A generalized parametric divisor method for political apportionment
Bibliographic record
Abstract
Abstract The political apportionment problem has been studied for more than 200 years. In this paper, we introduce a generalized parametric divisor method (GPDM), which generalizes most of the classical and widely used apportionment methods from the literature as special cases. Moreover, it allows for very flexible interpolation between previous methods by appropriately setting two parameters in the GPDM. We identify an inequity measure that the GPDM globally optimizes. We also identify two natural inequity measures for which an apportionment given by the GPDM is locally optimal. These results generalize similar results for classical apportionment methods, and justify the use of a large class of new apportionment methods given by the GPDM. From this class, we identify and recommend specific new methods. Our numerical experiments compare the apportionments given by the new methods with those given by existing methods using real data for the United States, Germany, Canada, Australia, England, and Japan. Explicit definition of the GPDM has enabled us to perform computational experiments for evaluating the unbiasedness of the GPDM using two standard measures while comparing with other traditional methods. Based upon our generalization technique and numerical experiments, we show that the GPDM outperforms all the traditional apportionment methods by selecting appropriate parameter values. Thus, we can conclude that the GPDM is the most “unbiased” and fairer if parameters can be agreed ex ante, and the GPDM is applicable to actual electoral voting systems.
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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 | 0.005 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
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".