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We analyse two-tier voting systems with voters described by a multi-groupmean-field model that allows for correlated voters both within groups as wellas across group boundaries. The objective is to determine the optimal weightseach group receives in the council to minimise the expected quadratic deviationof the council vote from a hypothetical referendum of the overall population.The mean-field model exhibits different behaviour depending on the intensity ofinteractions between voters. When interaction is weak, we obtain optimalweights given by the sum of a constant term equal for all groups and a termproportional to the square root of the group's population. When interaction isstrong, the optimal weights are in general not uniquely determined. Indeed,when all groups are positively coupled, any assignation of weights is optimal.For two competing clusters of groups, the difference in total weights must be aspecific number, but the assignation of weights within each cluster isarbitrary. We also obtain conditions for both interaction regimes under whichit is impossible to reach the minimal democracy deficit due to the negativityof weights.

Optimal Weights in a Two-Tier Voting System with Mean-Field Voters