Centralization and Stability in Formal Constitutions

Consider a social-choice function (SCF) is chosen to decide votes in a formal system, including votes to replace the voting method itself. Agents vote according to their ex-ante belief over what decisions are considered, and whether they prefer them to be decided by the incumbent SCF or the suggested replacement. The existing SCF then aggregates the agents' votes and arrives at a decision of whether it should itself be replaced. An SCF is self-maintaining if it can not be replaced in such fashion by any other SCF. Our focus is on the implications of self-maintenance for centralization. For this purpose, unlike [Barbera and Jackson, 2004], we do not generally restrict attention to anonymous SCFs. We also do not restrict attention to neutral SCFs, unlike [Koray, 2000]. We present results considering optimistic, pessimistic and i.i.d. approaches with respect to agent beliefs, different tie-breaking rules, and different SCF domains. To highlight two of the results, (i) for the i.i.d. unbiased case with arbitrary tie-breaking and general Boolean functions, we prove an Arrow-Style Theorem for Dynamics: We show that only a dictatorship is self-maintaining, and any other SCF has a path of changes that arrives at a dictatorship. (ii) With a pessimistic approach, tie-breaking that prefers the status quo, and WMGs, we provide a tight characterization of the self-maintaining rules, which are exactly all games with minimal winning coalitions of size at most 2. We then consider two extensions, (i) forward-looking voters, (ii) Where the voter utility depends on wisdom of the crowd effects. In both cases, less centralized SCFs become self-maintaining. All in all we provide a basic framework and body of results for centralization dynamics and stability, applicable for institution design, especially in formal De-Jure systems, such as Blockchain Decentralized Autonomous Organizations (DAOs).