Bibliographic record
Abstract
$ \newcommand{\eps}{\varepsilon} $ In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an $\eps$-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size $O(\log n)$ in the random graph $G(n,\frac12)$. This raises the question of whether a similar intractability holds for approximate Nash equilibrium without side constraints such as high value. We give evidence that asking for near-optimal value makes the problem distinctly harder: a simple algorithm finds a $\frac{1}{2}$-approximate equilibrium of optimal value, but getting below $\frac{1}{2}$ is as hard as the Hidden Clique problem. This is in contrast to the basic problem (finding a Nash equilibrium with no optimization criteria) where more sophisticated algorithms, achieving better approximations, are known. Unlike basic Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than $\eta$ (for any fixed $\eta>0$, even when the optimal Nash equilibrium has value $1-\eta$), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes-Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over players' types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem. A preliminary version of this work appeared in the Proceedings of APPROX 2011.
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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 | 0.004 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.018 | 0.006 |
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".