Majority Efficient Representation of the Citizens in a Federal Union

All the federal unions, like the United States of America or the European Union, face the issue of finding a “good” indirect voting mechanism. In particular, a crucial question is to know how many mandates should be given to each country or state in a two-tier voting system, given that the majority rule is used at each level. We here propose a new normative criterion to evaluate these voting rules: An apportionment of the seats among the states is majority ecient if the probability of electing the candidate who receives less than 50% of the votes in a two candidate competition over the whole union is minimized. Using computer simulations, we suggest that either the proportional or the square root rule can emerge as an optimal apportionment method depending on the probability model we use to describe the electoral process.

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