Emergence of Cooperation in Mobile Populations: Finite-Size and Initial-Condition Effects

Abstract

The “Vicsek Game” framework—where the Vicsek model of self-propelled particles is coupled with evolutionary game dynamics—provides a fertile ground for examining how mobility, interaction rules, and strategic behavior intertwine. In this setting, agents not only align their motion with neighbors but also adapt strategies based on payoffs from local game interactions, bridging nonequilibrium statistical physics with structural reciprocity in evolutionary game theory (EGT). Previous studies have examined diverse parameters such as cost–benefit ratios, communication overheads, and even machine learning–driven adaptations. Yet, a common simplifying assumption persists: particles are typically initialized in uniformly random positions and orientations. Such homogeneity, however, may obscure crucial early-time effects. For instance, uniformly dispersed particles tend to favor defectors, while cooperators benefit from spatial clustering that sustains mutual support. In this work, we systematically investigate how initial spatial configurations and alignment patterns shape long-term evolutionary outcomes. Our findings reveal that initial heterogeneity can decisively tilt the balance between cooperation and defection, underscoring the Vicsek Game as a minimal yet versatile model for studying the emergence of cooperation in mobile and active matter systems.