Characteristic functions for cooperative interval games
Bibliographic record
Abstract
Games where players' payoffs are given by intervals, instead of scalars, provide a conceptually attractive framework to account for uncertainty and vagueness in decision-making processes. In a cooperative game, one needs to determine the characteristic function values for all possible coalitions. In this paper, we extend the classical $ \alpha $ and $ \beta $ characteristic functions (CFs), initially defined for scalar payoffs, to cooperative interval games. Both characteristic functions are based on a solution of zero-sum interval games with the coalition being the maximizer player and the anti-coalition the minimizer player. We propose an algorithm to define the interval values of a cooperative game in the form of $ \alpha $ and $ \beta $ CFs and illustrate its use on an example with three players. Further, we discuss some properties of cooperative interval games and calculate an interval Shapley value. As expected, different characteristic functions lead to different Shapley values.
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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 | 0.000 | 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".