Symmetry breaking and social coordination in children: Using group theory to understand emergent patterns of multi-agent coordination

Predicting patterns of behavioural coordination that emerge in small interacting groups is challenging because goal-directed social action is shaped by complex reciprocal and compensatory dynamics. In this study, we examined whether formal symmetry principles derived from group theory could explain coordination patterns among children performing a triadic jumping task. We investigated how geometric symmetries of the task environment and dispositional (a)symmetries associated with leader-follower tendencies jointly constrain collective behaviour. Forty-seven children were classified into symmetric or asymmetric triads based on teacher evaluations of leadership dispositions. Each triad completed multiple trials of a synchronized jumping game requiring movement between adjacent hoops arranged in triangular or square configurations. Results showed that temporal asymmetries in inter-child movement (first, second, or last to jump) were consistent with group-theoretic predictions. In triangular configurations, observed asymmetries corresponded to the highest-order subgroup defined by task and dispositional symmetries. These findings demonstrate that environmental symmetry exerts a hierarchically dominant constraint on collective coordination, within which actor dispositional (a)symmetries further modulate emerging patterns.