Spatial public goods games with queueing and reputation

TL;DR

This paper tackles how asynchronous decision timing affects cooperation in public goods by embedding an $M/M/1$ queue into spatial public goods games and augmenting it with a reputation mechanism. The model combines queueing dynamics (arrival rate $\lambda$, service rate $\mu$, capacity $N$) with a birth-death process to yield stationary distributions and sojourn times, which in turn shape payoffs through a time-dependent enhancement. Reputation biases neighbor selection and, together with a Fermi update rule ($P(s_i\leftarrow s_j)=\frac{1}{1+e^{(\Pi_i-\Pi_j)/\kappa}}$ with $\kappa=0.5$), promotes cooperative behavior; explicit expressions show how payoffs depend on sojourn times $T_i=W_i+S_i$ and cumulative cooperation in neighborhoods. Simulations on square lattices and small-world networks demonstrate that high arrival rates, low service rates, and reputation jointly expand the cooperative regime and reveal rich spatiotemporal dynamics, with practical implications for public goods provisioning in time-sensitive and reputation-aware systems.

Abstract

In real-world social and economic systems, the provisioning of public goods generally entails continuous interactions among individuals, with decisions to cooperate or defect being influenced by dynamic factors such as timing, resource availability, and the duration of engagement. However, the traditional public goods game ignores the asynchrony of the strategy adopted by players in the game. To address this problem, we propose a spatial public goods game that integrates an M/M/1 queueing system to simulate the dynamic flow of player interactions. We use a birth-death process to characterize the stochastic dynamics of this queueing system, with players arriving following a Poisson process and service times being exponentially distributed under a first-come-first-served basis with finite queue capacity. We also incorporate reputation so that players who have cooperated in the past are more likely to be chosen for future interactions. Our research shows that a high arrival rate, low service rate, and the reputation mechanism jointly facilitate the emergence of cooperative individuals in the network, which thus provides an interesting and new perspective for the provisioning of public goods.

Figures (6)

• Figure 1: Schematic diagram of the proposed continuous SPGG model based on a queueing system. The model consists of three main stages: (a) the queueing process of network players, (b) the sequence of public goods games, and (c) the strategy update process. Firstly, the initial strategies of players are randomly assigned. Secondly, players enter an $M/M/1$ queueing system with an arrival rate $\lambda$ and are served with an exponential service rate $\mu$. Upon completion of service, they participate in a PGG. Finally, in the strategy update stage, the focal node selects a neighbor with the highest reputation with probability $P_r$ or randomly selects a neighbor with probability $1 - P_r$, then strategy update follows the Fermi function.
• Figure 2: The curve of cooperation ratio $\rho_{c}$ with $r$ under different service rates $\mu$. The arrival rate is fixed at $\lambda = 2$. Different panels illustrate the cooperation level for varying probabilities $P_{r}$ of selecting the highest-reputation player: (a) $P_{r} = 0$, (b) $P_{r} = 0.1$. These curves depict the transition dynamics from cooperation to defection within the network under different service rates.
• Figure 3: Evolutionary dynamics, stationary distribution, and payoff distribution of players. The arrival rate is fixed at $\lambda = 2$ and $P_r=0$. (a) The evolution of cooperation ratio $\rho_c$ over time $t$ for different values of $r$. (b) The stationary distribution of cooperative players for $r=2.7$ under different values of $\mu$. The data point represents the distribution of the number of cooperators over the last 4000 steps. (c) The probability density distribution of payoff in the final stage for $r=2.7$ . The red dashed lines in (b) and (c) represent the mean values of the number of cooperators and the payoff under the corresponding parameters, respectively.
• Figure 4: Evolutionary snapshots under different service rates $\mu$. Each row from top to bottom represents a situation corresponding to different service rates $\mu$. Each column from left to right represents different time steps $t$. Blue pixels represent cooperators, while red pixels represent defectors. The parameters are fixed with $r=2.2$, $P_r=0$, and $\lambda=2$. (a) $\mu=2.4$; (b) $\mu=2.5$; (c) $\mu=2.6$.
• Figure 5: The heatmap of cooperation level under different parameter settings on square lattice network. The density of cooperators $\rho_{c}$ is shown on different parameter planes. The illustration on the right side of the panel explains the meaning of color. (a) The impact of service rate $\mu$ and arrival rate $\lambda$ for fixed $r=4$ and $P_r=0$. (b) The influence of enhancement factor $r$ and service rate $\mu$ for fixed $\lambda=2$ and $P_r=0$.