From the bankruptcy problem and its Concede-and-Divide solution to the assignment problem and its Fair Division solution
Bibliographic record
Abstract
We revisit two classic problems: the assignment problem, in which agents create value when matched with a partner, and the bankruptcy problem, in which we need to share an endowment among agents with conflicting claims. We show that since Core Selection constrains us to exactly divide the value created by a pair of matched agents, the assignment problem can be seen as a two-player bankruptcy problem. This interpretation allows us to show that the classic Concede-and-Divide (Aumann and Maschler, 1985) sharing method for the bankruptcy problem is equivalent to the Fair Division solution (Thompson, 1981) for the assignment problem, itself the average of the extreme points of the core of Demange (1982) and Leonard (1983). We then exploit the link between the two problems to offer two characterizations of the Fair Division solution. The key property is an adapation of the Minimal Rights First property (Curiel, Maschler and Tijs, 1987) for the bankruptcy problem. The minimal rights of a claimant is what is left of the endowment, if any, when all claimants but himself have received their full claims. The property states that we obtain the same shares if we distribute the minimal rights first, adjust the claims and endowment and proceed on the reduced problem or simply ignore them and proceed on the original problem. In assignment problems, the conceptual equivalent of minimal rights are the minimal core allocations. Given the important role that minimal core allocations play in this link between assignment and bankruptcy problems, it is important to be able to compute them efficiently. We provide a new algorithm to compute them.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".