A Markov Model for Hockey: Manpower Differential and Win Probability Added
Bibliographic record
Abstract
We extend the classic Poisson model of hockey based on score differential and time remaining in the game to include the effect of penalties, and derive the associated Markov win probability model given the goal/manpower differential state at any point in a hockey game. Given data from the 2008/9–2011/12 National Hockey League seasons (a total of 4,920 games) reporting second-by-second goal and manpower differentials (which results in roughly 17.7 million observations), we estimate the state dependent transition rates and win probabilities. The data reveal that even after controlling for the home edge afforded by visiting teams being penalized more frequently than home teams, the goal scoring rate for the home team is higher than for visiting teams at most equivalent manpower differential levels. We use the model to develop a new win probability added metric for evaluating individual players based on their incremental contribution to the probability of winning and illustrate its use and conservation properties.
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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 | 0.005 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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".