Impartial games with entailing moves
Bibliographic record
Abstract
Combinatorial game theory has also been called “additive game theory” whenever the analysis involves sums of independent game components. Such disjunctive sums invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with entailing moves that break the alternating play axiom and/or restrict the other player’s options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as nimstring, a normal play variation of the classical children’s game dots & boxes, and top entails, an elegant ruleset introduced in the classical work Winning Ways by Berlekamp, Conway, and Guy. Such rulesets fall outside the scope of the established normal play theory. Here we axiomatize normal play via two new terminating games, ∞(Left wins) and∞(Right wins), and achieve a more general theory. We define affine impartial, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory with an accompanying minimum excluded rule. Solutions of nimstring and top entails are given to illustrate the theory.
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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 | 0.000 | 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 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 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".