also alter the environment. Furthermore, the evolution of strategies, shaped by reputation, can diverge due to variations in multiple factors. To explore the potential consequences of the mentioned situations, this paper studies how game and reputation dynamics alter the evolution of cooperation. Concretely, game transitions are determined by individuals' behaviors and external uncontrollable factors. The cooperation level of its neighbors reflects individuals' reputation, and further, a general fitness function regarding payoff and reputation is provided. Within the context of the donation game, we investigate the relevant outcomes associated with the aforementioned evolutionary process, considering various topologies for distinct interactions. Additionally, a biased mutation is introduced to gain a deeper insight into the strategy evolution. We detect a substantial increase in the cooperation level through intensive simulations, and some important phenomena are observed, e.g., the unilateral increase of the value of prosocial behavior limits promotion in cooperative behavior in square-lattice networks.

Figures (13)

• Figure 1: Illustration of the proposed model. Subplot (a) exhibits the general rule of the game transition. Concretely, the game state starting from $G_i$, given there is one cooperator, i.e., $l=1$ ($l$ indicating the number of cooperators in each pairwise interaction, the formal definition is given in the main text), has a probability $p^{[1]}_{ij}$ to enter $G_j$, where $\sum_{j \in E}p^{[1]}_{ij} = 1$ for each $G_i \in E$. Subplots (b) and (c) show the game transitions on graphs, where each player has two strategies to choose from, i.e., cooperation (blue) and defection (red), and the edge color indicates the game state. Subplots (b) and (c) exhibit interactive relationships' diversity and uniformity, respectively. The diversity indicates that each individual can face different social dilemmas at the same time (depicted by different colors in (b)). In contrast, the uniformity means all individuals face the same social dilemma at the same time (depicted by the same color in (c)), and the social dilemma changes as time progresses. Additionally, the dotted circle in subplot (b) indicates the homophily's effect in the cluster of cooperators. In each time step, players game with their neighbors and accumulate their payoff from all interactions. The diversity in interactions can manifest distinct game types, which are accentuated by the different hues of the connecting edges and their corresponding payoff matrices. At the end of each time step, individuals' strategies update, and then all games and reputations vary according to their corresponding rules.
• Figure 2: Plots of cooperation frequency against evolutionary time. The ranges of $x$-axis are set as $[0,2000]$ in both subplots, whereas the ranges of $y$-axis in the two subplots are set as $[0.2,1]$. For DSIGTs, it is under the parameter pair $(b_0,b_1) = (6,3)$ in two subplots. Other curves correspond to PBIGTs under $\sigma = 10^{-4}$ and $\mu = \sigma^2 / 2$ with different initial values. Other parameters are: $c=1$, $\delta=0.2$, $a=4$, and $B_1=4$ under SL and $B_1=3$ under WS.
• Figure 3: Illustration of the particular dilemma. The center player is the defector and the others cooperate. This condition is expected to persist for an extended period, especially under strong selection. Parameters are the same as DSIGTs in Fig. \ref{['fig:2']} during simulations.
• Figure 4: Plots of cooperation frequency against differences between two games. We set $\Delta b = b_0 - b_1$ and values of $b_1$ in $\{1.1, 2, 3, 4, 5, 10\}$ in SL and WS. The ranges of $x$-axis are set as $[0.8,1.8]$ and $[0.4,1.6]$ in subplots (a) and (b), respectively, with steps equal to $0.2$, whereas the ranges of $y$-axis in the two subplots are set as $[0,1]$. Other parameters are: $c=1$, $\delta=0.2$, $\kappa=1$, and $a=4$.
• Figure 5: Plots of cooperation frequency under different noise values $\kappa$. The ranges of $x$-axis are set as $[0.6,1.1]$ and $[0.3,0.9]$ in subplots (a) and (b), with steps equal to $0.1$, whereas the ranges of $y$-axis in two subplots are set as $[0.0,1.0]$. $b_1$ is fixed as $2$ in both subplots. Other parameters are: $c=1$, $\delta=0.2$, and $a=4$.