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 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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".