When aggregating individual preferences through the majority rule in an n-dimensional spatial voting model, the ‘worst-case’ scenario is a social choice configuration where no political equilibrium exists unless a super majority rate as high as 1 − 1/n is adopted. In this paper we assume that a lower d-dimensional (d < n) linear map spans the possible candidates’ platforms. These d ‘ideological’ dimensions imply some linkages between the n political issues. We randomize over these linkages and show that there almost surely exists a 50%-majority equilibria in the above worst-case scenario, when n grows to infinity. Moreover the equilibrium is the mean voter. The speed of convergence (toward 50%) of the super majority rate guaranteeing existence of equilibrium is computed for d = 1 and 2.