Governance as a complex, networked, democratic, satisfiability problem

TL;DR

The paper tackles how to structure governance to produce coherent decisions under complex interdependencies and diverse opinions. It combines a SAT-based encoding of decision interdependencies with a higher-order hypergraph voter model to capture how information flows through overlapping decision groups, enabling analysis across governance regimes from direct democracy to dictatorship. Key contributions include formalizing governance as a democratic satisfiability problem and identifying an effective governance regime where large overlapping groups yield high coherence with modest costs to democratic satisfaction, even in polarized or incoherent populations; results extend from simple toy networks to randomized constraint landscapes with greedy solvers, illustrating robustness of the regime. The framework offers a principled, bottom-up approach to explore governance architectures and can guide empirical studies and organizational design for rapid, crisis-relevant decision-making.

Abstract

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.

Figures (5)

• Figure 1: Schematic representation of our governance problem using signed networks with positive association denoted "AND" or negative association denoted "XOR" (disjunction/exclusive or). A. In this example, the decisions represent converting land into a hydroelectric reservoir (top), public farmland (right), and increasing local property taxes (left). Land conversion requires funding ("AND"), but the land can only be converted for a single use ("XOR"). B. Two simple solutions to the problem. C. We now represent the opinions of the 13 incoherent residents as a histogram. A direct vote would result in an incoherent set of decisions as the majority favors building a dam and farmland without raising taxes. The yellow agent is the most incoherent. D. A similar issue can arise in a coherent but polarized population, where one might end up in an incoherent state due to random fluctuations alone. Effective governance systems have to find the trade-offs and produce a coherent set of decisions that minimize democratic frustrations.
• Figure 2: Governance model with a coherent majority and incoherent zealots. The simple decision network is shown in panel (A) and admits two complementary solutions. The population making the decision is described in the main text, and the distributions of their opinions, with a coherent majority and incoherent zealots, are shown in cartoon form. In panels (B&C), we show the average democratic satisfaction and coherence of the decisions made by a given governance structure. Panels (D&E) show two slices of the previous panels, where error bars represent the standard deviation of the governance outcomes. (B&D) We see that increasing the size of the decision groups provides better statistical sampling of the population, increasing satisfaction but hurting coherence as more and more zealots get involved in decisions. (C&E) Importantly, adding a small amount of overlap between decision groups can dramatically increase coherence, counteracting the role of zealots without hurting democratic satisfaction as much. This region, highlighted in the right panels, is what we call effective governance, where solutions with both high satisfaction and effective governance are achieved.
• Figure 3: Governance model with a polarized population. The simple decision network is illustrated in panel A and admits two complementary solutions. The population making the decision is described in the main text and the distributions of their opinions, with asymmetric polarization, are shown in cartoon form. Increasing the size of the decision groups provides better statistical sampling of the population, increasing satisfaction (panel B) and coherence (panel C). Importantly, compared to a scenario with no overlap (panel D), adding a small amount of overlap between decision groups (panel E) can dramatically increase coherence and surprisingly increase satisfaction. Panel F shows an example of a hypergraph structure of decision groups produced around the effective governance regime of panel E (group size of 51 with overlap of 6), visualized with XGI landry2023xgi. Blobs represent decision groups. Nodes are shown in white if part of a single decision group, blue if part of two, and orange if part of three.
• Figure 4: Governance model with an incoherent population. The simple decision network is illustrated in panel A and admits two complementary solutions. The population making the decision is an incoherent population with random independent opinions drawn uniformly from positive values with a probability of 0.6 and from negative values with a probability of 0.4 (standard deviation of 0.1). Increasing the overlap between decision groups can decrease satisfaction (panel B) but increase the coherence of the decisions (panel C), given that the population itself is homogeneous and incoherent (panel D). This suggests that the governance system can either be democratic or coherent but not both. However, with significant overlap between large decision groups, we can dramatically increase coherence while maintaining some positive satisfaction. This is again the effective governance region, highlighted in panel E.
• Figure 5: We use a population of 500 agents, each with their own greedy coherent opinions, about 20 decisions connected with random constraints of random signs based on a fixed network density. This leads to a population polarized around every exclusive (or negative) constraint. We fix the size of decision groups to 20 and test the ability of this population to govern as we vary overlap (curves) and density of random constraints. We shift the points slightly to avoid overlapping error bars. Without overlap, satisfaction is always maximized as we are simply sampling the greedy agents (panel A), but coherence falls dramatically as the density of constraints is increased (panel B). By tuning the overlap we can balance democratic satisfaction and coherence of the decisions (panel C). Especially at low but non-zero overlap, around 3, we find a large gain in coherence at a very low satisfaction cost (always within a standard deviation of the top-performing average).