Majority Efficient Representation of the Citizens in a Federal Union

International audience; In federal unions, how many weights should be given to each state in a two-tier voting system, given that the majority rule is used at each level? We apply a majority criterion to evaluate these voting rules: An apportionment of the seats among the states is majority efficient if it minimizes the probability of electing the candidate who receives less than 50% of the votes in a two candidate competition over the whole union. Depending on the assumptions that we use to describe the electoral process, either the proportional or the square root rule can emerge as an optimal solution.