Active-Absorbing Phase Transitions in the Parallel Minority Game

TL;DR

This work investigates how microscopic decision rules in the Parallel Minority Game determine the universality class of active-absorbing transitions, focusing on the critical point $g_c=1$. Through extensive numerical simulations, two rule families are analyzed: instantaneous updates and threshold-based activation, with static and dynamical scaling of $A(t)$ and $F(t)$ near criticality. Instantaneous updates exhibit mean-field directed percolation scaling with exponents $β\approx1.00$, $δ\approx0.50$, and $ν_{∥}\approx2.00$, while threshold rules yield a distinct non-mean-field class with $β\approx0.75$ and non-MF-DP dynamical scaling, indicating a relevant perturbation to DP. The results highlight that minimal cognitive features can qualitatively alter collective critical behavior, with broad implications for socio-economic and active systems and caution in mapping agent-based dynamics to established universality classes, especially when thresholds introduce memory-like effects.

Abstract

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 $β\approx1.00$, $δ\approx0.5$, and $ν_{\parallel}\approx2.0$. Threshold rules, however, produce a distinct non-mean-field universality class with $β\approx0.75$ and a systematic failure of MF-DP dynamical scaling. We show that thresholding acts as a relevant perturbation to DP. The results highlight how minimal cognitive features at the agent level fundamentally alter large-scale critical behaviour in socio-economic and active systems.

Figures (3)

• Figure 1: Static scaling fits for the steady-state activity and overcrowding fraction near $g_c=1$. Log-log plots show power-law behavior $O_\infty \sim (g_c - g)^\beta$. Top row: Instantaneous rules yield $\beta \approx 0.95\pm0.05$, consistent with mean-field directed percolation. Bottom row: Threshold rules yield $\beta \approx 0.75\pm0.05$, indicating a different universality class.
• Figure 2: Dynamical scaling collapses for instantaneous strategy using MF-DP exponents $(\delta,\nu_{\parallel})=(0.5,2.0)$. Log-log show excellent superposition into a single master curve across $g$ values, with broad power-law windows and negligible scatter. This confirms single-parameter finite-time scaling and MF-DP universality, as instantaneous updates eliminate correlated fluctuations vemula2025.
• Figure 3: Dynamical scaling collapses for threshold strategies using non-MF-DP exponents. (a-b) Supercritical regime: $(\delta,\nu_{\parallel})=(1.0,0.75)$. (c-d) Subcritical regime: $(\delta,\nu_{\parallel})=(0.2,1.0)$. Log-log plots reveal systematic fanning when using MF-DP exponents, with partial up/down overlaps indicating weak hysteresis and multiple timescales. Failure across variants and observables confirms deviation from MF-DP, suggesting a distinct universality class.