Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses
Bibliographic record
Abstract
Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines {\em bilinear games}, and proposes efficient algorithms for its rank based subclasses. Bilinear games are two-player non-cooperative single-shot games with compact polytopal strategy sets and two payoff matrices (A,B) such that when (x,y) is the played strategy profile, the payoffs of the players are xAy and xBy respectively. We show that bilinear games are very general and capture many interesting classes of games like bimatrix games, two player Bayesian games, polymatrix games, two-player extensive form games with perfect recall etc. as special cases, and hence are hard to solve in general. Existence of a (symmetric) Nash equilibrium for (symmetric) bilinear games follow directly from the known results. For a given bilinear game, we define its {\em Best Response Polytopes} (BRPs) and characterize the set of Nash equilibria as {\em fully-labeled} pairs in the BRPs. We consider a rank based hierarchy of bilinear games, where rank of a game (A,B) is defined as rank(A+B). In this paper, we give polynomial time algorithms to compute Nash equilibrium for special classes of bilinear games: (i) Rank-1 games (i.e., rank(A+B)=1). (ii) FPTAS for constant rank games (i.e., rank(A+B) is constant). (iii) When rank(A) or rank(B) is constant. This improves the results by Lipton et al. (EC'03) and Kannan et al. (ET'09), for bimatrix games with low rank matrices.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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; both teacher heads agree on what is shown here.
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".