Consider a setting where a social choice function (SCF) is chosen to decide votes in a formal system, including votes to replace the voting method itself. Voters 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 voters' 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, where we combine the approaches of Barbera \& Jackson (2004) and Koray (2000). However, unlike Barbera and Jackson [2004], we do not generally restrict attention to \textit{anonymous} SCFs. Unlike Koray [2000], we do not restrict attention to \textit{neutral} SCFs. We present results that combine (1) optimistic, pessimistic and i.i.d. approaches with respect to voter beliefs, (2) different tie-breaking rules, and (3) different SCF domains. To highlight two of the results, (i) for the i.i.d. unbiased case with worst-case tie-breaking and Simple Games, we prove an Arrow-Style Theorem for stability: We show that only a dictatorship is self-maintaining. (ii) With a pessimistic approach and best-case tie-breaking, we provide a tight characterization of self-maintaining Weighted Majority Games (WMGs), which are exactly those with minimal winning coalitions of size at most 2. We then consider three extensions, (i) forward-looking voters, (ii) bi-level rules, (iii) an application of our framework to expansion and retraction of voting rights. All in all we provide a basic framework and body of results for centralization and stability, applicable for institution design.

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