Abstract
<title>Abstract</title> <p>Crowd-sourced continuous control systems enable multiple distributed participants to collectively generate control commands for a shared agent in real time, yet whether larger crowds control better under adversarial conditions remains insufficiently understood. Using a controlled simulation framework spanning four trajectory geometries (circle, square, lemniscate, and zigzag), three adversary models, adversarial ratios between 5% and 40%, and crowd sizes from 5 to 200, this study shows that crowd-size scaling follows a conditional structure rather than a monotonic improvement law. For smooth trajectories, a gradual adversarial transition was empirically observed around tr ≈ 10–20% (centered near 15%), beyond which increasing crowd size produced a measurable RMSE reduction. The crowd size × adversarial ratio interaction was substantial (η²_interaction = 21.8% for circle and 9.6% for square, two-way ANOVA on raw Monte Carlo realizations). In contrast, rapidly reversing zigzag trajectories exhibited practical crowd-size independence (ΔRMSE = 0.07–1.96%) across all tested conditions, indicating a geometry-imposed structural limit that additional participants do not overcome. As a simulation-based characterization without human-participant validation, these findings position crowd size as a secondary, conditional design factor jointly governed by adversarial participation, trajectory geometry, and adversary structure, with sizing implications for collective robotics, teleoperation, and large-scale interactive control platforms.</p>