Active-Absorbing Phase Transitions in the Parallel Minority Game

The Parallel Minority Game (PMG) is a synchronous adaptive multi-agent model that exhibits active-absorbing transitions characteristic of non-equilibrium statistical systems. We perform a comprehensive numerical study of the PMG under two families of microscopic decision rules: (i) agents update their choices based on instantaneous population in their alternative choices, and (ii) threshold-based activation that activates agents movement only after overcrowding density crossing a threshold. We measure time-dependent and steady state limits of activity $A(t)$, overcrowding fraction $F(t)$ as functions of the control parameter $g=N/D$, where $N$ is the number of agents and $D$ is the total number of sites. Instantaneous rules display mean-field directed-percolation (MF-DP) scaling with $\beta\approx1.00$, $\delta\approx0.5$, and $\nu_{\parallel}\approx2.0$. Threshold rules, however, produce a distinct non-mean-field universality class with $\beta\approx0.75$ and a systematic failure of dynamical scaling. We show that thresholding acts as a relevant perturbation to the critical behavior of the model. The results highlight how minimal cognitive features at the agent level fundamentally alter large-scale critical behavior in socio-economic and active systems.