Descending the Stable Matching Lattice: How many Strategic Agents are\n required to turn Pessimality to Optimality?
Bibliographic record
Abstract
The set of stable matchings induces a distributive lattice. The supremum of\nthe stable matching lattice is the boy-optimal (girl-pessimal) stable matching\nand the infimum is the girl-optimal (boy-pessimal) stable matching. The\nclassical boy-proposal deferred-acceptance algorithm returns the supremum of\nthe lattice, that is, the boy-optimal stable matching. In this paper, we study\nthe smallest group of girls, called the {\\em minimum winning coalition of\ngirls}, that can act strategically, but independently, to force the\nboy-proposal deferred-acceptance algorithm to output the girl-optimal stable\nmatching. We characterize the minimum winning coalition in terms of stable\nmatching rotations and show that its cardinality can take on any value between\n$0$ and $\\left\\lfloor \\frac{n}{2}\\right\\rfloor$, for instances with $n$ boys\nand $n$ girls. Our main result is that, for the random matching model, the\nexpected cardinality of the minimum winning coalition is\n$(\\frac{1}{2}+o(1))\\log{n}$. This resolves a conjecture of Kupfer \\cite{Kup18}.\n
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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 | 0.003 | 0.030 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".