Abstract
The current agenda of reapportionment (summer 2012), in which the less populous prefecture (Osaka) has more representatives than the more populous one (Kanagawa), is unfair. Under an unequal representation, the welfare of people with more representatives is worse than it is under a fair representation. Therefore, this agenda is not only inequitable but also inefficient. Using the Nash Social Welfare Function, we create a decomposable index to evaluate the unfairness of representation in electoral districts. We then provide some apportionments, which can be obtained by the divisor methods, by minimizing the objective function made from the Atkinson Social Welfare Function, which includes the Rawlsian, Nash, and Benthamian Social Welfare Functions. We also indicate the problem of man-made districting. We conclude the essay with an indication of the problems of proportional representation and geographic districting.
