Degrees of Freedom in Multi-Agent Coordination: Between Centralized and Decentralized Regimes

Large language model (LLM)-based multi-agent systems (MAS) are commonly organized around two coordination regimes. Centralized coordination provides explicit control but can create a bottleneck at a single coordination locus, whereas decentralized coordination offers local autonomy but must achieve effective collaboration through distributed decisions and experience. These trade-offs motivate a systematic study of the space between the two regimes. We introduce Degrees of Freedom (DoF), a continuous configuration framework that quantifies the size of the decision space delegated to downstream agents. DoF controls the granularity of delegated work and the target probability that each task unit is assigned to the branches. Its endpoints recover the two coordination regimes. Across multiple task domains, DoF reveals a hierarchy of organizational effects. Delegated granularity and within-regime routing produce structured but bounded profile reallocations, whereas task domain and retained experience are associated with stronger differences between the centralized- and decentralized-branch profiles. The resulting performance landscape is task-dependent: different domains favor different DoF configurations, while atomic-target conditions are more consistent across runs. In-domain retained experience brings the two branch profiles closer together. The same controlled sweep reveals a task-dependent coordination landscape: some domains exhibit higher-performing configurations, whereas others show flatter responses and no single configuration dominates across domains. In five of the six evaluated state, an intermediate DoF configuration achieves higher mean accuracy than both endpoints, demonstrating the value of exploring the coordination space between them. Together, these findings provide empirical guidance for studying and designing intermediate coordination regimes in LLM-based MAS.