Congressional Apportionment: A Multiobjective Optimization Approach

Two events, with major implications for U.S. voters, occur after each decennial census. First, congressional “apportionment” takes place, followed by congressional “districting.” Apportionment determines how to allocate the 435 seats in the House of Representatives across the 50 states, whereas districting determines the geographic boundaries assigned to representatives within each state. Although districting and the practice of gerrymandering often receive great attention in the media and courts, the best way to apportion representatives across states has been debated for nearly 250 years. Historical methods (including the current method) each satisfy some desirable optimality criteria that the others are not guaranteed to satisfy. Moreover, none are guaranteed to optimize certain reasonable fairness measures (e.g., minimum range, minimum bias). To our knowledge, we are the first to formulate and analyze a multiobjective optimization approach to apportionment, allowing policymakers to identify Pareto-optimal allocations and quantify their trade-offs between several competing criteria. Some of these models can be formulated and solved as mixed-integer linear programs, whereas others require the solution of mixed-integer, nonconvex, quadratically constrained quadratic programs. We take advantage of recent software advances that allow one to solve these problems with optimality guarantees. Policy implications of our work include Pareto curves from historical censuses and simulations, which suggest opportunities for improvement in some objectives at little sacrifice to others.