一个单鞍点1/5近似算法用于公共核双矩阵博弈：识别、精确段优化、尖锐选择器界与认证鲁棒性
A One-Saddle 1/5 Approximation Algorithm for Common-Kernel Bimatrix Games: Recognition, Exact Segment Optimization, Sharp Selector Bounds, and Certified Robustness
- Business and Technology University (BTU)（商业与技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对公共核对称双矩阵博弈，提出单鞍点计算实现1/5近似均衡，并给出识别、核恢复、精确后处理及线性规划投影方法，获得认证近似保证，明确约简架构局限。
AI中文摘要：
我们研究由公共核生成的一类固定归一化的对称双矩阵博弈，并证明该类博弈尽管精确对称均衡计算仍为PPAD难问题，但允许高效近似。对于该类中的每个有理博弈，单次辅助零和鞍点计算即可在多项式时间内产生一个有理对称1/5近似纳什均衡。我们还给出了精确识别和唯一核恢复，以及一个精确的多项式时间后处理算法，该算法在所选鞍点策略连接的线段上最小化遗憾。对于任意有理方博弈，我们将最近公共核投影表述为一个带显式对偶证书的线性规划，获得认证的(1/5+2η*)近似保证。一个独立的选取器结果证明了全子集构造的尖锐遗憾-均匀性常数。这些结果提供了一个可处理且可认证的结构化博弈机制，并阐明了哪些公共核约简架构无法支持某些细粒度近似困难目标。
英文摘要：
We study a fixed-normalization class of symmetric bimatrix games generated by a common kernel and show that it admits efficient approximation despite exact symmetric-equilibrium computation remaining PPAD-hard. For every rational game in the class, a single auxiliary zero-sum saddle computation yields a rational symmetric \(1/5\)-approximate Nash equilibrium in polynomial time. We also give exact recognition and unique kernel recovery, and an exact polynomial-time post-processing algorithm that minimizes regret along the segment joining the selected saddle strategies. For arbitrary rational square games, we formulate the nearest common-kernel projection as a linear program with an explicit dual certificate, obtaining a certified \((1/5+2η^*)\)-approximation guarantee. A separate selector result proves a sharp regret-to-uniformity constant for the full-subset construction. The results provide a tractable and certifiable structured-game regime and clarify which common-kernel reduction architectures cannot support certain fine-grained approximation-hardness objectives.
