Evaluating the Unfairness of Representation With the Nash Social Welfare Function

Using the Nash product (Nash Social Welfare Function) as a micro foundation, we create a decomposable index to evaluate the unfairness of representation in electoral districts. Using this index, we decompose the factors of such unfairness into apportionment and districting. We explore the situations of New Zealand, the United States, Australia, the United Kingdom, Japan, and Canada. We then provide an integer solution that minimizes the index of the apportionment, which can be obtained by the divisor method (Balinski and Young, 1982) using a logarithmic mean.