Collective Bias Models in Two-Tier Voting Systems and the Democracy Deficit

In this article, we analyse a two-tier voting system in which the overall population is split into several different groups and the democracy deficit, the expected quadratic deviation of the council vote from the referendum involving the entire population. The proposals voted on fall in the category of yes/no voting. The voting behaviour is described by a probabilistic model in which there is a central influence or bias that affects the voters' decision. We study different versions of the model and the correlation between voters belonging to different groups is allowed to be weaker than within groups. The two main questions we analyse are the asymptotic behaviour of the (normalised) democracy deficit and the optimal voting weights the groups receive in the council vote chosen to minimise the democracy deficit.