Do legislative bodies’ voting rules affect the diversity of equilibrium policies observed across them, and if so, to what extent? To what degree do these voting rules affect legislative compromise and the stability of the social optimum? Using a spatial model of political competition with single-peaked preferences, we examine these questions in settings where changing incumbent or proposed policies requires supermajority consensus. We develop three findings pertaining to equilibrium policies that are immune to change by any supermajority coalition. First, we prove that at least one equilibrium policy exists and then find the maximum number of equilibrium policies that exist as a function of the supermajority’s size. Policy diversity increases in the size of the supermajority coalition needed to change the status quo, and the Median Voter Theorem is a special case of the result with minimal policy diversity. Second, we find the optimal level of compromise needed by a leader to ensure that her proposed policy is not defeated and establish that compromise decreases in the size of the coalition neededto change policy. Third, we identify the minimal supermajority rule that ensures the stability of the social optimum. The robustness of these findings and their theoretical and policy implications are further investigated in the paper.