Efficiency of fairness in voting systems

Fair representation of voters in a committee representing different voters' groups is being broadly discussed during last few years. Assuming we know what the fair representation is, there exists a problem of optimal quota: given a fair distribution of voting weights, how to set up voting rule (quota) in such a way that distribution of relative a priori voting power is as close as possible to distribution of relative voting weights. Together with optimal quota problem a problem of trade-off between fairness and efficiency (ability of a voting body to change status quo) is formalized by a fairness-efficiency matrix.