Towards a Characterisation of Lottery Set Overlapping Structures.
Bibliographic record
Abstract
Consider a lottery scheme consisting of randomly selecting a winning t–set from a universal m–set, while a player participates in the scheme by purchasing a playing set of any number of n–sets from the universal set prior to the draw, and is awarded a prize if k or more elements of the winning t–set occur in at least one of the player’s n–sets (1 ≤ k ≤ {n, t} ≤ m). This is called a k–prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a k–prize, no matter which winning t–set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by L(m, n, t; k), and the number of such non–isomorphic sets is called the lottery characterisation number, denoted by η(m, n, t; k). In this paper an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges 2 ≤ k ≤ 4, k ≤ t ≤ 11, k ≤ n ≤ 12 and max{n, t} ≤ m ≤ 20. In the process 32 new lottery numbers are found, and bounds on a further 31 lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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".