Unbalanced Random Matching Markets with Partial Preferences
Bibliographic record
Abstract
Properties of stable matchings in the popular random-matching-market model have been studied for over 50 years. In a random matching market, each agent has complete preferences drawn uniformly and independently at random. Wilson (1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching markets, the proposers are matched to their ln nth choice on average. In this paper, we consider competitive markets with n jobs and n+k candidates, and partial lists where each agent only ranks their top d choices. Despite the long history of the problem, the following fundamental question remains unanswered for these generalized markets: what is the tight threshold on list length d that results in a perfect stable matching with high probability? In this paper, we answer this question exactly - we prove a sharp threshold d₀ = ln n ⋅ ln (n+k)/(k+1) on the existence of perfect stable matchings when k = o(n). That is, we show that if d < (1-ε) d₀, then no stable matching matches all jobs; moreover, if d > (1+ ε) d₀, then all jobs are matched in every stable matching with high probability. This bound improves and generalizes recent results by Kanoria, Min and Qian (2021). Furthermore, we extend the line of work studying the effect of imbalance on the expected rank of the proposers (termed the "stark effect of competition"). We establish the regime in unbalanced markets that forces this stark effect to take shape in markets with partial preferences.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".