Inverse optimization for divisor methods of apportionment

The apportionment problem is to allocate a fixed number of seats among regions in proportion to their populations. In this paper, we propose an inverse optimization approach to evaluate how fairly an observed apportionment treats each region. It is well known that every divisor method can be formulated as a separable discrete convex minimization problem. We introduce a weighted divisor method, in which a nonnegative weight is assigned to each region, as the forward problem. Then we formulate an inverse optimization problem that finds the weight vector closest to the all-one vector in the $\ell_1$-norm such that the observed apportionment is optimal for the forward problem. We show that the optimal weight vector is always unique and has a closed-form expression as the median of three values that depend on a single parameter. Based on this result, the inverse optimization problem is reduced to a one-dimensional monotone equation, and it can be solved in $O(n \log n)$ time, where $n$ is the number of regions. We also apply the proposed approach to the apportionment of the single-member districts of the House of Representatives of Japan.