Inequality in population weights,majority threshold and the inversion probability in the case of three states

Two-tier voting systems are prone to majority inversions, when the outcome of an election is not backed by a majority of popular vote. We study the inversion probability in a model with two candidates, three states and uniformly distributed fractions of supporters for each candidate. We show that the inversion probability in a two-tier voting system with three states eventually decreases with a majority threshold in the states and increases with the inequality in the size distribution of the states.