Democratic governments comprise a subset of a population whose goal is to produce coherent decisions, solving societal challenges while respecting the will of the people. New governance frameworks represent this as a social network rather than as a hierarchical pyramid with centralized authority. But how should this network be structured? We model the decisions a population must make as a satisfiability problem and the structure of information flow involved in decision-making as a social hypergraph. This framework allows to consider different governance structures, from dictatorships to direct democracy. Between these extremes, we find a regime of effective governance where small overlapping decision groups make specific decisions and share information. Effective governance allows even incoherent or polarized populations to make coherent decisions at low coordination costs. Beyond simulations, our conceptual framework can explore a wide range of governance strategies and their ability to tackle decision problems that challenge standard governments.