How to Solve the Apportionment Paradox

Abstract

In the ideal world, the number of seats that each region or each community gets in a representative body should be exactly proportional to the population of this region or community. However, since the number of seats allocated to each region or community is whole, we cannot maintain the exact proportionality. Not only this leads to a somewhat unfair situation, when residents of one region get more votes per person than residents of another one, it also leads to paradoxes—e.g., sometimes a region that gained the largest number of people loses a number of seats. To avoid this unfairness (and thus, to avoid the resulting paradoxes), we propose to assign, to each representative, a fractional number of votes, so that the overall number of votes allocated to a region will be exactly proportional to the region’s population. This approach resolves the fairness problem, but it raises a new problem: in a secret vote, if we—as it is usually done—disclose the overall numbers of those who voted For and those who voted Against, we may reveal who voted how. In this paper, we propose a way to avoid this disclosure.