Congressional Apportionment: A Multiobjective Optimization Approach
Bibliographic record
Abstract
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. This paper was accepted by David Simchi-Levi, operations management. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02472 .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".