We analyse optimal voting weights in two-tier voting systems. In our model, the overall population (or union) is split in groups (or member states) of different sizes. The individuals comprising the overall population constitute the first tier, and the council is the second tier. Each group has a representative in the council that casts votes on their behalf. By `optimal weights, we mean voting weights in the council which minimise the democracy deficit, i.e. the expected deviation of the council vote from a (hypothetical) popular vote. We assume that the voters within each group interact via what we call a local collective bias or common belief (through tradition, common values, strong religious beliefs, etc.). We allow in addition an interaction across group borders via a global bias. Thus, the voting behaviour of each voter depends on the behaviour of all other voters. This correlation is stronger between voters in the same group, but in general not zero for voters in different groups. We call the respective voting measure a Collective Bias Model (CBM). The `simple CBM introduced by Kirsch (2007) and in particular the Impartial Culture and the Impartial Anonymous Culture are special cases of our general model. We compute the optimal weights for large groups rather explicitly. Those optimal weights are unique as long as there is no `complete correlation between the groups. If the correlation between voters in different groups is extremely strong, then the optimal weights are not unique at all. In fact, in this case, the weights are essentially arbitrary.
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